A054410 Susceptibility series H_3 for 2-dimensional Ising model (divided by 2).

1, 12, 52, 148, 328, 620, 1052, 1652, 2448, 3468, 4740, 6292, 8152, 10348, 12908, 15860, 19232, 23052, 27348, 32148, 37480, 43372, 49852, 56948, 64688, 73100, 82212, 92052, 102648, 114028, 126220, 139252, 153152, 167948, 183668, 200340, 217992, 236652

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 (4,-6,4,-1).

Formula

G.f.: (1 +8*x +10*x^2 +8*x^3 +x^4)/(1-x)^4.

From Colin Barker, Dec 09 2016: (Start)

a(n) = 2*n*(11 + 7*n^2)/3 for n>0.

a(0)=1, a(1)=12, a(2)=52, a(3)=148, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4. (End)

E.g.f.: (3 + 2*x*(18 + 21*x + 7*x^2)*exp(x))/3. - G. C. Greubel, Jul 31 2019

Mathematica

CoefficientList[Series[(1+8*x+10*x^2+8*x^3+x^4)/(1-x)^4, {x, 0, 40}], x] (* or *) a[0]=1; a[n_]:= 2*n*(11+7*n^2)/3; Table[a[n], {n, 0, 40}] (* Indranil Ghosh, Feb 24 2017 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 12, 52, 148, 328}, 50] (* Harvey P. Dale, Nov 24 2024 *)

Prog

(PARI) Vec((1+8*x+10*x^2+8*x^3+x^4)/(1-x)^4 + O(x^40)) \\ Colin Barker, Dec 09 2016
(PARI) vector(40, n, n--; if(n==0, 1, 2*n*(11+7*n^2)/3)) \\ G. C. Greubel, Jul 31 2019
(Python)
def A054410(n):
    if n == 0: return 1
    return 2*(n*(11 + 7*n**2))/3 # Indranil Ghosh, Feb 24 2017
(Magma) [1] cat [2*n*(11+7*n^2)/3: n in [1..40]]; // G. C. Greubel, Jul 31 2019
(Sage) [1]+[2*n*(11+7*n^2)/3 for n in (1..40)] # G. C. Greubel, Jul 31 2019
(GAP) Concatenation([1], List([1..40], n-> 2*n*(11+7*n^2)/3)); # G. C. Greubel, Jul 31 2019

Crossrefs

Cf. A008574, A054275, A054389, A054764.

Sequence in context: A317466 A045219 A325379 * A185355 A339029 A297757

Adjacent sequences: A054407 A054408 A054409 * A054411 A054412 A054413

Keyword

nonn,easy

Author

N. J. A. Sloane, May 09 2000

Status

approved