A081340 (5^n+(-1)^n)/2.

1, 2, 13, 62, 313, 1562, 7813, 39062, 195313, 976562, 4882813, 24414062, 122070313, 610351562, 3051757813, 15258789062, 76293945313, 381469726562, 1907348632813, 9536743164062, 47683715820313, 238418579101562

Offset: 0

Author

Paul Barry, Mar 18 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A003665. 2nd binomial transform of (1,0,9,0,81,0,729,0,..). Case k=2 of family of recurrences a(n)=2k*a(n-1)-(k^2-9)*a(n-2), a(0)=0, a(1)=k. A003665 is case k=1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (4,5).

Crossrefs

Cf. A081341, A087404.

Programs

Mathematica

CoefficientList[Series[(1 - 2 x) / ((1 + x) (1 - 5 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 08 2013 *)

PARI

a(n)=(5^n+(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[lucas_number2(n,4,-5)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

Formula

a(n) = 4*a(n-1) + 5*a(n-2), a(0)=1, a(1)=2.

G.f.: (1-2*x)/((1+x)*(1-5*x)).

E.g.f.: exp(2*x) * cosh(3*x).

a(n) = ((2+sqrt(9))^n+(2-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008

a(n) = sum( k=0..n, A201730(n,k)*8^k ). - Philippe Deléham, Dec 06 2011