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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.
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%I #17 Jan 01 2024 01:59:26

%S 1,5,11,7,17,23,13,29,35,19,41,47,25,53,59,31,65,71,37,77,83,43,89,95,

%T 49,101,107,55,113,119,61,125,131,67,137,143,73,149,155,79,161,167,85,

%U 173,179,91,185,191,97,197,203,103,209,215,109,221,227,115,233,239

%N 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.

%C Based on an idea of _Pierre CAMI_.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).

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

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

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

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

%F 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

%t 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 *)

%t LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 5, 11, 7, 17, 23}, 60] (* _Amiram Eldar_, Nov 28 2023 *)

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

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Nov 27 2023