A182191 a(n) = 6*a(n-1) - a(n-2) + 12 with n>1, a(0)=-1, a(1)=5.

-1, 5, 43, 265, 1559, 9101, 53059, 309265, 1802543, 10506005, 61233499, 356895001, 2080136519, 12123924125, 70663408243, 411856525345, 2400475743839, 13990997937701, 81545511882379, 475282073356585, 2770146928257143, 16145599496186285, 94103450048860579

Offset: 0

Author

Kenneth J Ramsey, Apr 17 2012

Keywords

• sign
• easy

Comments

If p is a prime of the form 8*r +/- 3 then a(p) == 1 (mod p); if p is a prime of the form 8*r +/- 1 then a(p) == 5 (mod p).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..100
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Cf. A000129, A002203, A038723.

Programs

Magma

I:=[-1,5]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)+12: n in [1..19]]; // Bruno Berselli, May 15 2012

Mathematica

m = 19;n = 1; c = 0;
list3 = Reap[While[c < 22, t = 6 n - m + 12; Sow[t];m = n; n = t;c++]][[2,1]]
Table[LucasL[2*n, 2] +Fibonacci[2*n, 2] -3, {n, 0, 40}] (* G. C. Greubel, May 24 2021 *)

Sage

[lucas_number1(2*n+2,2,-1) - 2*lucas_number1(2*n,2,-1) -3 for n in (0..40)] # G. C. Greubel, May 24 2021

Formula

G.f.: -(1-12*x-x^2)/((1-x)*(1-6*x+x^2)). - Bruno Berselli, May 15 2012

a(n) = 2*A038723(n) - 3. -Bruno Berselli, May 16 2012

a(n) = -3 + (1/4)*( (4+sqrt(2))*(3+2*sqrt(2))^n + (4-sqrt(2))*(3-2*sqrt(2))^n ). - Colin Barker, Mar 05 2016

From G. C. Greubel, May 24 2021: (Start)

a(n) = A000129(2*n+2) - 2*A000129(2*n) - 3.

a(n) = A000129(2*n) + A002203(2*n) - 3. (End)