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A106803
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Expansion of x(1-x)/(x^3-2x-x^2+1).
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4
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0, 1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| A Bombieri dual sequence.
a(n) appears in the formula for the nonnegative powers of sigma, the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with rho = 2*cos(Pi/7), the ratio of the smaller heptagon diagonal to the side length, as follows. sigma^n = a(n-1)*1 + B(n)*rho + a(n)*sigma, n>=0, with B(n)=A006054(n). Put a(-1):= 1. See the Steinbach reference, and a comment under A052547.
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REFERENCES
| P. Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, 70,1 (1997) 22-31.
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FORMULA
| a(n)=A077998(n-1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 07 2008
a(n)=A187070(2*n), a(n)=A187068(2*n+2), from L. Edson Jeffery, Mar 10, 2011.
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MATHEMATICA
| M = {{0, 0, 1}, {1, 2, 0}, {1, 1, 0}}; Det[M - x*IdentityMatrix[3]] v[0] = {0, 1, 1} v[n] = M.v[n - 1] a = Table[v[n][[1]], {n, 0, 50}]
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CROSSREFS
| Sequence in context: A091601 A063119 * A199853 A006356 A077998 A090165
Adjacent sequences: A106800 A106801 A106802 * A106804 A106805 A106806
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 17 2005
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 08 2008
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