A164539 a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 13.

1, 13, 33, 157, 545, 2189, 8193, 31709, 120769, 463501, 1772385, 6789277, 25985249, 99495437, 380887617, 1458243293, 5582699905, 21373102861, 81825105057, 313261930141, 1199299595681, 4591432702349, 17577962574465, 67295954065373

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A164675. Inverse binomial transform of A164540.

Pisano period lengths: 1, 1, 8, 1, 24, 8, 3, 2, 24, 24, 15, 8, 168, 3, 24, 2, 4, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Links

• Vincenzo Librandi and Harvey P. Dale, Table of n, a(n) for n = 0..1000 (Vincenzo Librandi to 177)
• Index entries for linear recurrences with constant coefficients, signature (2,7).

Crossrefs

Cf. A164675, A164540.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(1+2*r)^n+(1-3*r)*(1-2*r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009

Mathematica

LinearRecurrence[{2,7},{1,13},50] (* Harvey P. Dale, Oct 16 2011 *)

Formula

a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 13.

G.f.: (1+11*x)/(1-2*x-7*x^2).

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

a(n) = 11*A015519(n) + A015519(n+1). - R. J. Mathar, Aug 10 2012

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009