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Expansion of (1+x)/(1-2x-x^2-x^3).
3

%I #16 Aug 06 2024 02:14:34

%S 1,3,7,18,46,117,298,759,1933,4923,12538,31932,81325,207120,527497,

%T 1343439,3421495,8713926,22192786,56520993,143948698,366611175,

%U 933692041,2377943955,6056191126,15424018248,39282171577,100044552528,254795294881,648917313867

%N Expansion of (1+x)/(1-2x-x^2-x^3).

%C Diagonal sums of number triangle A116412. A078058 is an alternating sign version.

%C 4*a(n) is the number of ways to tile a strip of length n+2 with red and blue squares, blue dominos, and blue trominos, where the first two tiles must have total length <= 3. - _Greg Dresden_ and Arnim Kuchhal, Aug 05 2024

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

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

%F a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(k+1,j)*C(n-k-j,k)*2^(n-2k-j).

%t CoefficientList[Series[(1+x)/(1-2x-x^2-x^3),{x,0,50}],x] (* or *) LinearRecurrence[{2,1,1},{1,3,7},50] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2012 *)

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 13 2006