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<a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (-2, -6, -11, -19, -27, -34, -38, -36, -30, -21, -12, -5, -1).

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Expansion of 1/((1 +x +x^2)^2 *(1 +x^2 +x^3)^3).

G. C. Greubel, <a href="/A167177/b167177.txt">Table of n, a(n) for n = 0..1000</a>

CoefficientList[Series[1/((1 + x + x^2)^2*(1 + x^2 + x^3)^3), {x, 0, 100}], x] (* G. C. Greubel, Jun 04 2016 *)

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_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Oct 29 2009

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Inverse toral expansion Expansion of:p 1/(x)=(1+x+x^2)^2*(1+x^2+x^3)^3).

1, -2, -2, 5, 5, -7, -13, 2, 29, 19, -47, -68, 43, 151, 31, -246, -237, 267, 611, -34, -1078, -707, 1327, 2149, -701, -4118, -1760, 5611, 6904, -4361, -14463, -3123, 21453, 20320, -20510, -47501, -426, 76389, 54711, -84119, -147200, 30748, 256922, 132152, -315913, -432648, 196632

The resoning in the Galois polynomial model:

1/alpha is about 137=Prime[33]

I realized that

33=2^5+2+1

was of the form

x^5+x+1

which is the Galois field polynomial for

GF[2^5] which has 32 elements.

Suppose that we take the standard model of physics

symmetry breaking as a GF[2^5] breaking;

GF[2^5]=>GF[2^2]^2*GF[2^3]^3

32->2*4+3*8

Since these are Abelian fields, this is really a different

way of looking at the standard model that is still 5d

and hyperbolic in time and space.

In polynomial terms that is one variable x0 going to five;

x0^5+x0+1-> {t^2+t+1,tau^2+tau+1,x^3+x+1,y^3+y+1,z^3+z+1}

I,then, substitute back x for all the variables

and make a product polynomial.

1/(1 + 2 x + 6 x^2 + 11 x^3 + 19 x^4 + 27 x^5 + 34 x^6 + 38 x^7 + 36 x^8 + 30 x^9 + 21 x^10 + 12 x^11 + 5 x^12 + x^13)

a(n) = -2*a(n-1) -6*a(n-2) -11*a(n-3) -19*a(n-4) -27*a(n-5) -34*a(n-6) -38*a(n-7) -36*a(n-8) -30*a(n-9) -21*a(n-10) -12*a(n-11) -5*a(n-12) -a(n-13).

Cf. A099254.

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