A116413 Expansion of (1+x)/(1-2x-x^2-x^3).

1, 3, 7, 18, 46, 117, 298, 759, 1933, 4923, 12538, 31932, 81325, 207120, 527497, 1343439, 3421495, 8713926, 22192786, 56520993, 143948698, 366611175, 933692041, 2377943955, 6056191126, 15424018248, 39282171577, 100044552528, 254795294881, 648917313867

Offset: 0

Author

Paul Barry, Feb 13 2006

Keywords

• easy
• nonn

Comments

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

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

Links

• Index entries for linear recurrences with constant coefficients, signature (2,1,1).

Programs

Mathematica

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 *)

Formula

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

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).