A054469 a(n) = a(n-1) + a(n-2) + (n+2)*binomial(n+3, 3)/2, with a(0) = 1, a(1) = 7.

1, 7, 28, 85, 218, 499, 1053, 2092, 3970, 7272, 12958, 22596, 38739, 65535, 109714, 182185, 300620, 493635, 807555, 1317360, 2144396, 3485032, 5657028, 9174560, 14869613, 24088399, 39009168, 63156437, 102233030, 165466347, 267786673

Offset: 0

Author

Barry E. Williams, Mar 31 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
• A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
• Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1).

Crossrefs

Cf. A000045, A001891, A001911.

Right-hand column 11 of triangle A011794.

Programs

Magma

A054469:= func< n | Fibonacci(n+12) -(1/12)*(1716 +802*n +173*n^2 +20*n^3 +n^4) >;
[A054469(n): n in [0..40]]; // G. C. Greubel, Oct 21 2024

Mathematica

RecurrenceTable[{a[0]==1,a[1]==7,a[n]==a[n-1]+a[n-2]+(n+2) Binomial[ n+3,3]/2},a,{n,30}] (* Harvey P. Dale, Sep 22 2013 *)
CoefficientList[Series[(1+x)/((1-x)^5*(1-x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 23 2013 *)

PARI

a(n) = sum(i=1,(n+2)\2,binomial(n+5-i,n+2-2*i))+2*sum(i=1,(n+1)\2,binomial(n+5-i,n+1-2*i)) \\ Jason Yuen, Aug 13 2024

SageMath

def A054469(n): return fibonacci(n+12) - (1716 + 802*n + 173*n^2 + 20*n^3 + n^4)//12
[A054469(n) for n in range(41)] # G. C. Greubel, Oct 21 2024

Formula

a(n) = a(n-1) + a(n-2) + (n+1)*(n+2)^2*(n+3)/12.

a(-n) = 0.

a(n) = (Sum_{i=1..floor((n+2)/2)} binomial(n+5-i, n+2-2*i)) + 2*(Sum_{i=1..floor((n+1)/2)} binomial(n+5-i, n+1-2*i)).

G.f.: (1+x) / ((1-x)^5*(1-x-x^2)). - Colin Barker, Jun 11 2013

From G. C. Greubel, Oct 21 2024: (Start)

a(n) = Fibonacci(n+12) - Sum_{j=0..4} Fibonacci(11-2*j) * binomial(n+j, j).

a(n) = Fibonacci(n+12) - (1/12)*(1716 + 802*n + 173*n^2 + 20*n^3 + n^4). (End)