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 1-3 of 3 results.

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).

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

Views

Author

Roman Witula, Sep 17 2012

Keywords

Comments

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).

Examples

			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.

References

  • Roman 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. A216540, A161905, A216861.

Programs

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

Formula

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

Views

Author

Roman Witula, Oct 06 2012

Keywords

Comments

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.

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. 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

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.
Showing 1-3 of 3 results.

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