cp's OEIS Frontend

A097333 a(n) = Sum_{k=0..n} C(n-k, floor(k/2)).

%I A097333 #41 Dec 29 2023 11:40:07
%S A097333 1,2,2,3,5,7,10,15,22,32,47,69,101,148,217,318,466,683,1001,1467,2150,
%T A097333 3151,4618,6768,9919,14537,21305,31224,45761,67066,98290,144051,
%U A097333 211117,309407,453458,664575,973982,1427440,2092015,3065997,4493437,6585452
%N A097333 a(n) = Sum_{k=0..n} C(n-k, floor(k/2)).
%C A097333 Partial sums of A058278.
%H A097333 Vincenzo Librandi, <a href="/A097333/b097333.txt">Table of n, a(n) for n = 0..1000</a>
%H A097333 Kassie Archer and Aaron Geary, <a href="https://arxiv.org/abs/2312.14351">Powers of permutations that avoid chains of patterns</a>, arXiv:2312.14351 [math.CO], 2023. See p. 15.
%H A097333 Engin Özkan, Bahar Kuloǧlu, and James Peters, <a href="https://hal.archives-ouvertes.fr/hal-03242990">k-Narayana sequence self-similarity</a>, hal-03242990 [math.CO], 2021. See p. 12.
%H A097333 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1).
%F A097333 G.f.: (1+x-x^2-x^3)/((1-x)*(1-x^2-x^3-x^4)) = (1+x)/(1-x-x^3);
%F A097333 a(n) = a(n-1) + a(n-3);
%F A097333 a(n) = a(n-1) + a(n-2) - a(n-5).
%F A097333 a(n) = A058278(n+3) = A000930(n-1)+A000930(n). - _R. J. Mathar_, Jul 07 2023
%F A097333 a(n) = A003410(n-1) for n >= 2. - _Jianing Song_, Aug 11 2023
%t A097333 LinearRecurrence[{1, 0, 1}, {1, 2, 2}, 70] (* _Vladimir Joseph Stephan Orlovsky_, Feb 26 2011*)
%o A097333 (PARI) a(n) = sum(k=0, n, binomial(n-k, k\2)); \\ _Michel Marcus_, Mar 02 2022
%Y A097333 Essentially the same as A003410 and A058278.
%K A097333 easy,nonn
%O A097333 0,2
%A A097333 _Paul Barry_, Aug 05 2004