OFFSET
0,3
COMMENTS
A_n + B_{n-1} + C_{n-2} in the notation of A137356.
Lim_{n->infinity} a(n+1)/a(n) = x ~= 1.7016..., with x given by the real root (A324498) of (x - 1)^3*x^2 = 1. - Hugo Pfoertner, Mar 15 2019
REFERENCES
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Hùng Việt Chu, Yubo Geng, Julian King, Steven J. Miller, Garrett Tresch, and Zachary Louis Vasseur, Linear Recurrences from Counting Schreier-Type Multisets, Integers 26 (2026), Article A53. See p. 4. See also arXiv:2509.05158 [math.CO], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1).
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5); a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - Harvey P. Dale, Aug 22 2011
G.f.: (1 - 2*x + 2*x^2 - x^3 + x^4) / (1 - 3*x + 3*x^2 - x^3 - x^5). - Colin Barker, Dec 14 2015
MAPLE
f:=n->add( binomial( floor(n-2*k/3), k), k=0..n);
MATHEMATICA
Table[Sum[Binomial[Floor[n-(2k)/3], k], {k, 0, n}], {n, 0, 40}] (* Harvey P. Dale, Aug 22 2011 *)
(* Alternative: *)
LinearRecurrence[{3, -3, 1, 0, 1}, {1, 1, 2, 3, 5}, 40] (* Harvey P. Dale, Aug 22 2011 *)
PROG
(PARI) Vec((1-2*x+2*x^2-x^3+x^4)/(1-3*x+3*x^2-x^3-x^5) + O(x^50)) \\ Colin Barker, Dec 14 2015
(PARI) a(n) = sum(k=0, n, binomial(floor(n-2*k/3), k)); \\ Altug Alkan, Dec 14 2015
(Magma) [(&+[Binomial(Floor(n-2*k/3), k): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Mar 15 2019
(SageMath) [sum(binomial(floor(n-2*k/3), k) for k in (0..n)) for n in (0..40)] # G. C. Greubel, Mar 15 2019
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Don Knuth, Apr 11 2008
STATUS
approved