A005676 - OEIS

%I M1610 #54 Jan 05 2025 19:51:33

%S 1,1,1,1,1,2,6,16,36,71,128,220,376,661,1211,2290,4382,8347,15706,

%T 29191,53824,99009,182497,337745,627401,1167937,2174834,4046070,

%U 7517368,13951852,25880583,48009456,89090436,165392856,307137901

%N a(n) = Sum_{k=0..n} C(n-k,4*k).

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

%H Vincenzo Librandi, <a href="/A005676/b005676.txt">Table of n, a(n) for n = 0..1000</a>

%H V. C. Harris, C. C. Styles, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-4/harris.pdf">A generalization of Fibonacci numbers</a>, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,4).

%H V. E. Hoggatt, Jr., <a href="/A005676/a005676.pdf">7-page typed letter to N. J. A. Sloane with suggestions for new sequences</a>, circa 1977.

%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,-6,4,-1,1).

%F From _Paul Barry_, Jul 23 2004: (Start)

%F G.f.: (1-3x+3x^2-x^3)/(1-4x+6x^2-4x^3+x^4-x^5) = (1-x)^3/((1-x)^4-x^5).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, 4k).

%F a(n) = 4a(n-1)-6a(n-2)+4a(n-3)-a(n-4)+a(n-5). (End)

%p A005676:=(z-1)**3/(-1+4*z-6*z**2+4*z**3-z**4+z**5); # _Simon Plouffe_ in his 1992 dissertation.

%t LinearRecurrence[{4, -6, 4, -1, 1}, {1, 1, 1, 1, 1}, 40] (* or *) CoefficientList[Series[(1 - x)^3 / ((1 - x)^4 - x^5), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 08 2017 *)

%o (Magma) [&+[Binomial(n-k, 4*k): k in [0..n]]: n in [0..40]]; // _Vincenzo Librandi_, Sep 08 2017

%Y Column k=4 of A306680.

%K nonn,easy

%O 0,6

%A _N. J. A. Sloane_.

%E More terms from _James A. Sellers_, Aug 21 2000