A054451 Third column of triangle A054450 (partial row sums of unsigned Chebyshev triangle A049310).

1, 1, 4, 5, 12, 17, 33, 50, 88, 138, 232, 370, 609, 979, 1596, 2575, 4180, 6755, 10945, 17700, 28656, 46356, 75024, 121380, 196417, 317797, 514228, 832025, 1346268, 2178293, 3524577, 5702870, 9227464, 14930334, 24157816, 39088150, 63245985, 102334135

Offset: 0

Author

Wolfdieter Lang, Apr 27 2000

Keywords

• easy
• nonn

Comments

Equals triangle A173284 * [1, 2, 3, ...]. - Gary W. Adamson, Mar 03 2010

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-3,1,1).

Crossrefs

Cf. A054450, A049310, A000045, A052952.

Cf. A007382.

Cf. A173284.

Programs

Maple

BB:=1/(1-k^2)^2/(1-k-k^2): seq(coeff(series(BB, k, n+1), k, n), n=0..50); # Zerinvary Lajos, May 16 2007

Mathematica

LinearRecurrence[{1,3,-2,-3,1,1},{1,1,4,5,12,17},40] (* Harvey P. Dale, Oct 06 2024 *)

PARI

Vec(-1/((x-1)^2*(x+1)^2*(x^2+x-1)) + O(x^100)) \\ Colin Barker, Jun 14 2015

Formula

a(n) = A054450(n+2, 2).

G.f.: Fib(x)/(1-x^2)^2, with Fib(x)=1/(1-x-x^2) = g.f. A000045 (Fibonacci numbers without 0).

a(2*k) = A027941(k)= F(2*k+3)-1; a(2*k+1)= F(2*(k+2))-(k+2)= A054452(k), k >= 0.

a(n-2) = Fibonacci(n+1) - binomial(n-floor(n/2), floor(n/2)), or a(n-2) = Sum_{i=0..floor(n/2)-1} binomial(n-i, i). - Jon Perry, Mar 18 2004

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, k). - Paul Barry, Oct 23 2004

Extensions

More terms from James Sellers, Apr 28 2000