A182435 a(n) = 6*a(n-1) - a(n-2) - 2 with n>1, a(0)=0, a(1)=1.

0, 1, 4, 21, 120, 697, 4060, 23661, 137904, 803761, 4684660, 27304197, 159140520, 927538921, 5406093004, 31509019101, 183648021600, 1070379110497, 6238626641380, 36361380737781, 211929657785304, 1235216565974041, 7199369738058940, 41961001862379597

Offset: 0

Author

Kenneth J Ramsey, Apr 28 2012

Keywords

• nonn
• easy

Comments

It appears that for n>0, A143608(n) divides a(n).

The sequence a(n)/A143608(n) appears to generate A001541 interleaved with A001653. - R. J. Mathar, Jul 04 2012

It also appears that if p equals a prime of the form 8*r +/- 3 then a(p + 1) == 0 (mod p); and that if p is a prime in the form of 8*r +/- 1 then a(p - 1) == 0 (mod p), inherited from A143608.

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. A001108, A143608, A001541 (first differences).

Essentially a duplicate of A046090.

Programs

Magma

[n le 2 select n-1 else 6*Self(n-1)-Self(n-2)-2: n in [1..24]]; // Bruno Berselli, May 15 2012

Mathematica

m = -20;
n = -3;
c = 0;
list3 = Reap[While[c < 20,t = 6 n - m - 2;Sow[t];m = n;n = t; c++]][[2,1]]
LinearRecurrence[{7,-7,1},{0,1,4},30] (* Harvey P. Dale, May 11 2018 *)

PARI

concat(0,Vec((1-3*x)/(1-x)/(1-6*x+x^2)+O(x^98))) \\ Charles R Greathouse IV, Jun 11 2013

Formula

a(n) = A046090(n-1), for n>=1.

G.f.: x*(1-3*x)/((1-x)*(1-6*x+x^2)). - Bruno Berselli, May 15 2012

a(n) = A001652(n-1)+1 with A001652(-1)=-1. - Bruno Berselli, May 16 2012

2*a(n)*(a(n)-1)+1 = A001653(n)^2 for n>0. - Bruno Berselli, Oct 23 2012