OFFSET
0,3
COMMENTS
A "Rogers-Ramanujan-Slater" type identity is an identity containing a variable q which equates an infinite product with an infinite series. A finite generalization of such an identity consists of two sequences of polynomials, such that corresponding terms in each sequence are equal and one sequence tends to the infinite sum and the other sequence tends to the infinite product. [From AMS Abstracts 2008 Eric Werley by Michael Somos, Mar 27 2014]
In statistical mechanics, the fermionic side of a Rogers-Ramanujan type identity is the infinite series side of the identity and the bosonic side is the infinite product side of the identity.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
George E. Andrews, The hard-hexagon model and Rogers-Ramanujan type identities, Proc. Nat. Acad. Sci. U.S.A., 78(1981), 5290-5292.
L. J. Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., 54(1952), 147-167.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
a(n) = (1/8)*(10*n^2 + 2*(1+(-1)^n)*n - (1-(-1)^n)).
From Colin Barker, Mar 26 2014: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(x^3+3*x^2+5*x+1) / ((x-1)^3*(x+1)^2). (End)
MAPLE
A239767:=n->(10*n^2 + 2*n*(1+(-1)^n) - (1-(-1)^n))/8; seq(A239767(n), n=0..100); # Wesley Ivan Hurt, Mar 27 2014
MATHEMATICA
Table[(10 n^2 + 2 n (1 + (-1)^n) - (1 - (-1)^n))/8, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 27 2014 *)
CoefficientList[Series[- x (x^3 + 3 x^2 + 5 x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 29 2014 *)
PROG
(PARI) concat(0, Vec(-x*(x^3+3*x^2+5*x+1)/((x-1)^3*(x+1)^2) + O(x^100))) \\ Colin Barker, Mar 26 2014
(Magma) [(1/8)*(10*n^2+2*(1+(-1)^n)*n-(1-(-1)^n)): n in [0..50]]; // Vincenzo Librandi, Mar 29 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric Werley, Mar 26 2014
EXTENSIONS
More terms from Colin Barker, Mar 26 2014
STATUS
approved