A216861 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, -2, -9, -44, -215, -1001.

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

Author

Roman Witula, Sep 18 2012

Keywords

• sign
• easy

Comments

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).

Examples

			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).
		

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

• 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).

Crossrefs

Cf. A216905, A216801, A216540.

Programs

Mathematica

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

Formula

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

Extensions

Name clarified by Robert C. Lyons, Feb 08 2025