A034478 a(n) = (5^n + 1)/2.

1, 3, 13, 63, 313, 1563, 7813, 39063, 195313, 976563, 4882813, 24414063, 122070313, 610351563, 3051757813, 15258789063, 76293945313, 381469726563, 1907348632813, 9536743164063, 47683715820313, 238418579101563

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Terms (with the offset changed to 1) are also the quotients arising from sequence A050621.

Partial sums of A020699. - Paul Barry, Sep 03 2003

Binomial transform of A081294. - Paul Barry, Jan 13 2005

Examples

			G.f. = 1 + 3*x + 13*x^2 + 63*x^3 + 313*x^4 + 1563*x^5 + 7813*x^6 + ...
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..250
• Index entries for linear recurrences with constant coefficients, signature (6,-5).

Crossrefs

Cf. A020699, A034474, A050621, A081294.

Programs

Maple

seq((5^n + 1)/2, n=0..20); # Zerinvary Lajos, Jun 16 2007

Mathematica

LinearRecurrence[{6, -5},{1, 3},22] (* Ray Chandler, May 25 2021 *)

Sage

[lucas_number2(n,6,5)/2 for n in range(0,22)] # Zerinvary Lajos, Jul 08 2008

Formula

E.g.f.: exp(3*x)*cosh(2*x). - Paul Barry, Mar 17 2003

G.f.: (1-3*x)/((1-x)*(1-5*x)). - Paul Barry, Sep 03 2003

a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n, k)*binomial(2*k, 2*j). - Paul Barry, Jan 13 2005

a(n) = 6*a(n-1) - 5*a(n-2) for n>1, a(0)=1, a(1)=3. - Philippe Deléham, Jul 11 2005

a(n)^2 + (a(n) - 1)^2 = a(2*n). E.g., 63^2 + 62^2 = 7813 = a(6). - Gary W. Adamson, Jun 17 2006

a(n) = 5*a(n-1) - 2 for n>0, a(0)=1. - Vincenzo Librandi, Aug 01 2010

a(n) = A034474(n)/2. - Elmo R. Oliveira, Dec 10 2023