A081190 8th binomial transform of (1,0,1,0,1,.....), A059841.

1, 8, 65, 536, 4481, 37928, 324545, 2803256, 24405761, 213887048, 1884629825, 16679193176, 148135411841, 1319377419368, 11777507763905, 105319346802296, 943126559710721, 8454906106826888, 75861524447454785, 681125306429182616

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A081189.

a(n) is also the number of words of length n over an alphabet of nine letters, of which a chosen one appears an even number of times. See a comment in A007582, also for the crossrefs. for the 1- to 11- letter word cases. For a formulation in terms of maps see a Geoffrey Critzer comment in A081189. - Wolfdieter Lang, Jul 17 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (16,-63).

Crossrefs

Cf. A007582, A060531, A081189.

Programs

Magma

[7^n/2 + 9^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

Mathematica

CoefficientList[Series[(1 - 8 x) / ((1 - 7 x) (1 - 9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{16,-63},{1,8},20] (* Harvey P. Dale, Apr 04 2017 *)

Formula

a(n) = 16*a(n-1) -63*a(n-2), a(0)=1, a(1)=8.

G.f.: (1-8*x)/((1-7*x)*(1-9*x)).

E.g.f. exp(8*x) * cosh(x).

a(n) = 7^n/2 + 9^n/2.