A216605 -id:A216605 - cp's OEIS frontend

A161905 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with a(1)..a(6) as shown.

2, 4, 13, 52, 221, 949, 4056, 17186, 72163, 300482, 1241981, 5100758, 20833813, 84695026, 342920942, 1383646433, 5566235714, 22334785486, 89420529809, 357319721889, 1425447435997, 5678246483273, 22590565547134, 89775857333032, 356428030609222, 1413891596961194, 5604509198580578

Offset: 1

Roman Witula, Sep 12 2012

a(n) is equal to the rational part (with respect to the field Q(sqrt(13))) of the product 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.
The Berndt-type sequence number 6 for the argument 2*Pi/13 defined by the relation a(n) + A216540(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.
We note that all numbers a(n+1)-4*a(n) for n=3,4,..., are divisible by 13. For example we have a(4)=4*a(3), a(5)-4*a(4)=13, a(6)-4*a(5)=5*13, a(7)-4*a(6)=20*13, and a(10)-4*a(9)=70*13^2.
a(n) is also equal to the rational part (with respect to the field Q(sqrt(13))) of the product sqrt(2*(13+3*sqrt(13))/13)*Y(2*n-1), 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. Let us observe that a(n) - A216540(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

			It can be shown that 4*X(5) - X(7) = sqrt(26*(13+3*sqrt(13))), 4*X(7) - X(9) = 13*(sqrt(13) - 1)*sqrt(2*(13 + 3*sqrt(13)))/4, and 4*X(11) - X(13) = 130*(sqrt(13) - 2)*sqrt(2*(13 + 3*sqrt(13)))/4, which implies
(4*X(7) - X(9))/(4*X(5) - X(7)) = 13*(sqrt(13) - 1) and
(4*X(11) - X(13))/(4*X(7) - X(9)) = 10*(sqrt(13) - 2)/(sqrt(13) - 1) = 5*(11 - sqrt(13))/6.
We also have a(6) - a(3) - a(1) = 4000, a(9) - 2*a(4) - a(3) + 3*a(1) = 300000, and a(11) - a(5) + a(4) - 2*a(2) - a(1) = 5100000.

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
R. 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, A216540.

LinearRecurrence[{13,-65,156,-182,91,-13}, {2,4,13,52,221,949}, 30]
CoefficientList[Series[(2-22 x+91 x^2-169 x^3+130 x^4-26 x^5)/(1-13 x+ 65 x^2- 156 x^3+182 x^4-91 x^5+13 x^6),{x,0,40}],x] (* Harvey P. Dale, Jun 05 2021 *)

G.f.: -x*(-2 + 22*x - 91*x^2 + 169*x^3 - 130*x^4 + 26*x^5) / (1 - 13*x + 65*x^2 - 156*x^3 + 182*x^4 - 91*x^5 + 13*x^6). - R. J. Mathar, Sep 18 2012

Better name from Joerg Arndt, Sep 17 2012

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.

A216486 a(n) is equal to the rational part (considering of the field Q(sqrt(13))) of the numbers A(n)/sqrt(13), where we have A(n) = ((sqrt(13) - 1)/2)A(n-1) + A(n-2) + ((3-sqrt(13))/2)A(n-3), with A(0) = 6, A(1) = sqrt(13) - 1, and A(2) = 11 - sqrt(13).

Original entry on oeis.org

0, 1, -1, 4, -3, 14, -10, 48, -37, 166, -144, 582, -570, 2067, -2260, 7421, -8923, 26878, -35020, 98039, -136612, 359649, -529990, 1325491, -2046310, 4903786, -7868991, 18199354, -30157768, 67720279, -115255425, 252540383, -439456837, 943488036

Offset: 0

Roman Witula, Sep 11 2012

The Berndt-type sequence number 2 for the argument 2*Pi/13 defined by the following relation: A216605(n) + a(n)*sqrt(13) = A(n) = 2*(c(1)^n + c(3)^n + c(4)^n), where c(j) := 2*cos(2*Pi*j/13), j=1..6. The numbers a(n), n=0,1,..., are all positive integers. We note that we also have A216605(n) - a(n)*sqrt(13) = B(n) = 2*(c(2)^n + c(5)^n + c(6)^n) and the following recurrence relation holds: B(n) = -((sqrt(13)+1)/2)*B(n-1) + B(n-2) + ((3+sqrt(13))/2)*B(n-3), with B(0) = 6, B(1) = -sqrt(13) - 1, and B(2) = 11 + sqrt(13).

We note that the sums a(2*n+1) + a(2*n+2) are nonnegative only for n = 0..5.

			We have a(5) + a(6) + a(4) + a(2) = a(7) + a(8) + a(6) + a(2) = a(9) + a(5) + a(1) + a(10) + a(8) = 0 and
  a(6) + a(9) + a(10) = a(11) + a(12) = 12.
Moreover, the following relations hold: A(3) = 4*A(1), B(3) = 4*B(1), A(5) = 4*A(3) + 2*sqrt(13), B(5) = 4*B(3)-2*sqrt(13), A(7) = 4*A(5) + 8*sqrt(13), B(7) = 4*B(5)-8*sqrt(13), A(4) = 3*A(2) - 2, B(4) = 3*B(2) + 2, 6 + A(6) = 3*A(4) + A(2), and A(8) - 3*A(6) = 25 - A(5)/2.

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..500
R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.
Index entries for linear recurrences with constant coefficients, signature (-1,5,4,-6,-3,1).

Cf. A216605.

LinearRecurrence[{-1, 5, 4, -6, -3, 1}, {0, 1, -1, 4, -3, 14}, 30]

concat([0],Vec((1-2*x^2+2*x^3+x^4)/(1+x-5*x^2-4*x^3+6*x^4+3*x^5-x^6) + O(x^30))) \\ Andrew Howroyd, Feb 25 2018

G.f.: x*(1 - 2*x^2 + 2*x^3 + x^4)/(1 + x - 5*x^2 - 4*x^3 + 6*x^4 + 3*x^5 - x^6).

a(n) = - a(n-1) + 5*a(n-2) + 4*a(n-3) - 6*a(n-4) - 3*a(n-5) + a(n-6), which from the respective polynomial-type formula follows given by Witula in section "Formula" in A216605.

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

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.

A217549 The Berndt-type sequence number 8 for the argument 2*Pi/13.

Original entry on oeis.org

0, -1, 21, 85, 365, -5707, -24935, -108872, 1713705, 7480420, 32652893, -513913649, -2243303605, -9792325686, 154118686736, 672748988550, 2936640671285, -46218967738367, -201752069488280, -880675175822422, 13860700755359325, 60503840705600655, 264107479466296733

Offset: 0

Roman Witula, Oct 06 2012

sign

a(n) is defined by the relation A217548(n) + a(n)*sqrt(13)= 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, A(1) = sqrt((13-3*sqrt(13))/26).
However 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).
The numbers of other Berndt-type sequences for the argument 2*Pi/13 in Crossrefs are given.

			We have A(1) = A(-1) = sqrt((13-3*sqrt(13))/2), A(2) = (7-sqrt(13))/2, A(3) = (2*sqrt(13)-3)*sqrt((13-3*sqrt(13))/26), A(4) = (21-5*sqrt(13))/2, A(5) = ((13*sqrt(13)-37)/2)*sqrt((13-3*sqrt(13))/26), 2*sqrt(13)*A(6)  = -295 + 85*sqrt(13), and 2*sqrt(13)*(A(6) - 4*A(4)) + 2*A(2) = -28. Furthermore it can be verified that  -a(5)/13 - a(4) - a(3) = A217548(5)/13 + A217548(4) + A217548(3) = -11.

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, A216597, A216508, A216540, A161905, A217548, A216450, A216801, A216861.

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 3, 1, 5, 1, 5, 4, 1, 6, 1, 7, 4, 7, 1, 8, 1, 11, 4, 19, 16, 18, 1, 9, 1, 13, 4, 25, 16, 38, 29, 1, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 12, 1, 19, 4, 43, 16, 98, 64, 187, 193, 123, 1

Offset: 1

L. Edson Jeffery, Jan 12 2013

Antidiagonal sums: {1,3,5,9,16,26,46,78,136,...}.

			Array begins as
.1..1...1..1...1...1
.2..1...3..4...7..11
.3..1...5..4..13..16
.4..1...7..4..19..16
.5..1...9..4..25..16
.6..1..11..4..31..16

Rows: cf. A000012, A000032, A094649, A189234, A216605, etc.

Cf. A185095, A186740.

T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.
Empirical g.f. for row n: F(x) = (Sum_{u=0..n-1} A122765(n,n-1-u)*x^u)/(Sum_{v=0..n} A108299(n,v)*x^v).
Empirical: odd column first differences tend to A000984 = {1, 2, 6, 20, 70, 252, ...} (central binomial coefficients).

A218655 The Berndt-type sequence number 10 for the argument 2*Pi/13.

Original entry on oeis.org

2, 4, 13, -176, -786, -3452, 54483, 237722, 1037569, -16329149, -71279530, -311145495, 4897036897, 21376227709, 93310132523, -1468582101731, -6410560285891, -27982966049682, 440416091468393, 1922476035761802, 8391868916275609

Offset: 0

Roman Witula, Nov 04 2012

sign

A211988(n) + a(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 could be deduced 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).
The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.

			Let us put b(n) = A211988(n) + a(n)*sqrt(13). Then we get b(0) = 2*sqrt(13), b(1) = -6 + 4*sqrt(13), b(2) = -37 + 13*sqrt(13), b(3) = 676 - 176*sqrt(13), b(4) = 2882 - 786*sqrt(13), b(5) = 12502 - 3452*sqrt(13).

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. A211988, A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.

cp's OEIS Frontend

A161905 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 a(1)..a(6) as shown.

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

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A216486 a(n) is equal to the rational part (considering of the field Q(sqrt(13))) of the numbers A(n)/sqrt(13), where we have A(n) = ((sqrt(13) - 1)/2)*A(n-1) + A(n-2) + ((3-sqrt(13))/2)*A(n-3), with A(0) = 6, A(1) = sqrt(13) - 1, and A(2) = 11 - sqrt(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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A217548 The Berndt-type sequences number 7 for the argument 2*Pi/13.

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A161905 a(n) = 13a(n-1) - 65a(n-2) + 156a(n-3) - 182a(n-4) + 91a(n-5) - 13a(n-6), with a(1)..a(6) as shown.

A216486 a(n) is equal to the rational part (considering of the field Q(sqrt(13))) of the numbers A(n)/sqrt(13), where we have A(n) = ((sqrt(13) - 1)/2)A(n-1) + A(n-2) + ((3-sqrt(13))/2)A(n-3), with A(0) = 6, A(1) = sqrt(13) - 1, and A(2) = 11 - sqrt(13).

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.

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.