A note on sum formulas $\sum_{k=0}^{n}kx^{k}W_{k}$ and $\sum_{k=1}^{n}kx^{k}W_{-k}$ of generalized Hexanacci numbers

[1] M. Akbulak, A.  Oteles, On the sum of Pell and Jacobsthal numbers by matrix method, Bulletin of the Iranian Mathematical Society, 40(4), (2014) 1017{1025.

[2] E. G. Alptekin, Pell, Pell-Lucas ve Modi ed Pell saylar ile tanml circulant ve semicirculant matrisler (Doctoral dissertation, Selcuk Universitesi Fen Bilimleri Enstitusu), (2005).

[3] Z. Cerin, Formulae for sums of Jacobsthal{Lucas numbers, Int. Math. Forum, 2(40), (2007) 1969{1984.

[4] Z. Cerin, Sums of Squares and Products of Jacobsthal Numbers, Journal of Integer Sequences, 10(2), Article 07.2.5, (2007) 1{15.

[5] L. Chen, X. Wang, The Power Sums Involving Fibonacci Polynomials and Their Applications, Symmetry, 11, (2019), doi.org/10.3390/sym11050635.

[6] C. K. Cook, M. R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41, (2013) 27{39.

[7] A. Coskun, N. Taskara, On the Some Properties of Circulant Matrices with Third Order Linear Recurrent Sequences, Mathematical Sciences and Applications E-Notes, 6(1), (2018) 12{18.

[8] P. J. Davis, Circulant Matrices, JohnWiley&Sons, New York, (1979).

[9] O Deveci, E. Karaduman, C.M. Campbell, The Fibonacci-Circulant Sequences and Their Applications, Iranian Journal of Science and Technology Transaction A-Science, 41(A4), (2007) 1033{1038.

[10] O Deveci, On The Fibonacci-Circulant p-Sequences, Utilitas Mathematica, 108, (2018) 107{124.

[11] O Deveci, On the connections among Fibonacci, Pell, Jacobsthal and Padovan numbers, Notes on Number Theory and Discrete Mathematics, 27(2), (2021) 111{128.

[12] O Erdag, O Deveci, A. G. Shannon, Matrix Manipulations for Properties of Pell p-Numbers and their Generalizations, Analele Stiinti ce ale Universitatii Ovidius, 28(3), (2020) 89{102.

[13] A. Esi, N. Subramanian, M. K. Ozdemir, Chlodowsky type (, q)-Bernstein Stancu operators of Pascal rough triple sequences, J. Mahani Math. Res. Cent. 2023; 12(1): 289-310.

[14] R. Frontczak, Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach, Notes on Number Theory and Discrete Mathematics, 24(2), (2018) 94{103.

[15] R. Frontczak, Sums of Cubes Over Odd-Index Fibonacci Numbers, Integers, 18, (2018) 1{9.

[16] R. Frontczak, Sums of Tribonacci and Tribonacci-Lucas Numbers, International Journal of Mathematical Analysis, 12(1), (2018) 19{24.

[17] A. Gnanam, B. Anitha, Sums of Squares Jacobsthal Numbers, IOSR Journal of Mathematics, 11(6), (2015) 62{64.

[18] H. Gokbas, H. Kose, Some Sum Formulas for Products of Pell and Pell-Lucas Numbers, Int. J. Adv. Appl. Math. and Mech. 4(4), (2017) 1{4.

[19] R.T. Hansen., General Identities for Linear Fibonacci and Lucas Summations, Fibonacci Quarterly, 16(2), (1978) 121{128.

[20] C. He, J. Ma, K. Zhang, Z.Wang, The Upper Bound Estimation on the Spectral Norm of r-Circulant Matrices with the Fibonacci and Lucas Numbers, J. Inequal. Appl. 2015:72, (2015).

[21] E. Kilic, D. Tasci, The Linear Algebra of The Pell Matrix, Boletn de la Sociedad Matematica Mexicana, 3(11), (2005).

[22] E. Klc, Sums of the squares of terms of sequence fung, Proc. Indian Acad. Sci. (Math. Sci.) 118(1), (2008) 27{41.

[23] C. Kzlates, N. Tuglu, On the Norms of Geometric and Symmetric Geometric Circulant Matrices with the Tribonacci Number, Gazi University Journal of Science, 31(2), (2018) 555{567.

[24] C. Kzlates, N. Tuglu On the Bounds for the Spectral Norms of Geometric Circulant Matrices, Journal of Inequalities and Applications, 2016(1), Article ID 312, (2016). DOI 10.1186/s13660-016-1255-1

[25] T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, New York, (2001).

[26] T. Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, (2014).

[27] J. K. Merikoski, P. Haukkanen, M. Mattila, T. Tossavainen, On the Spectral and Frobenius Norm of a Generalized Fibonacci r-Circulant Matrix, Special Matrices, 6, (2018) 23{36.

[28] L. R. Natividad, On Solving Fibonacci-Like Sequences of Fourth, Fifth and Sixth Order, International Journal of Mathematics and Computing, 3(2), (2013) 38{40.

[29] A. Ozkoc, E. Ardyok, Circulant and Negacyclic Matrices Via Tetranacci Numbers, Honam Mathematical J., 38(4), (2016) 725{738.

[30] A.  Oteles, M. Akbulak, A Note on Generalized k-Pell Numbers and Their Determinantal Representation, Journal of Analysis and Number Theory, 4(2), (2016) 153{158.

[31] T. Parpar, k'nc Mertebeden Rekurans Bagntsnn  Ozellikleri ve Baz Uygulamalar, Selcuk Universitesi, Fen Bilimleri Enstitusu, Yuksek Lisans Tezi, (2011).

[32] A. Pacheenburawana, W. Sintunavarat, On the Spectral Norms of r-Circulant Matrices with the Padovan and Perrin Sequences, Journal of Mathematical Analysis, 9(3), (2018) 110{122.

[33] E. Polatl, On the Bounds for the Spectral Norms of r-Circulant Matrices with a Type of Catalan Triangle Numbers, Journal of Science and Arts, 3(48), (2019) 575{578.

[34] H. Prodinger, Sums of Powers of Fibonacci Polynomials, Proc. Indian Acad. Sci. (Math. Sci.), 119(5), (2009) 567{570.

[35] H. Prodinger, S. J. Selkirk, Sums of Squares of Tetranacci Numbers: A Generating Function Approach, (2019), http://arxiv.org/abs/1906.08336v1.

[36] B. Radicic, On k-Circulant Matrices Involving the Jacobsthal Numbers, Revista De La Union Matematica Argentina, 60(2), (2009) 431{442.

[37] Z. Raza, M. A. Ali, On the Norms of Circulant, r-Circulant, Semi-Circulant and Hankel Matrices with Tribonacci Sequence, (2014) arxiv, http://arxiv.org/abs/1407.1369v1.

[38] Z. Raza, M. A. Ali, On the Norms of Some Special Matrices with Generalized Fibonacci Sequence, J. Appl. Math. & Informatics, 33(5-6), (2015) 593{605.

[39] Z. Raza, M. Riaz, M. A. Ali, Some Inequalities on the Norms of Special Matrices with Generalized Tribonacci and Generalized Pell-Padovan Sequences, arXiv, (2015), http://arxiv.org/abs/1407.1369v2

[40] G. P. S. Rathore, O. Sikhwal, R. Choudhary, Formula for  nding nth Term of Fibonacci-Like Sequence of Higher Order, International Journal of Mathematics And its Applications, 4 (2-D), (2016) 75{80.

[41] R. Schumacher, How to sum the squares of the Tetranacci numbers and the Fibonacci m-step numbers, Fibonacci Quarterly, 57, (2019) 168{175.

[42] W. Sintunavarat, The Upper Bound Estimation for the Spectral Norm of r-Circulant and Symmetric r-Circulant Matrices with the Padovan Sequence, J. Nonlinear Sci. Appl.9, (2016) 92{101.

[43] S. Shen, On the Norms of Circulant Matrices with the (k,h)-Fibonacci and (k,h)-Lucas Numbers, Int. J. Contemp. Math. Sciences, 6(18), (2018) 887{894.

[44] S. Shen, The Spectral Norms of Circulant Matrices Involving (k,h)-Fibonacci and (k,h)-Lucas Numbers, Int. J. Contemp. Math. Sciences, 9(14), (2014) 661{665.

[45] S. Shen, J. Cen, On the Spectral Norms of r-Circulant Matrices with the k-Fibonacci and k-Lucas Numbers, Int. J. Contemp. Math. Sciences, 5(12), (2010) 569{578.

[46] S. Shen, J. Cen, On the Bounds for the Norms of r-Circulant Matrices with the Fibonacci and Lucas Numbers, Applied Mathematics and Computation 216, (2010) 2891{2897.

[47] B. Shi, The Spectral Norms of Geometric Circulant Matrices with the Generalized k-Horadam Numbers, Journal of Inequalities and Applications, 2018:14, (2018). https://doi.org/10.1186/s13660-017-1608-4

[48] S. Solak, On the Norms of Circulant Matrices with the Fibonacci and Lucas Numbers, Applied Mathematics and Computation, 160, (2005) 125{132.

[49] S. Solak, Erratum to On the Norms of Circulant Matrices with the Fibonacci and Lucas Numbers" [Appl. Math. Comput. 160 (2005) 125{132], Applied Mathematics and Computation, 190, (2007) 1855{1856.

[50] Y. Soykan, Simson Identity of Generalized m-step Fibonacci Numbers, Int. J. Adv. Appl. Math. and Mech. 7(2), (2019) 45{56.

[51] Y. Soykan, Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports, 9(1), (2020) 23{39. https://doi.org/10.9734/ajarr/2020/v9i130212

[52] Y. Soykan, Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of and
Pn k=0W3 k and Pn k=1W3 􀀀  k, Archives of Current Research International, 20(2), (2020) 58{69. DOI: 10.9734/ACRI/2020/v20i230177

[53] Y. Soykan, A Closed Formula for the Sums of Squares of Generalized Tribonacci numbers, Journal of Progressive Research in Mathematics, 16(2), (2020) 2932-2941.

[54] Y. Soykan, A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Pn k=0 xkW3 k and Pn k=1 xkW3 􀀀 k , Preprints (2020), 2020040437 (doi:10.20944/preprints202004.0437.v1).

[55] Y. Soykan, P On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of n k=0 kW3 k and Pn k=1 kW3 􀀀 k , Asian Research Journal of Mathematics, 16(6), (2020) 37{52. DOI: 10.9734/ARJOM/2020/v16i630196

[56] Y. Soykan, On the Sums of Squares of Generalized Tribonacci Numbers: Closed Formulas of Pn k=0 xkW2 k , Archives  of Current Research International, 20(4), (2020) 22{47. DOI: 10.9734/ACRI/2020/v20i430187

[57] Y. Soykan, Formulae For The Sums of Squares of Generalized Tribonacci Numbers: Closed Form Formulas of Pn k=0 kW2 k , IOSR Journal of Mathematics, 16(4), (2020) 1{18. DOI: 10.9790/5728-1604010118

[58] Y. Soykan, A Study on Generalized Fibonacci Numbers: Sum Formulas Pn k=0 kxkW3 k and Pn k=1 kxkW3 􀀀 k for the Cubes of Terms, Earthline Journal of Mathematical Sciences, 4(2), (2020) 297{331. https://doi.org/10.34198 ejms.4220.297331

[59] Y. Soykan, Generalized Fibonacci Numbers: Sum Formulas of the Squares of Terms, MathLAB Journal, 5, (2020) 46{62.

[60] Y. Soykan, Horadam Numbers: Sum of the Squares of Terms of Sequence, Int. J. Adv. Appl. Math. and Mech. 7(4), (2020) 34{50.

[61] Y. Soykan, A Study on the Sums of Squares of Generalized Fibonacci Numbers: Closed Forms of the Sum Formulas
Pn k=0 kxkW2 k and Pn k=1 kxkW2 􀀀 k, Asian Journal of Advanced Research and Reports, 12(1), (2020) 44{67. DOI: 10.9734/AJARR/2020/v12i130280

[62] Y. Soykan, On Generalized P Tetranacci Numbers: Closed Form Formulas of the Sum n k=0W2 k of the Squares of  Terms, International Journal of Advances in Applied Mathematics and Mechanics, 8(1), (2020) 15{26.

[63] Y. Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research, 20(2), (2019) 1{15.

[64] Y. Soykan, Corrigendum: On Summing Formulas for Generalized Fibonacci and Gaussian Generalized Fibonacci  Numbers, Advances in Research, 21(10), (2020) 66{82. DOI: 10.9734/AIR/2020/v21i1030253 [65] Y. Soykan, On Summing Formulas for Horadam Numbers, Asian Journal of Advanced Research and Reports 8(1), (2020) 45{61. DOI: 10.9734/AJARR/2020/v8i130192.

[66] Y. Soykan, Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science, 35(1), (2020) 89{104. DOI: 10.9734/JAMCS/2020/v35i130241.

[67] Y. Soykan, Generalized Tribonacci Numbers: Summing Formulas, Int. J. Adv. Appl. Math. and Mech. 7(3), (2020) 57{76.

[68] Y. Soykan, Summing Formulas For Generalized Tribonacci Numbers, Universal Journal of Mathematics and Applications, 3(1), (2020) 1{11, 2020. DOI: https://doi.org/10.32323/ujma.637876

[69] Y. Soykan, On Sum Formulas for Generalized Tribonacci Sequence, Journal of Scienti c Research & Reports, 26(7), (2020) 27{52. DOI: 10.9734/JSRR/2020/v26i730283

[70] Y. Soykan, Summation Formulas For Generalized Tetranacci Numbers, Asian Journal of Advanced Research and Reports, 7(2), (2019) 1{12. doi.org/10.9734/ajarr/2019/v7i230170.

[71] Y. Soykan, Sum Formulas For Generalized Fifth-Order Linear Recurrence Sequences, Journal of Advances in Mathematics and Computer Science, 34(5), (2019) 1{14. DOI: 10.9734/JAMCS/2019/v34i530224.

[72] Y. Soykan, Linear Summing Formulas of Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers, Journal of Advanced in Mathematics and Computer Science, 33(3), (2019) 1{14.

[73] Y. Soykan, On Summing Formulas of Generalized Hexanacci and Gaussian Generalized Hexanacci Numbers, Asian Research Journal of Mathematics, 14(4), (2019) 1{14.

[74] Y. Soykan, A Study On Sum Formulas of Generalized Sixth-Order Linear Recurrence Sequences, Asian Journal of Advanced Research and Reports, 14(2), (2020) 36{48.DOI:10.9734/AJARR/2020/v14i230329

[75] Y. Soykan, Matrix Sequences of Tribonacci and Tribonacci-Lucas Numbers, Communications in Mathematics and Applications, 11(2), (2020) 281-295. DOI:10.26713/cma.v11i2.1102

[76] Y. Soykan, A Study On the Sums of Squares of Generalized Tribonacci Numbers: Closed Form Formulas of Pn k=0 kxkW2 k , Journal of Scienti c Perspectives, 5(1), (2021) 1{23.DOI:https://doi.org/10.26900/jsp.5.1.02

[77] YP. Soykan, A Study on Generalized Tetranacci Numbers: Closed Form Formulas n k=0 xkW2 k of Sums of the Squares of Terms, Asian Research Journal of Mathematics, 16(10), (2020) 109{136. DOI: 10.9734/ARJOM/2020/v16i1030234

[78] Y. Soykan, Generalized Fibonacci Numbers with Indices in Arithmetic Progression and Sum of Their Squares: the Sum Formula Pn k=0 xkW2 mk+j ; Journal of Advances in Mathematics and Computer Science, 36(6), (2021) 30{62. DOI:
10.9734/JAMCS/2021/v36i630371

[79] Y. Soykan, A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Pn k=0 xkW3 k and Pn k=1 xkW3 􀀀 k, Int. J. Adv. Appl. Math. and Mech. 8(3), (2021) 9{23.

[80] Y. Soykan, Sums of Cubes of Generalized Fibonacci Numbers with Indices in Arithmetic Progression: the Sum Formulas Pn k=0 xkW3 mk+j ; Int. J. Adv. Appl. Math. and Mech. 9(1), (2021) 6{41.

[81] Soykan, Y., Sum of Generalized Tribonacci Sequence: The Sum Formulas of Pn k=0 xkWk via Generating Functions, IOSR Journal of Mathematics (IOSR-JM), 18(1), 39-47, 2022. DOI: 10.9790/5728-1801033947

[82] Y. Soykan, Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas Pn k=0 xkWmk+j ; Archives of Current Research International, 21(3), (2021) 11{38. DOI: 10.9734/ACRI/2021/v21i330235

[83] Y. Soykan, Sum Formulas For Generalized Tetranacci Numbers: Closed Forms of the Sum Formulas Pn k=0 xkWk and Pn k=1 xkW􀀀k, Journal of Progressive Research in Mathematics, 18(1), (2021) 24{47.

[84] Y. Soykan, A Study on Sum Formulas of Generalized Pentanacci Sequence: Closed Forms of the Sum Formulas Pn k=0 xkWk and Pn k=1 xkW􀀀k; Journal of Progressive Research in Mathematics, 18(2), (2021) 20{38.

[85] Y. Soykan, A Study on Sum Formulas of Generalized Hexanacci Numbers: Closed Forms of the Sum Formulas Pn k=0 xkWk and Pn k=1 xkW􀀀k, Asian Research Journal of Mathematics, 17(3), (2021) 93{118. DOI: 10.9734/ARJOM /2021/v17i330285

[86] Y. Soykan, Sum of the Cubes of Generalized Mersenne Numbers: the Sum Formula Pn k=0 xkW3 mk+j ; IOSR Journal of Mathematics, 17(5), (2021) 25{41. DOI:10.9790/5728-1705022541

[87] Y. Soykan, P On the Sum of the Cubes of Generalized Oresme Numbers: the Sum Formula n k=0 xkW3 mk+j , Asian Research Journal of Current Science, 3(1), (2021) 295{308.

[88] Y. Soykan, E. Tasdemir, C. M. Dikmen, On the Sum of the Cubes of Generalized Balancing Numbers: the Sum Formula
Pn k=0 xkW3 mk+j ; Open Journal of Mathematical Sciences (OMS), 6, (2022) 152-167. doi:10.30538/oms2022.0184

[89] Y. Soykan, A Study on the Sum of the Squares of Generalized Oresme Numbers: The Sum Formula Pn k=0 xkW2 mk+j , Asian Journal of Pure and Applied Mathematics, 4(1), (2022) 16{27.

[90] Y. Soykan, A Study on the Sum of the Squares of Generalized p-Oresme Numbers: the Sum Formula Pn k=0 xkW2 mk+j , Asian Journal of Advanced Research and Reports, 16(1), (2022) 1{24. DOI: 10.9734/AJARR/2022/v16i130444

[91] Y. Soykan, On the Sum of the Squares of Generalized Mersenne Numbers: the Sum Formula Pn k=0 xkW2 mk+j ; International Journal of Advances in Applied Mathematics and Mechanics, 9(2), (2021) 28{37.

[92] Y. Soykan, E. Tasdemir, C. M. Dikmen, A Study on the Sum of the Squares of Generalized Balancing Numbers: the Sum Formula Pn k=0 xkW2 mk+j , Journal of Innovative Applied Mathematics and Computational Sciences, 1(1), (2021) 16{30.

[93] Y. Soykan, On k-circulant Matrices with the Generalized Third-Order Pell Numbers, Notes on Number Theory and Discrete Mathematics, 27(4), (2021) 187{206.DOI:10.7546/nntdm.2021.27.4.187-206.

[94] Y. Soykan, Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with the Generalized Tribonacci Numbers, Earthline Journal of Mathematical Sciences, 6(1), (2021) 131{151. https://doi.org/10.34198/ejms.6121.131151

[95] Y. Soykan, A Study On Generalized (r,s,t,u,v,y)-Numbers, Journal of Progressive Research in Mathematics, 17(1), (2020) 54{72.

[96] Y. Soykan, N. Ozmen, On Generalized Hexanacci and Gaussian Generalized Hexanacci Numbers, Turk. J. Math. Comput. Sci. 13(1), (2021) 25{43. DOI:10.47000/tjmcs.787578

[97] Y. Soykan, On Generalized Sixth-Order Pell Sequence, Journal of Scienti c Perspectives, 4(1), (2020) 49{70. DOI: https://doi.org/10.26900/jsp.4.005.

[98] Y. Soykan, E. E. Polatl, On Generalized Sixth-Order Jacobsthal Sequence, Int. J. Adv. Appl. Math. and Mech. 8(3), (2021) 24{40.

[99] Y. Soykan, Properties of Generalized 6-primes Numbers, Archives of Current Research International, 20(6), (2020) 12{30. DOI: 10.9734/ACRI/2020/v20i630199

[100] Y. Soykan, Sum Formulas of Generalized Hexanacci Sequence: Closed Forms of the Sum Formulas Pn k=0 kWk and Pn k=1 kW􀀀k, International Journal of Mathematics Trends and Technology- IJMTT, 67(4), (2021) 67{78. /doi:10.14445 /22315373/IJMTTV67I4P510

[101] N. Tuglu, C. Kzlates, On the Norms of Circulant and r-Circulant Matrices with the Hyperharmonic Fibonacci Numbers, Journal of Inequalities and Applications, 2015, Article ID 253, (2015). http://dx.doi.org/10.1186/s13660-015-0778-1

[102] R. Turkmen, H. Gokbas, On the Spectral Norm of r-Circulant Matrices with the Pell and Pell-Lucas Numbers, J. Inequal. Appl. 2016:65, (2016).

[103] S Uygun, Some Bounds for the Norms of Circulant Matrices with the k-Jacobsthal and k-Jacobsthal Lucas Numbers, Journal of Mathematics Research, 8(6), (2016) 133{138.

[104] S Uygun, S. Yasamal, On the Bounds for the Norms of Circulant Matrices with the Jacobsthal and Jacobsthal{Lucas Numbers, Notes on Number Theory and Discrete Mathematics, 23(1), (2017) 91-98.

[105] S Uygun, S. Yasamal, On the Bounds for the Norms of r-Circulant Matrices with the Jacobsthal and Jacobsthal{Lucas Numbers, International Journal of Pure and Applied Mathematics, 112(1), (2017) 93{102.

[106] M. E. Waddill, The Tetranacci Sequence and Generalizations, Fibonacci Quarterly, 30(1), (1992) 9{20.

[107] W. Wamiliana, S. Suharsono, P. E. Kristanto, Counting the sum of cubes for Lucas and Gibonacci Numbers, Science and Technology Indonesia, 4(2), (2019) 31{35.