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A054469
A second-order recursive sequence.
3
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
0,2
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
LINKS
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.
FORMULA
a(n) = a(n-1) + a(n-2) + (n+2)*binomial(n+3, 3)/2.
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)
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 *)
PROG
(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
(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
(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
CROSSREFS
Right-hand column 11 of triangle A011794.
Sequence in context: A369807 A221141 A144900 * A369806 A156928 A117473
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Mar 31 2000
STATUS
approved