A161905 -id:A161905 - cp's OEIS frontend

A186685 Total number of n-digit numbers requiring 19 positive biquadrates in their representation as sum of biquadrates.

0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Martin Renner, Feb 25 2011

Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A046050, A052268, A099591.

Cf. A161905, A186650, A186652, A186654, A186656, A186658, A186660(n) + A186662, A186664, A186666, A186668, A186670, A186672, A186674, A186676, A186678, A186681, A186683, A186684.

PadRight[{0, 1, 6}, 100] (* Paolo Xausa, Jul 26 2024 *)

a(n) = A186684(n) - A186684(n-1).
A161905(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + a(n) = A052268(n).
a(n) = 0 for n >= 4. - Nathaniel Johnston, May 09 2011

A186681 Total number of n-digit numbers requiring 17 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 3, 30, 30, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Martin Renner, Feb 25 2011

A161905(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + a(n) + A186683(n) + A186685(n) = A052268(n)

a(n) = 0 for n >= 6. - Nathaniel Johnston, May 09 2011

Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A046048, A099591, A186683, A186685.

a(n) = A186680(n) - A186680(n-1).

A186683 Total number of n-digit numbers requiring 18 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 2, 17, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Martin Renner, Feb 25 2011

A161905(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + A186660(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + a(n) + A186685(n) = A052268(n)

a(n) = 0 for n >= 5. - Nathaniel Johnston, May 09 2011

Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A046049, A099591.

PadRight[{0, 2, 17, 5}, 100] (* Paolo Xausa, Jul 30 2024 *)

a(n) = A186682(n) - A186682(n-1).

A211988 The Berndt-type sequence number 9 for the argument 2*Pi/13.

Original entry on oeis.org

0, -6, -37, 676, 2882, 12502, -196209, -856850, -3740697, 58876883, 257003504, 1121852777, -17656510365, -77073076671, -336434457597, 5295048110651, 23113603862267, 100894018986142, -1587942800101489, -6931585922526870, -30257313674299627, 476211413709501353

Offset: 0

Roman Witula, Oct 25 2012

sign

a(n) + A218655(n)*sqrt(13) = A(2*n+1)*13^((1+floor(n/3))/2)*sqrt(2*(13 + 3*sqrt(13))/13), where A(n) is defined below.
The sequence A(n) from the name of a(n) is defined by the relation A(n) = s(1)^(-n) + s(3)^(-n) + s(9)^(-n), where s(j) := 2*sin(2*Pi*j/13). The sequence with respective positive powers is discussed in A216508 (see sequence Y(n) in Comments to A216508).
It follows that A(n) = sqrt((13-3*sqrt(13))/2)*A(n-1) + (sqrt(13)-3)*A(n-2)/2 - sqrt((13-3*sqrt(13))/26)*A(n-3), with A(-1) = sqrt((13-3*sqrt(13))/2), A(0)=3, and A(1) = sqrt((13-3*sqrt(13))/2).
We note that s(1) + s(3) + s(9) = s(1)^(-1) + s(3)^(-1) + s(9)^(-1) = sqrt((13-3*sqrt(13))/2), sqrt(2*sqrt(13))*(s(1)^(-3) + s(3)^(-3) + s(9)^(-3)) = sqrt(97*sqrt(13)-339), and s(1)^(-9) + s(3)^(-9) + s(9)^(-9) = (131/13)*sqrt(2834 - 786*sqrt(13)).
The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.
R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.

Cf. A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.

A216540 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with initial terms 0, 0, -1, -8, -45, -221.

Original entry on oeis.org

0, 0, -1, -8, -45, -221, -1014, -4472, -19227, -81224, -338767, -1399320, -5736705, -23377770, -94804944, -382930847, -1541565610, -6188513994, -24784429501, -99058333803, -395227906723, -1574536914951, -6264614281978, -24896955293210, -98848880984490

Offset: 1

Roman Witula, Sep 12 2012

a(n) is equal to the rational part (with respect of the field Q(sqrt(13))) of the product sqrt(2(13-3*sqrt(13)))*X(2*n-1)/13, where X(n) = sqrt((13 + 3*sqrt(13))/2)*X(n-1) - sqrt(13)*X(n-2) + sqrt((13 - 3*sqrt(13))/2)*X(n-3), with X(0)=3, X(1)=sqrt((13 + 3*sqrt(13))/2), and X(2)=(13 - sqrt(13))/2.
The Berndt-type sequence number 5 for the argument 2Pi/13 defined by the relation A161905(n) + a(n)*sqrt(13) = sqrt(2*(13-3*sqrt(13))/13)*X(2*n-1), where X(n) := s(2)^n + s(5)^n + s(6)^n, and s(j) := 2*sin(2*Pi*j/13), j=1,2,...,6.
It follows that s(2) + s(5) + s(6) = s(1)*s(3)*s(4) = sqrt((13 + 3*sqrt(13))/2) and s(2)*s(5)*s(6) = s(1) + s(3) - s(4) =  sqrt((13 - 3*sqrt(13))/2).
a(n) is equal to the negated rational part (with respect of the field Q(sqrt(13))) of the product sqrt(2(13+3*sqrt(13)))*Y(2*n-1)/13, where Y(n) = sqrt((13 - 3*sqrt(13))/2)*Y(n-1) + sqrt(13)*Y(n-2) - sqrt((13 + 3*sqrt(13))/2)*Y(n-3), with Y(0)=3, Y(1)=sqrt((13 - 3*sqrt(13))/2), and Y(2)=(13 + sqrt(13))/2. Moreover we have A161905(n) - a(n)*sqrt(13) = sqrt(2*(13+3*sqrt(13))/13)*Y(2*n-1) and Y(n) = s(1)^n + s(3)^n + s(9)^n (we have s(9) = -s(4)) - Roman Witula, Sep 22 2012

			We note that: s(2)^3 + s(5)^3 + s(6)^3 = 2*(s(2) + s(5) + s(6)),  s(2)^5 + s(5)^5 + s(6)^5 = 5* sqrt((13 + 3*sqrt(13))/2) - sqrt((13 - 3*sqrt(13))/2).

Roman Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and Their Applications, Congressus Numerantium, 201 (2010), 89-107.
Roman Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

Roman Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.
Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-182,91,-13).

Cf. A216605, A216486, A216597, A216508, A161905, A216801.

LinearRecurrence[{13,-65,156,-182,91,-13}, {0,0,-1,-8,-45,-221}, 30]

G.f.: -x^3*(2*x-1)*(3*x-1)/(13*x^6-91*x^5+182*x^4-156*x^3+65*x^2-13*x+1). - Colin Barker, Sep 23 2012

Better name from Joerg Arndt, Sep 17 2012

Name clarified by Robert C. Lyons, Feb 08 2025

A216597 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with initial terms 0, -1, -5, -22, -91, -364.

Original entry on oeis.org

0, -1, -5, -22, -91, -364, -1430, -5564, -21541, -83200, -321100, -1239446, -4787770, -18514119, -71683040, -277913233, -1078918139, -4194134516, -16324764560, -63616690111, -248187382924, -969250588865, -3788814577730, -14823325196459, -58040165033110, -227415509487686

Offset: 0

Roman Witula, Sep 11 2012

a(n) is equal to the rational part of 2*X(2*n)/sqrt(13) (with respect of the field Q(sqrt(13))), where X(n) = sqrt((13 + 3*sqrt(13))/2)*X(n-1) - sqrt(13)*X(n-2) + sqrt((13 - 3*sqrt(13))/2)*X(n-3), with X(0)=3, X(1)=sqrt((13 + 3*sqrt(13))/2), and X(2)=(13 - sqrt(13))/2.
The Berndt-type sequence number 4 for the argument 2Pi/13 defined by the relation A216508(n) + a(n)*sqrt(13) = 2*X(2*n), where X(n) := s(2)^n + s(5)^n + s(6)^n, where s(j) := 2*sin(2*Pi*j/13).
I observe that all numbers of the form (a(6*n + k + 4) - 4*a(6*n + k + 3))*13^(-n), where k = 1,...,6, n = 0,1,... are integers. For example we have a(10)-4*a(9)=900*13 and a(11)-4*a(10)=266*13^2.
We note that a(n) = -A050185(n) for n=0,1,...,5 and a(6) + A050185(6) = -2. - Roman Witula, Sep 22 2012
a(n) is equal to the negative rational part of 2*Y(2*n)/sqrt(13) (with respect of the field Q(sqrt(13))), where Y(n) = sqrt((13 - 3*sqrt(13))/2)*Y(n-1) + sqrt(13)*Y(n-2) - sqrt((13 + 3*sqrt(13))/2)*Y(n-3), with Y(0)=3, Y(1)=sqrt((13 - 3*sqrt(13))/2), and Y(2)=(13 + sqrt(13))/2. It can be proved that Y(n) = s(1)^n + s(3)^n + s(9)^n (we have s(9) = -s(4)), and 2*Y(2*n) = A216508(n) - a(n)*sqrt(13). - Roman Witula, Sep 24 2012

			We have s(2)^4 + s(5)^4 + s(6)^4 + sqrt(13) = s(2)^2 + s(5)^2 + s(6)^2 = (13 - sqrt(13))/2.
We note that 2*a(1) - a(2) = 1, 4*a(2) - a(3) = 2, 4*a(3) - a(4) = 3, 4*a(4) = a(5) and 4*a(n) - a(n+1) < 0 for every n = 5,6,...

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.
R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
Roman Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.
Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-182,91,-13).

Cf. A216605, A216486, A216508, A216540, A161905, A216801.

LinearRecurrence[{13,-65,156,-182,91,-13}, {0,-1,-5,-22,-91,-364}, 30]

concat([0], Vec(-x*(13*x^4 -26*x^3 +22*x^2 -8*x +1) / (13*x^6 -91*x^5 +182*x^4 -156*x^3 +65*x^2 -13*x +1) + O(x^30))) \\ Andrew Howroyd, Feb 25 2018

G.f.: -x*(13*x^4 - 26*x^3 + 22*x^2 - 8*x + 1) / (13*x^6 - 91*x^5 + 182*x^4 - 156*x^3 + 65*x^2 - 13*x + 1). - Colin Barker, Jun 01 2013

a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n,n+k)*(k|13), where (k|13) represents the Legendre symbol. - Greg Dresden, Oct 09 2022

Better name from Joerg Arndt, Sep 17 2012

Name clarified by Robert C. Lyons, Feb 08 2025

A216508 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with initial terms 6, 13, 39, 130, 455, 1638.

Original entry on oeis.org

6, 13, 39, 130, 455, 1638, 6006, 22308, 83655, 316030, 1200914, 4585308, 17577014, 67603887, 260757536, 1008258225, 3906958055, 15167837542, 58983478554, 229708325847, 895760071050, 3497141791455, 13667427167576, 53464307173927, 209315686335366, 820090746381088, 3215215287887889

Offset: 0

Roman Witula, Sep 11 2012

a(n) is equal to the rational part of 2*X(2*n) (with respect to the field Q(sqrt(13))), where X(n) = sqrt((13 + 3*sqrt(13))/2)*X(n-1) - sqrt(13)*X(n-2) + sqrt((13 - 3*sqrt(13))/2)*X(n-3), with X(0)=3, X(1)=sqrt((13 + 3*sqrt(13))/2), and X(2)=(13 - sqrt(13))/2.
The Berndt-type sequence number 3 for the argument 2Pi/13 defined by the relation a(n) + A216597(n)*sqrt(13) = 2*X(2*n), where X(n) := s(2)^n + s(5)^n + s(6)^n, where s(j) := 2*sin(2*Pi*j/13).
We note that all numbers of the form a(6*n+k)*13^(-n), where k = 1,...,6, n = 0,1,... are integers, and even the number a(13)*13^(-4) is an integer.
a(n) is also equal to the rational part of 2*Y(2*n) (with respect to the field Q(sqrt(13))), where Y(n) = sqrt((13 - 3*sqrt(13))/2)*Y(n-1) + sqrt(13)*Y(n-2) - sqrt((13 + 3*sqrt(13))/2)*Y(n-3), with Y(0)=3, Y(1)=sqrt((13 - 3*sqrt(13))/2), and Y(2)=(13 + sqrt(13))/2. Moreover we can deduce the following decompositions:
  2*Y(2*n) = a(n) - A216597(n)*sqrt(13) and Y(n) = s(1)^n + s(3)^n + s(9)^n (we have s(9) = -s(4)) - Roman Witula, Sep 22 2012

			We have a(7)/2 + 2*A216597(7) = 26, 4*X(4) - X(6) = 13 + sqrt(13), 4*X(8) - X(10) = 91, 4*X(10) - X(12) = 13*(21 - sqrt(13)), 4*X(12) - X(14)= 78*(11 - sqrt(13)), 8*X(14) - 2*X(16) = 11*13*sqrt(13)*(3*sqrt(13) - 5) and X(6) - 10*X(2) = -6*sqrt(13) since 2*X(2) = 13 - sqrt(13), 2*X(4) = 39 - 5*sqrt(13), X(6) = 65 - 11*sqrt(13), 2*X(8) = 91*(5 - sqrt(13)), X(10) = 91*(9 - 2*sqrt(13)), X(12) = 3003 - 715*sqrt(13) = 13*(3*77 - 55*sqrt(13)), X(14) = 11154 - 2782*sqrt(13), 2*X(16) = 83655 - 21541*sqrt(13).

Roman Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.
Roman Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

Roman Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.
Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-182,91,-13).

Cf. A216605, A216486, A216597, A216540, A161905.

LinearRecurrence[{13,-65,156,-182,91,-13}, {6,13,39,130,455,1638}, 30]

G.f.: -(91*x^5-364*x^4+468*x^3-260*x^2+65*x-6) / (13*x^6-91*x^5+182*x^4-156*x^3+65*x^2-13*x+1). - Colin Barker, Jun 01 2013

Better name from Joerg Arndt, Sep 17 2012

Name clarified by Robert C. Lyons, Feb 08 2025

A216801 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6).

Original entry on oeis.org

2, -22, -117, -468, -1755, -6513, -24336, -91988, -351689, -1357408, -5277363, -20625774, -80909257, -318173258, -1253243498, -4941450657, -19495914360, -76945654032, -303737001009, -1199041027587, -4733273752831, -18683644465447, -73743457866962

Offset: 1

Roman Witula, Sep 17 2012

a(n) is equal to the rational part of the number sqrt(2*(13 + 3*sqrt(13))/13)*X(2*n-1), where X(n) = sqrt((13 -3*sqrt(13))/2)*X(n-1) + sqrt(13)*X(n-2) - sqrt((13 + 3*sqrt(13))/2)*X(n-3), with X(0) = 3, X(1) = sqrt((13 - 3*sqrt(13))/2), and X(2) = -(13 + sqrt(13))/2.
Let us observe that all numbers of the form a(n)*13^(-floor((n+3)/6)) are integers.
We note that the sequence X(n) is defined "similarly" to sequence Y(n) in the comments to A216540. The only difference between them is in initial condition: X(2) = -Y(2).

			We have 4*a(3)=a(4), 4*a(4)=a(5)+a(3). The 3-valuation of a(n) for n=1,...,10 is contained in A167366. Moreover it can be obtained X(7) - 22*X(3) = 4*sqrt(2*(13-3*sqrt(13))), 4*X(5) - X(7) = 2*sqrt(26(13-3*sqrt(13))), and 15*X(5) - X(9) = 20*sqrt(26(13-3*sqrt(13))), which implies (15*X(5) - X(9))/(4*X(5) - X(7)) = 10.

Roman Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-182,91,-13).

Cf. A216540, A161905, A216861.

LinearRecurrence[{13, -65, 156, -182, 91, -13}, {2, -22, -117, -468, -1755, -6513}, 25] (* Paolo Xausa, Feb 23 2024 *)

G.f.: -x*(52*x^5-520*x^4+689*x^3-299*x^2+48*x-2) / (13*x^6-91*x^5+182*x^4-156*x^3+65*x^2-13*x+1). - Colin Barker, Jun 01 2013

A217548 The Berndt-type sequences number 7 for the argument 2*Pi/13.

Original entry on oeis.org

6, 7, -65, -295, -1303, 20631, 89967, 392616, -6178549, -26970688, -117731275, 1852943703, 8088348131, 35306734632, -555682818080, -2425630962790, -10588208505263, 166644858132571, -727427431532172, 3175319503526856, -49975467287014789

Offset: 0

Roman Witula, Oct 06 2012

sign

a(n) is the rational component (with respect to the field Q(sqrt(13))) of the number A(2*n)*2*13^(floor((n+1)/3)/2), where A(n) = sqrt((13-3*sqrt(13))/2)*A(n-1) + (sqrt(13)-3)*A(n-2)/2 - sqrt((13-3*sqrt(13))/26)*A(n-3), with A(-1) = sqrt((13-3*sqrt(13))/2), A(0)=3, and A(1) = sqrt((13-3*sqrt(13))/2).
The basic sequence A(n) is defined  by the relation A(n) = s(1)^(-n) + s(3)^(-n) + s(9)^(-n), where s(j) = 2*sin(2*Pi*j/13). The sequence with respective positive powers is discussed in A216508 (see sequence Y(n) in Comments to A216508).
We note that s(1) + s(3) + s(9) = s(1)^(-1) + s(3)^(-1) + s(9)^(-1) = sqrt((13-3*sqrt(13))/2).
The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.
R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.

Cf. A019698, A216605, A216486, A216508, A216597, A216540, A161905, A216450, A216801, A216861, A217548, A217549, A211988.

A216861 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with initial terms 0, -2, -9, -44, -215, -1001.

Original entry on oeis.org

0, -2, -9, -44, -215, -1001, -4446, -19058, -79677, -327418, -1329601, -5355272, -21446945, -85548138, -340268656, -1350664731, -5353389340, -21195056584, -83846301409, -331483318257, -1309872510973, -5174049465897, -20431456722794, -80660347594658

Offset: 1

Roman Witula, Sep 18 2012

a(n) is equal to the rational part (with respect of the field Q(sqrt(13))) of the product sqrt(2*(13 + 3*sqrt(13)))*X(2*n-1)/13, where X(n) = sqrt((13-3*sqrt(13))/2)*X(n-1) + sqrt(13)*X(n-2) - sqrt((13+3*sqrt(13))/2)*X(n-3), with X(0)=3, X(1)=sqrt((13-3*sqrt(13))/2), and X(2)=-(13+sqrt(13))/2.

The sequence X(n) is defined in almost the same way as sequence Y(n) from the comments to A161905. The only difference is in the initial condition X(2) = -Y(2).

			We have a(3)-5*a(2)=a(4)-5a(3)=1, a(5)-5*a(4)=5, and 19000 + a(8) = a(4) + 2*a(3) - 2*a(2).

Roman Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-182,91,-13).

Cf. A216905, A216801, A216540.

LinearRecurrence[{13, -65, 156, -182, 91, -13}, {0, -2, -9, -44, -215, -1001}, 25] (* Paolo Xausa, Feb 23 2024 *)

G.f.: -x^2*(26*x^4-84*x^3+57*x^2-17*x+2) / (13*x^6-91*x^5+182*x^4-156*x^3+65*x^2-13*x+1). - Colin Barker, Jun 01 2013

Name clarified by Robert C. Lyons, Feb 08 2025

cp's OEIS Frontend

A186685 Total number of n-digit numbers requiring 19 positive biquadrates in their representation as sum of biquadrates.

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A186681 Total number of n-digit numbers requiring 17 positive biquadrates in their representation as sum of biquadrates.

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A186683 Total number of n-digit numbers requiring 18 positive biquadrates in their representation as sum of biquadrates.

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A211988 The Berndt-type sequence number 9 for the argument 2*Pi/13.

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A216540 a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 182*a(n-4) + 91*a(n-5) - 13*a(n-6), with initial terms 0, 0, -1, -8, -45, -221.

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A216597 a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 182*a(n-4) + 91*a(n-5) - 13*a(n-6), with initial terms 0, -1, -5, -22, -91, -364.

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A216508 a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 182*a(n-4) + 91*a(n-5) - 13*a(n-6), with initial terms 6, 13, 39, 130, 455, 1638.

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A216801 a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 182*a(n-4) + 91*a(n-5) - 13*a(n-6).

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A216540 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with initial terms 0, 0, -1, -8, -45, -221.

A216597 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with initial terms 0, -1, -5, -22, -91, -364.

A216508 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with initial terms 6, 13, 39, 130, 455, 1638.

A216801 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6).

A216861 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with initial terms 0, -2, -9, -44, -215, -1001.