A099098 - OEIS

%I #20 May 29 2025 16:33:48

%S 1,1,4,12,37,114,351,1081,3329,10252,31572,97229,299426,922111,

%T 2839729,8745217,26931732,82938844,255418101,786584466,2422362079,

%U 7459895657,22973462017,70748973084,217878227876,670976837021,2066337330754

%N Quadrisection of a Padovan sequence.

%C Quadrisection of sequence with g.f. 1/(1-x^2-x^3), or A000931(n+3).

%H Harvey P. Dale, <a href="/A099098/b099098.txt">Table of n, a(n) for n = 0..1000</a>

%H Sela Fried, <a href="https://arxiv.org/abs/2505.14196">Even-up words and their variants</a>, arXiv:2505.14196 [math.CO], 2025. See p. 4.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,1).

%F G.f.: (1-x-x^2)/(1-2x-3x^2-x^3);

%F a(n)=sum{k=0..2n, binomial(k, 4n-2k)};

%F a(n)=2a(n-1)+3a(n-2)+a(n-3);

%F a(n)=A000931(4n+3).

%F a(n) = Sum [k=0..n, C(2n-k, 2k) ].

%e 1 + x + 4*x^2 + 12*x^3 + 37*x^4 + 114*x^5 + 351*x^6 + ...

%t LinearRecurrence[{2,3,1},{1,1,4},40] (* _Harvey P. Dale_, Aug 23 2011 *)

%Y Bisection of A005251.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Sep 29 2004