A162771 a(n) = ((2+sqrt(5))*(3+sqrt(5))^n + (2-sqrt(5))*(3-sqrt(5))^n)/2.

2, 11, 58, 304, 1592, 8336, 43648, 228544, 1196672, 6265856, 32808448, 171787264, 899489792, 4709789696, 24660779008, 129125515264, 676109975552, 3540157792256, 18536506851328, 97058409938944, 508204432228352

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

Keywords

• nonn

Comments

Binomial transform of A001077 without initial 1. Third binomial transform of A162963. Inverse binomial transform of A162772.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (6,-4).

Crossrefs

Cf. A001077, A002878, A162963, A162772.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((2+r)*(3+r)^n+(2-r)*(3-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 19 2009

Mathematica

LinearRecurrence[{6,-4},{2,11},30] (* Harvey P. Dale, Aug 15 2013 *)
CoefficientList[Series[(2 - x) / (1 - 6 x + 4 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 16 2013 *)

Formula

a(n) = 6*a(n-1) - 4*a(n-2) for n > 1; a(0) = 2, a(1) = 11. [corrected by Harvey P. Dale, Aug 15 2013]

G.f.: (2-x)/(1-6*x+4*x^2).

a(n) = 2^(n-1) * A002878(n+1). - Diego Rattaggi, Jun 16 2020

a(n) = Sum_{k>=1} binomial(k+n-1,n) * A000032(k) / 2^(k+1). - Diego Rattaggi, Aug 02 2020

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009