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Search: id:A122885
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| A122885 |
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Third of three sequences exhibiting an algebraic relationship. |
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+0
4
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| 1, 13, 61, 385, 2185, 12853, 74677, 435721, 2538625, 14798077, 86245741, 502684561, 2929845241, 17076419653, 99528607141, 580095354265, 3381043256305, 19706164707853, 114855943942237, 669429501042721
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OFFSET
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1,2
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COMMENT
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Let M^n * [0,0,1] = [p,q,r] = [A122883(n), A122884(n, a(n)]; then f(x),x=1,2,3: px^2 + qx + r = M^(n+1) * [0,0,1] = [A122883(n+1), A122884(n+1), a(n+1)]. Example: M^2 * [0,0,1] = [3, 7, 13]. Then f(x), x=1,2,3: 3x^2 + 7x + 13 = the three terms in M^3 * [0,0,1] = [23, 39, 61] = [A122883(3), A122884(3), a(3)].
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FORMULA
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a(1)= 1, a(2) = 13, a(n+1) = 4*a(n) + 11*a(n-1) - 2*a(n-2). a(n) = right term in M^n * [0,0,1] where M = the 3 X 3 matrix [1,1,1; 4,2,1; 9,3,1].
a(n)=-(45/68)*sqrt(2)*[3-2*sqrt(2)]^(n-1)+(33/34)*[3+2*sqrt(2)]^(n-1)+(45/68)*[3+2*sqrt(2)]^(n-1) *sqrt(2)+(33/34)*[3-2*sqrt(2)]^(n-1)-(16/17)*(-2)^(n-1), with n>=1 - Paolo P. Lava (ppl(AT)spl.at), Jul 09 2008
G.f.: -x*(-1-9*x+2*x^2)/((2*x+1)*(1-6*x+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 12 2009]
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EXAMPLE
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a(6) = 12853 = 4*a(5) + 11*a(4) - 2*a(3) = 4*2185 + 11*385 - 2*61.
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CROSSREFS
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Cf. A122883, A122884, A122886.
Sequence in context: A047673 A141725 A147185 this_sequence A135535 A158870 A145044
Adjacent sequences: A122882 A122883 A122884 this_sequence A122886 A122887 A122888
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KEYWORD
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nonn,uned
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AUTHOR
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Gary W. Adamson and Roger L. Bagula (qntmpkt(AT)yahoo.com), Sep 17 2006
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EXTENSIONS
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More terms from Paolo P. Lava (ppl(AT)spl.at), Jul 09 2008
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