A054470 Partial sums of A054469.

1, 8, 36, 121, 339, 838, 1891, 3983, 7953, 15225, 28183, 50779, 89518, 155053, 264767, 446952, 747572, 1241207, 2048762, 3366122, 5510518, 8995550, 14652578, 23827138, 38696751, 62785150, 101794318, 164950755, 267183785, 432650132

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 (7,-20,29,-20,1,8,-5,1).

Crossrefs

Cf. A000045, A001891, A001911, A054469.

Right-hand column 13 of triangle A011794.

Programs

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

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 *)

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

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)