A099571 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, k).

1, 1, 5, 6, 17, 23, 50, 73, 138, 211, 370, 581, 979, 1560, 2575, 4135, 6755, 10890, 17700, 28590, 46356, 74946, 121380, 196326, 317797, 514123, 832025, 1346148, 2178293, 3524441, 5702870, 9227311, 14930334, 24157645, 39088150, 63245795

Offset: 0

Author

Paul Barry, Oct 23 2004

Keywords

• easy
• nonn

Comments

Fourth column of triangle A054450.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-6,3,4,-1,-1).

Crossrefs

Cf. A000045, A052952, A054450, A054451, A099572.

Programs

Magma

[Fibonacci(n+4) +(-1)^n*(n^2+4*n+7)/16 -(n^2+12*n+39)/16: n in [0..40]]; // G. C. Greubel, Jul 25 2022

Mathematica

Table[Sum[Binomial[n-k+3,k],{k,0,Floor[n/2]}],{n,0,40}] (* or *) LinearRecurrence[{1,4,-3,-6,3,4,-1,-1},{1,1,5,6,17,23,50,73},40] (* Harvey P. Dale, Jun 04 2021 *)

SageMath

[fibonacci(n+4) +(-1)^n*(n^2+4*n+7)/16 -(n^2+12*n+39)/16 for n in (0..40)] # G. C. Greubel, Jul 25 2022

Formula

G.f.: 1/((1-x^2)^3*(1-x-x^2)).

a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 6*a(n-4) + 3*a(n-5) + 4*a(n-6) - a(n-7) - a(n-8);

a(n) = Sum_{k=0..n} Fibonacci(n-k+1)*binomial(k/2 +2, 2)*(1+(-1)^k)/2.

a(n) = Fibonacci(n+4) + (-1)^n*(n^2 + 4*n + 7)/16 - (n^2 + 12*n + 39)/16. - G. C. Greubel, Jul 25 2022