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A095264
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a(n) = 2^(n+2) - 3*n - 4.
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2
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1, 6, 19, 48, 109, 234, 487, 996, 2017, 4062, 8155, 16344, 32725, 65490, 131023, 262092, 524233, 1048518, 2097091, 4194240, 8388541, 16777146, 33554359, 67108788, 134217649, 268435374, 536870827, 1073741736, 2147483557
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OFFSET
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1,2
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COMMENTS
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A sequence derived from a 3rd-order matrix generator.
The number of positive 3-strand braids of degree at most n. - R. J. Mathar, May 04 2006
Define a triangle T by T(n,n) = n*(n+1)/2, T(n,1) = n*(n-1) + 1, and T(r,c) = T(r-1,c-1) + T(r-1,c). Its rows are 1; 3,3; 7,6,6; 13,13,12,10; 21,26,25,22,15; etc. The sum of the terms in the n-th row is a(n). - J. M. Bergot, May 03 2013
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LINKS
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FORMULA
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Let M = [1 0 0 / 1 1 0 / 1 3 2], then M^n * [1 0 0] = [1 n a(n)]. The characteristic polynomial of M is x^3 - 4x^2 + 5x - 2.
a(n+3) = 4*a(n+2) - 5*a(n+1) + 2*a(n).
G.f.: -x*(1+2*x)/((2*x-1)*(x-1)^2). - R. J. Mathar, Nov 18 2007
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EXAMPLE
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a(5) = 109 = 2^7 - 3*5 - 4.
a(5) = 109 since M^5 * [1 0 0] = [1 5 109].
a(7) = 487 = 4*234 - 5*109 + 2*48.
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MATHEMATICA
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a[n_] := (MatrixPower[{{1, 0, 0}, {1, 1, 0}, {1, 3, 2}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 30}] (* Robert G. Wilson v, Jun 05 2004 *)
Table[2^(n+2)-3n-4, {n, 40}] (* or *) LinearRecurrence[{4, -5, 2}, {1, 6, 19}, 40] (* Harvey P. Dale, Sep 24 2021 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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