A054455 Row sums of triangle A054453.

1, 3, 7, 16, 34, 70, 140, 274, 527, 999, 1871, 3468, 6371, 11613, 21023, 37826, 67688, 120530, 213670, 377252, 663607, 1163361, 2033101, 3542808, 6157045, 10673703, 18460759, 31859716, 54872158, 94326622

Offset: 0

Author

Wolfdieter Lang, Apr 27 2000

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A054453, A000045.

Programs

GAP

a:=[1,3,7,16,34,70];; for n in [7..30] do a[n]:=3*a[n-1]-5*a[n-3] +3*a[n-5]+a[n-6]; od; a; # G. C. Greubel, Jan 31 2019

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x^2)^2/(1-x-x^2)^3 )); // G. C. Greubel, Jan 31 2019

Mathematica

LinearRecurrence[{3,0,-5,0,3,1}, {1,3,7,16,34,70}, 40] (* or *) CoefficientList[Series[(1-x^2)^2/(1-x-x^2)^3, {x,0,40}],x] (* G. C. Greubel, Jan 31 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-x^2)^2/(1-x-x^2)^3) \\ G. C. Greubel, Jan 31 2019

Sage

((1-x^2)^2/(1-x-x^2)^3).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

Formula

a(n) = Sum_{m=0..n} A054453(n, m).

a(n) = ((5*n^2 + 27*n + 50)*F(n+1) + 34*(n+1)*F(n))/50, F(n)= A000045(n) (Fibonacci numbers).

G.f.: ((Fib(x))^3)*(1-x^2)^2, with Fib(x)=1/(1-x-x^2) g.f. for A000045(n+1) (Fibonacci numbers without F(0)).