A110526 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.

0, 1, 3, 14, 58, 247, 1045, 4428, 18756, 79453, 336567, 1425722, 6039454, 25583539, 108373609, 459077976, 1944685512, 8237820025, 34895965611, 147821682470, 626182695490, 2652552464431, 11236392553213, 47598122677284

Offset: 0

Author

Creighton Dement, Jul 24 2005

Keywords

• easy
• nonn

Comments

A001076(n) = a(n) + a(n+1). Program "Superseeker" finds: A033887(n+1) = a(n+2) - a(n); Elements of even index in the sequence: A049661(n) = (F(6n+1)-1)/4; A015448(n+2) = a(n+2) + 2*a(n+1) + a(n)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3, 5, 1).

Crossrefs

Cf. A110526, A110527, A033887, A001076, A049661, A033887, A000045.

Programs

Maple

seriestolist(series(-x/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1jbaseseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')]

Mathematica

Table[(Fibonacci[3n+1]-(-1)^n)/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

PARI

concat(0, Vec(x/((1+x)*(1-x^2-4*x)) + O(x^100))) \\ Altug Alkan, Oct 28 2015

Formula

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

a(n) = (-1)^n/2 * Sum_{k=0..n} (-1)^k*Fibonacci(3*k). - Gary Detlefs, Jan 03 2013

a(n) = (Fibonacci(3*n+1)-(-1)^n)/4, where Fibonacci(n) = A000045(n). - Vladimir Reshetnikov, Oct 28 2015