A367705 Coefficients of expansion of (1 + 5*x + 11*x^2 + 5*x^3 + 7*x^4 + x^5)/(1 - x^3)^2 in powers of x.

1, 5, 11, 7, 17, 23, 13, 29, 35, 19, 41, 47, 25, 53, 59, 31, 65, 71, 37, 77, 83, 43, 89, 95, 49, 101, 107, 55, 113, 119, 61, 125, 131, 67, 137, 143, 73, 149, 155, 79, 161, 167, 85, 173, 179, 91, 185, 191, 97, 197, 203, 103, 209, 215, 109, 221, 227, 115, 233, 239

Offset: 0

Author

Philippe Deléham, Nov 27 2023

Keywords

• nonn
• easy

Comments

Based on an idea of Pierre CAMI.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A006369, A007310, A016921, A017581, A017653, A130196.

Programs

Mathematica

CoefficientList[Series[(1 + 5*x + 11*x^2 + 5*x^3 + 7*x^4 + x^5)/(1 - x^3)^2, {x, 0, 60}], x] (* or *)
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 5, 11, 7, 17, 23}, 60] (* Amiram Eldar, Nov 28 2023 *)

Formula

a(n) = 3*A006369(n) + A130196(n).

a(n) = A007310(A006369(n) + 1).

a(n) = 2*a(n-3) - a(n-6) for n >= 6.

a(3*n) = 6*n+1, a(3*n+1) = 12*n+5, a(3*n+2) = 12*n+11.

Sum_{n>=0} (-1)^n/a(n) = ((2+sqrt(2))*Pi + sqrt(3)*log(7+4*sqrt(3)) + sqrt(6)*log(5-2*sqrt(6)))/12. - Amiram Eldar, Nov 28 2023