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Search: id:A122884
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| A122884 |
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Second of three sequences exhibiting an algebraic relationship. |
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+0
4
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| 1, 7, 39, 231, 1339, 7819, 45543, 265503, 1547347, 9018835, 52565151, 306373095, 1785671371, 10407659227, 60660275799, 353554011951, 2060663763139, 12010428632419, 70001907900303, 408001019031543, 2378004205764667
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OFFSET
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1,2
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COMMENT
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Given M^n * [0,0,1] = [A122883(n), a(n), A122885(n)] = [p, q, r], then M^(n+1) * [0,0,1] = the three terms in f(x), 1,2,3: p*A122883(n)*x^2 + q*a(n)*x + r*A122885(n) = A122883(n+1) + a(n+1) + A122885(n+1). Example: M^3 * [0,0,1] = [3, 7, 13]. Then (f(x),x=1,2,3]: 3x^2 + 7x + 13 = the three terms in M^4 * [0,0,1] = [23, 39, 61].
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FORMULA
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a(n) = middle term in M^n * [0, 0, 1] where M = the 3 X 3 matrix [1,1,1; 4,2,1; 9,3,1] a(1) = 1, a(2) = 7; a(n+1) = 4*a(n) + 11*a(n-1) - 2*a(n-3).
a(n)=-(29/68)*sqrt(2)*[3-2*sqrt(2)]^(n-1)+(19/34)*[3+2*sqrt(2)]^(n-1)+(29/68)*[3+2*sqrt(2)]^(n-1) *sqrt(2)+(19/34)*[3-2*sqrt(2)]^(n-1)-(2/17)*(-2)^(n-1), with n>=1 - Paolo P. Lava (ppl(AT)spl.at), Jul 09 2008
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EXAMPLE
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a(3) = 39 since M^3 * [0,0,1] = [23, 39, 61] = [A122883(3), a(3), A122885(3)].
a(6) = 7819 = 4*a(5) + 11*a(4) - 2*a(3) = 4*1339 + 11*231 - 2*39.
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CROSSREFS
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Cf. A122883, A122885, A122886.
Sequence in context: A125786 A155589 A071082 this_sequence A139173 A138417 A035357
Adjacent sequences: A122881 A122882 A122883 this_sequence A122885 A122886 A122887
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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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