A188021 Expansion of (x^2)/[(1-x)*(1-3*x^2-x^3)].

0, 0, 1, 1, 4, 5, 14, 20, 48, 75, 165, 274, 571, 988, 1988, 3536, 6953, 12597, 24396, 44745, 85786, 158632, 302104, 561683, 1064945, 1987154, 3756519, 7026408, 13256712, 24835744, 46796545, 87763945, 165225380, 310088381, 583440086, 1095490524

Offset: 0

Author

L. Edson Jeffery, Mar 18 2011

Keywords

• nonn

Comments

Sequence is related to rhombus substitution tilings. For the tridiagonal unit-primitive matrix U_1= (0 1 0 0)

(1 0 1 0)

(0 1 0 1)

(0 0 1 1),

let M=(m_(i,j))=(U_1)^n, i,j=1,2,3,4. Then a(n) = m_(2,4).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..3651
• Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
• Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

Programs

Mathematica

LinearRecurrence[{1,3,-2,-1},{0,0,1,1},40] (* Harvey P. Dale, Jan 26 2013 *)

Formula

a(n)=a(n-1)+3*a(n-2)-2*a(n-3)-a(n-4), for n>=4, with {a(k)}={0,0,1,1}, k=0,1,2,3.

a(n)=A187498(3*n).

G.f.: x^2/(1 - x - 3*x^2 + 2*x^3 + x^4) -Michael De Vlieger, Aug 21 2019