A053739 - OEIS

%I #42 Dec 30 2023 23:49:09

%S 1,6,22,63,155,344,709,1383,2587,4685,8273,14323,24416,41119,68595,

%T 113590,187030,306605,500950,816410,1327986,2157046,3499982,5674578,

%U 9195035,14893364,24115804,39040633,63192397,102273950,165512723,267839033,433410661,701315739,1134800215

%N Partial sums of A014166.

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

%H G. C. Greubel, <a href="/A053739/b053739.txt">Table of n, a(n) for n = 0..1000</a>

%H Hung Viet Chu, <a href="https://arxiv.org/abs/2106.03659">Partial Sums of the Fibonacci Sequence</a>, arXiv:2106.03659 [math.CO], 2021.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,15,-5,-4,4,-1).

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

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

%F G.f.: 1/((1-x-x^2)*(1-x)^5). - _R. J. Mathar_, May 22 2013

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

%p 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

%t 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 *)

%o (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

%o (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

%o (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

%o (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

%Y Cf. A014166 and A000045.

%Y Right-hand column 10 of triangle A011794.

%Y Cf. A228074.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Feb 13 2000

%E Terms a(28) onward added by _G. C. Greubel_, Sep 06 2019