cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Roman Witula, Oct 25 2012

Keywords

Comments

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.

References

  • 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).

Links

Crossrefs

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

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

Views

Author

Roman Witula, Oct 06 2012

Keywords

Comments

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.

Examples

			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.

References

  • 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).

Links

Crossrefs

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

A019698 Decimal expansion of 2*Pi/13.

Original entry on oeis.org

4, 8, 3, 3, 2, 1, 9, 4, 6, 7, 0, 6, 1, 2, 2, 0, 3, 6, 6, 8, 6, 5, 6, 0, 5, 2, 0, 5, 0, 4, 5, 3, 8, 9, 0, 5, 2, 6, 1, 1, 0, 2, 9, 8, 4, 5, 1, 9, 2, 4, 7, 0, 4, 9, 3, 8, 0, 7, 6, 0, 7, 0, 6, 5, 0, 8, 8, 9, 4, 8, 3, 1, 7, 3, 6, 3, 3, 9, 8, 4, 5, 9, 4, 2, 7, 7, 4, 5, 8, 8, 5, 1, 4, 1, 7, 1, 8, 5, 6
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			0.4833219467...

Links

Crossrefs

Equals twice A019680.
Cf. A217548.

Programs

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

Views

Author

Roman Witula, Nov 04 2012

Keywords

Comments

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.

Examples

			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).

References

  • 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).

Links

Crossrefs

Cf. A211988, A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.
Showing 1-4 of 4 results.

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