A129346 a(2n) = A100525(n), a(2n+1) = A001653(n+1); a Pellian-related sequence.

4, 5, 22, 29, 128, 169, 746, 985, 4348, 5741, 25342, 33461, 147704, 195025, 860882, 1136689, 5017588, 6625109, 29244646, 38613965, 170450288, 225058681, 993457082, 1311738121, 5790292204, 7645370045, 33748296142, 44560482149, 196699484648, 259717522849

Offset: 0

Author

Creighton Dement, Apr 10 2007

Keywords

• nonn
• easy

Comments

Summation of -a(n) and A129345 returns twice Pell numbers A000129 (apart from signs; starting from 2nd term of A000129).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Crossrefs

Cf. A129345, A000129, A001542, A038761.

Programs

Mathematica

LinearRecurrence[{0,6,0,-1},{4,5,22,29},30] (* Harvey P. Dale, Apr 08 2018 *)

PARI

Vec((4+5*x-2*x^2-x^3)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^40)) \\ Colin Barker, May 26 2016

Formula

O.g.f.: (4 + 5*x - 2*x^2 - x^3) / ((x^2 - 2*x - 1)*(x^2 + 2*x - 1)).

From Colin Barker, May 26 2016: (Start)

a(n) = (-(-1-sqrt(2))^(1+n)+(-1+sqrt(2))^(1+n)+(1-sqrt(2))^n*(-4+3*sqrt(2))+(1+sqrt(2))^n*(4+3*sqrt(2)))/(2*sqrt(2)).

a(n) = 6*a(n-2)-a(n-4) for n>3. (End)

E.g.f.: 2*cosh(sqrt(2)*x)*(sinh(x) + 2*cosh(x)) + (sinh(sqrt(2)*x)*(5*sinh(x) + 3*cosh(x)))/sqrt(2). - Ilya Gutkovskiy, May 26 2016