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A122023 Vacuum Virtual Particle 10 vertex graph as Feynman diagram seen as a 10 X 10 bonding graph vector Matrix Markov: characteristic Polynomial: x^4(-3 + x^2)(8 - 7 x^2 + x^4). 1
0, 1, 3, 11, 29, 109, 283, 795, 2061, 5053, 13099, 30091, 78013, 173453, 449723, 983163, 2549229, 5523677, 14322635, 30887915, 80092061, 172288429, 446745691, 959703003, 2488530381, 5341975549, 13851888235, 29723290699, 77073397885 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Calculation of relative energy of ten particle states as secular equation: aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[10]] == 0, x][[n]], {n, 1, 10}] Sum[2*aaa[[n]], {n, 1, 5}]=-10.5794 The excess energy is about 8*alpha if each vertex is taken as one unit.

LINKS

Table of n, a(n) for n=1..29.

FORMULA

M = {{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 1, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 1, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 1, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}} v[1] = Table[Fibonacci[n], {n, 0, 9}] v[n_] := v[n] = M.v[n - 1] a(n) = v[n][[1]]

Empirical G.f.: x^2*(1+3*x+x^2-x^3+28*x^4+80*x^5)/(1-10*x^2+29*x^4-24*x^6). [Colin Barker, Jan 04 2012]

MATHEMATICA

M = {{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 1, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 1, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 1, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}} v[1] = Table[Fibonacci[n], {n, 0, 9}] v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]

CROSSREFS

Sequence in context: A000251 A159229 A239713 * A259594 A293010 A236467

Adjacent sequences:  A122020 A122021 A122022 * A122024 A122025 A122026

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Sep 12 2006

STATUS

approved

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Last modified May 26 18:08 EDT 2020. Contains 334630 sequences. (Running on oeis4.)