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A137356 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k). 9

%I

%S 1,1,1,1,1,2,5,11,21,36,58,92,149,250,431,750,1299,2227,3784,6401,

%T 10828,18364,31236,53228,90741,154603,263178,447702,761403,1295022,

%U 2203162,3749001,6380241,10858285,18478155,31443013,53501860,91034937,154900529,263576791

%N a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k).

%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

%H Robert Israel, <a href="/A137356/b137356.txt">Table of n, a(n) for n = 0..4329</a>

%H V. C. Harris, C. C. Styles, <a href="http://www.fq.math.ca/Scanned/2-4/harris.pdf">A generalization of Fibonacci numbers</a>, Fib. Quart. 2 (1964) 277-289, sequence u(n,2,3).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,1).

%F Let A_n = Sum_{k<=n/2}binomial(n-2k,3k) (the present sequence), B_n= Sum_{k<=n/2}binomial(n-2k, 3k+1)(A137357), C_n= Sum_{k<=n/2}binomial(n-2k, 3k+2) (A137358).

%F Then A_n = A_{n-1} + C_{n-3} + \delta_{n0}, B_n=B_{n-1} + A_{n-1}, C_n=C_{n-1} + B_{n-1};

%F so the generating functions are A = (1-z)^2/p(z), B=z(1-z)/p(z), C=z^2/p(z),

%F where p(z) = (1-z)^3 - z^5 = 1 - 3z + 3z^2 - z^3 - z^5.

%F The growth ratio is the real root of r^2(r-1)^3 = 1, approximately 1.70161.

%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+a(n-5). - _Vincenzo Librandi_, Aug 09 2015

%F a(n) = hypergeom([-(1/5)*n, -(1/5)*n+1/5, 2/5-(1/5)*n, 3/5-(1/5)*n, -(1/5)*n+4/5], [1/3, 2/3, -(1/2)*n, -(1/2)*n+1/2], -3125/108). - _Robert Israel_, May 26 2017

%F G.f.: -(x-1)^2/(-1+3*x-3*x^2+x^3+x^5) . - _R. J. Mathar_, May 29 2017

%p f:= gfun:-rectoproc({a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+a(n-5),seq(a(i)=1,i=0..4)},a(n),remember):

%p map(f, [$0..50]); # _Robert Israel_, May 26 2017

%t LinearRecurrence[{3, -3, 1, 0, 1}, {1, 1, 1, 1, 1}, 50] (* _Vincenzo Librandi_, Aug 09 2015 *)

%t CoefficientList[Series[(1-x)^2/(1 - 3 x + 3 x^2 - x^3 - x^5), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Aug 09 2015 *)

%o (MAGMA) I:=[1,1,1,1,1]; [n le 5 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-5): n in [1..45]]; // _Vincenzo Librandi_, Aug 09 2015

%Y Cf. A137357-A137361, A136444, A137402.

%K nonn,easy

%O 0,6

%A _Don Knuth_, Apr 11 2008

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Last modified October 16 13:12 EDT 2018. Contains 316263 sequences. (Running on oeis4.)