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

%I A137356 #27 Jan 05 2025 19:51:38
%S A137356 1,1,1,1,1,2,5,11,21,36,58,92,149,250,431,750,1299,2227,3784,6401,
%T A137356 10828,18364,31236,53228,90741,154603,263178,447702,761403,1295022,
%U A137356 2203162,3749001,6380241,10858285,18478155,31443013,53501860,91034937,154900529,263576791
%N A137356 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k).
%D A137356 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.
%H A137356 Robert Israel, <a href="/A137356/b137356.txt">Table of n, a(n) for n = 0..4329</a>
%H A137356 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,2,3).
%H A137356 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,1).
%F A137356 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 A137356 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 A137356 so the generating functions are A = (1-z)^2/p(z), B=z(1-z)/p(z), C=z^2/p(z),
%F A137356 where p(z) = (1-z)^3 - z^5 = 1 - 3z + 3z^2 - z^3 - z^5.
%F A137356 The growth ratio is the real root of r^2(r-1)^3 = 1, approximately 1.70161.
%F A137356 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+a(n-5). - _Vincenzo Librandi_, Aug 09 2015
%F A137356 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 A137356 G.f.: -(x-1)^2/(-1+3*x-3*x^2+x^3+x^5) . - _R. J. Mathar_, May 29 2017
%p A137356 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 A137356 map(f, [$0..50]); # _Robert Israel_, May 26 2017
%t A137356 LinearRecurrence[{3, -3, 1, 0, 1}, {1, 1, 1, 1, 1}, 50] (* _Vincenzo Librandi_, Aug 09 2015 *)
%t A137356 CoefficientList[Series[(1-x)^2/(1 - 3 x + 3 x^2 - x^3 - x^5), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Aug 09 2015 *)
%t A137356 Table[Sum[Binomial[n-2k,3k],{k,0,n/2}],{n,0,50}] (* _Harvey P. Dale_, Nov 07 2021 *)
%o A137356 (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 A137356 Cf. A137357-A137361, A136444, A137402.
%K A137356 nonn,easy
%O A137356 0,6
%A A137356 _Don Knuth_, Apr 11 2008