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.

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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.

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

Views

Author

Roman Witula, Sep 11 2012

Keywords

Comments

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

Examples

			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,...

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, A216540, A161905, A216801.

Programs

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

Formula

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

Extensions

Better name from Joerg Arndt, Sep 17 2012
Name clarified by Robert C. Lyons, Feb 08 2025

A130513 Subtriangle of triangle in A051168: remove central column of A051168 and all columns to the right; now read by upwards diagonals.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 2, 1, 0, 14, 7, 3, 1, 0, 42, 20, 9, 3, 1, 0, 132, 66, 30, 12, 4, 1, 0, 429, 212, 99, 40, 15, 4, 1, 0, 1430, 715, 333, 143, 55, 18, 5, 1, 0, 4862, 2424, 1144, 497, 200, 70, 22, 5, 1, 0, 16796, 8398, 3978, 1768, 728, 273, 91, 26, 6, 1, 0, 58786, 29372, 13995
Offset: 1

Views

Author

Philippe Deléham, Aug 08 2007

Keywords

Examples

			Triangle T(n,k), 1<=k<=n, begins:
1;
1, 0;
2, 1, 0;
5, 2, 1, 0;
14, 7, 3, 1, 0;
42, 20, 9, 3, 1, 0;
132, 66, 30, 12, 4, 1, 0;
429, 212, 99, 40, 15, 4, 1, 0;

References

  • A. Errera, Analysis situs: Un problème d'énumération, Memoires Acad. Bruxelles (1931), Serie 2, Vol. 11, No. 6, 26pp.

Crossrefs

Cf. A000108, A000150, A050181, A050182, A050183, A050184, A050185.

Programs

  • Mathematica
    Table[1/(2n-k) Plus@@ (MoebiusMu[ # ]Binomial[(2n-k)/#,(n-k)/# ]&/@ Divisors[GCD[2n-k,n-k]]),{n,12},{k,n}] (* Wouter Meeussen, Jul 20 2008 *)

Formula

Sum_{k, 1<=k<=n} T(n,k) = A022553(n); Sum_{k, 1<=k<=n}k*T(n,k) = A002996(n).
T(n,k) = 1/(2n-k) Sum( d | gcd(2n-k,n-k) = mu(d) C((2n-k)/d,(n-k)/d) ). - Wouter Meeussen, Jul 20 2008

Extensions

Edited by N. J. A. Sloane, Oct 08 2007
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