A053739 Partial sums of A014166.

1, 6, 22, 63, 155, 344, 709, 1383, 2587, 4685, 8273, 14323, 24416, 41119, 68595, 113590, 187030, 306605, 500950, 816410, 1327986, 2157046, 3499982, 5674578, 9195035, 14893364, 24115804, 39040633, 63192397, 102273950, 165512723, 267839033, 433410661, 701315739, 1134800215

Offset: 0

Author

Barry E. Williams, Feb 13 2000

Keywords

• easy
• nonn

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Hung Viet Chu, Partial Sums of the Fibonacci Sequence, arXiv:2106.03659 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1).

Crossrefs

Cf. A014166 and A000045.

Right-hand column 10 of triangle A011794.

Cf. A228074.

Programs

GAP

List([0..35], n-> Fibonacci(n+11)-(n^4+22*n^3+203*n^2+974*n + 2112)/24); # G. C. Greubel, Sep 06 2019

Magma

[Fibonacci(n+11) - (n^4+22*n^3+203*n^2+974*n+2112)/24: n in [0..35]]; // G. C. Greubel, Sep 06 2019

Maple

with(combinat); seq(fibonacci(n+11)-(n^4 + 22*n^3 + 203*n^2 + 974*n + 2112)/4!, n = 0..35); # G. C. Greubel, Sep 06 2019

Mathematica

Table[Fibonacci[n+11] -(n^4+22*n^3+203*n^2+974*n+2112)/4!, {n,0,35}] (* G. C. Greubel, Sep 06 2019 *)

PARI

vector(35, n, m=n-1; fibonacci(n+10) - (m^4+22*m^3+203*m^2+974*m +2112)/4!) \\ G. C. Greubel, Sep 06 2019

Sage

[fibonacci(n+11) - (n^4+22*n^3+203*n^2+974*n+2112)/24 for n in (0..35)] # G. C. Greubel, Sep 06 2019

Formula

a(n) = Sum_{i=0..floor(n/2)} binomial(n+5-i, n-2*i) for n >= 0.

a(n) = a(n-1) + a(n-2) + C(n+4,4); n >= 0; a(-1)=0.

G.f.: 1/((1-x-x^2)*(1-x)^5). - R. J. Mathar, May 22 2013

a(n) = Fibonacci(n+11) - (n^4 + 22*n^3 + 203*n^2 + 974*n + 2112)/4!. - G. C. Greubel, Sep 06 2019

Extensions

Terms a(28) onward added by G. C. Greubel, Sep 06 2019