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