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Search: id:A094256
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| A094256 |
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Let M = the 4 X 4 matrix [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 10 -15 7]. Perform M^n * [1 0 0 0] = [p q r s]. Then a(n-3), a(n-2), a(n-1), a(n) = -p, -q, -r, -s respectively. |
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+0
3
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| 1, 7, 34, 143, 560, 2108, 7752, 28101, 100947, 360526, 1282735, 4552624, 16131656, 57099056, 201962057, 714012495, 2523515514, 8916942687, 31504028992, 111295205284, 393151913464, 1388758662221, 4905479957435, 17327203698086
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OFFSET
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1,2
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COMMENT
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a(n)/a(n-1) tends to 3.53208888624... = 4Cos^2 Pi/9 an eigenvalue of the matrix and a root of the polynomial x^4 - 6x^3 + 15x^2 -10x + 1 = 0 (having roots 4Cos^2 rPi/9, with r = 1,2,3,4.
Number of (s(0), s(1), ..., s(2n+4)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+4, s(0) = 1, s(2n+4) = 7. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 13 2004
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REFERENCES
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C. V. Durell and A. Robson, "Advanced Trigonometry", Dover 2003, p. 216.
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FORMULA
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a(n)=2/9*Sum(r, 1, 8, Sin(r*Pi/9)Sin(7*r*Pi/9)(2Cos(r*Pi/9))^(2n+4)) a(n)=7a(n-1)-15a(n-2)+10a(n-3)-a(n-4) G.f.: x/(1-7x+15x^2-10x^3+x^4) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 13 2004
a(n)=A005023(n-1), n>1. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 05 2008]
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EXAMPLE
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a(2), a(3), a(4), a(5) = 7, 34, 143, 560, since M^5 * [1 0 0 0] = [ -7 -34 -143 -560].
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MATHEMATICA
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Table[ (MatrixPower[{{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {-1, 10, -15, 7}}, n].{-1, 0, 0, 0})[[4]], {n, 24}] (from Robert G. Wilson v Apr 28 2004)
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CROSSREFS
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Sequence in context: A014915 A137747 A005023 this_sequence A094891 A052161 A080960
Adjacent sequences: A094253 A094254 A094255 this_sequence A094257 A094258 A094259
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 25 2004
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 28 2004
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