OFFSET
0,2
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 (7,-20,29,-20,1,8,-5,1).
FORMULA
a(n) = a(n-1) + a(n-2) + (2*n+5)*C(n+4, 4)/5, with a(-n) = 0.
a(n) = Sum_{j=1..[(n+2)/2]} binomial(n+6-j, n+2-2*j) + 2*Sum_{j=1..[(n+1)/2]} binomial(n+6-j, n+1-2*j), where [x]=greatest integer in x.
G.f.: (1+x) / ((1-x)^6*(1-x-x^2)). - Colin Barker, Jun 11 2013
From G. C. Greubel, Oct 21 2024: (Start)
a(n) = Fibonacci(n+14) - Sum_{j=0..5} Fibonacci(13-2*j)*binomial(n+j,j).
a(n) = Fibonacci(n+14) - (1/120)*(45120 + 21458*n + 4925*n^2 + 680*n^3 + 55*n^4 + 2*n^5). (End)
MATHEMATICA
Accumulate[RecurrenceTable[{a[0]==1, a[1]==7, a[n]==a[n-1]+a[n-2]+(n+2) Binomial[n+3, 3]/2}, a, {n, 40}]] (* Harvey P. Dale, Sep 22 2013 *)
CoefficientList[Series[(1+x)/((1-x)^6*(1-x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 23 2013 *)
PROG
(Magma)
A054470:= func< n | Fibonacci(n+14) - (45120 +21458*n +4925*n^2 +680*n^3 +55*n^4 +2*n^5)/120 >;
[A054470(n): n in [0..40]]; // G. C. Greubel, Oct 21 2024
(SageMath)
def A054470(n): return fibonacci(n+14) -(45120 +21458*n +4925*n^2 +680*n^3 +55*n^4 +2*n^5)//120
[A054470(n) for n in range(41)] # G. C. Greubel, Oct 21 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Mar 31 2000
STATUS
approved