A053808 - OEIS

%I #55 Jan 06 2025 10:57:28

%S 1,5,15,36,76,148,273,485,839,1424,2384,3952,6505,10653,17383,28292,

%T 45964,74580,120905,195885,317231,513600,831360,1345536,2177521,

%U 3523733,5701983,9226500,14929324,24156724,39087009,63244757,102332855,165578768,267912848

%N Partial sums of A001891.

%C Antidiagonal sums of the convolution array A213579 and row 1 of the convolution array A213590. - _Clark Kimberling_, Jun 18 2012

%C Also number CG(n,2) of complete games with n players of 2 types. - _N. J. A. Sloane_, Dec 29 2012

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

%H G. C. Greubel, <a href="/A053808/b053808.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Freixas and S. Kurz, <a href="http://www.wm.uni-bayreuth.de/fileadmin/Sascha/Publikationen2/Fibonacci.pdf">The golden number and Fibonacci sequences in the design of voting structures</a>, 2012. - From _N. J. A. Sloane_, Dec 29 2012

%H W. Lang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-4/elementary36-4.pdf">Problem B-858</a>, Fibonacci Quarterly, 36,3 (1998) 373-374; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/37-2/elementary37-2.pdf">Solution</a>, ibid. 37,2 (1999) 183-184.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).

%F a(n) = a(n-1) + a(n-2) + (n+1)^2, a(-n)=0.

%F G.f.: (1+x)/((1-x-x^2)*(1-x)^3).

%F a(n) = Fibonacci(n+6) - (n^2 + 4*n + 8), n >= 2 (see p. 184 of FQ reference).

%F a(n-2) = Sum_{i=0..n} Fibonacci(i)*(n-i)^2. - _Benoit Cloitre_, Mar 06 2004

%t Table[Fibonacci[n+8] -(n^2 +8*n+20), {n,0,40}] (* _G. C. Greubel_, Jul 06 2019 *)

%t LinearRecurrence[{4,-5,1,2,-1},{1,5,15,36,76},40] (* _Harvey P. Dale_, Apr 14 2022 *)

%o (PARI) vector(40, n, n--; fibonacci(n+8) - (n^2 +8*n+20)) \\ _G. C. Greubel_, Jul 06 2019

%o (Magma) [Fibonacci(n+8) - (n^2+8*n+20): n in [0..40]]; // _G. C. Greubel_, Jul 06 2019

%o (Sage) [fibonacci(n+8) - (n^2 +8*n+20) for n in (0..20)] # _G. C. Greubel_, Jul 06 2019

%o (GAP) List([0..40], n-> Fibonacci(n+8) - (n^2 +8*n+20)); # _G. C. Greubel_, Jul 06 2019

%Y Convolution of A000290 (squares) with A000045, n >= 1. (Fibonacci) - _Wolfdieter Lang_, Apr 10 2000

%Y Right-hand column 7 of triangle A011794.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Mar 27 2000