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A nonlinear binomial sum.
(Formerly M1120 N0428)
2

%I M1120 N0428 #60 Sep 23 2023 16:17:19

%S 1,2,4,8,16,31,58,105,185,319,541,906,1503,2476,4058,6626,10790,17537,

%T 28464,46155,74791,121137,196139,317508,513901,831686,1345888,2177900,

%U 3524140,5702419,9226966,14929821,24157253,39087571,63245353,102333486

%N A nonlinear binomial sum.

%D Ralph P. Grimaldi, A generalization of the Fibonacci sequence. Proceedings of the seventeenth Southeastern international conference on combinatorics, graph theory, and computing (Boca Raton, Fla., 1986). Congr. Numer. 54 (1986), 123-128. MR0885268 (89f:11030). - _N. J. A. Sloane_, Apr 08 2012

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000128/b000128.txt">Table of n, a(n) for n=1..201</a>

%H D. A. Lind, <a href="http://www.fq.math.ca/Scanned/3-4/lind.pdf">On a class of nonlinear binomial sums</a>, Fib. Quart., 3 (1965), 292-298.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

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

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

%F a(n) = F(n+4) - n*(n+1)/2 - 3, with F(n) = A000045(n). - _Ralf Stephan_, Aug 19 2004

%F a(n) = 2*a(n-1) - a(n-3) + n - 3, n > 3, and a(1) = 1, a(2) = 2, a(3) = 4. - _Chunqing Liu_, Sep 23 2023

%p A000128:=(1-2*z+z**2+z**3)/(z**2+z-1)/(z-1)**3; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t LinearRecurrence[{4, -5, 1, 2, -1}, {1, 2, 4, 8, 16}, 40] (* _Jean-François Alcover_, Feb 04 2016 *)

%Y Differences are A000126.

%Y Second differences are A000071 (Fibonacci -1).

%Y Cf. A000045.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Michel ten Voorde_, Oct 06 2002