OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
G.f.: (1+6*x+8*x^2+6*x^3+x^4) / ((1-x)^3*(1+x)).
From Colin Barker, Dec 09 2016: (Start)
a(n) = (11*n^2+4)/2 for n>0 and even.
a(n) = (11*n^2+5)/2 for n odd.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4. (End)
E.g.f.: ((9 + 22*x + 22*x^2)*exp(x) - 4 - exp(-x))/4. - G. C. Greubel, Jul 31 2019
MATHEMATICA
CoefficientList[Series[(1+6*x+8*x^2+6*x^3+x^4)/((1-x)^3*(1+x)), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {1, 8, 24, 52, 90}, 51] (* Indranil Ghosh, Feb 24 2017 *)
Table[If[n==0, 1, (22*n^2+9-(-1)^n)/4], {n, 0, 50}] (* G. C. Greubel, Jul 31 2019 *)
PROG
(PARI) Vec((1+6*x+8*x^2+6*x^3+x^4)/((1-x)^3*(1+x)) + O(x^50)) \\ Colin Barker, Dec 09 2016
(PARI) a(n)=if(n, 11*n^2+5, 2)\2 \\ Charles R Greathouse IV, Feb 24 2017
(Magma) [n eq 0 select 1 else (22*n^2+9-(-1)^n)/4: n in [0..50]]; // G. C. Greubel, Jul 31 2019
(Sage) [1]+[(22*n^2+9-(-1)^n)/4 for n in (1..50)] # G. C. Greubel, Jul 31 2019
(GAP) Concatenation([1], List([1..50], n-> (22*n^2+9-(-1)^n)/4)); # G. C. Greubel, Jul 31 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 09 2000
STATUS
approved