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Search: id:A101990
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| A101990 |
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a(n) = 3*a(n-1) - 3*a(n-2) + 9*a(n-3), a(1) = a(2) = 1, a(3) = 9. |
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1
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| 1, 1, 9, 33, 81, 225, 729, 2241, 6561, 19521, 59049, 177633, 531441, 1592865, 4782969, 14353281, 43046721, 129127041, 387420489, 1162300833, 3486784401, 10460235105, 31381059609, 94143533121, 282429536481, 847287546561
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OFFSET
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1,3
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COMMENT
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Alternate terms are powers of nine (A001019): a(2b+1) = 9^b; b = 0,1,2...
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FORMULA
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a(n) = left term in M^n * [1 0 0], where M = the 3X3 matrix [1 0 2 / 2 1 0 / 0 2 1].
a(n)=(24*I)*[72-(72*I)*sqrt(3)]^(-1)*sqrt(3)*[I*sqrt(3)]^n-(72*I)*3^n*[72-(72*I) *sqrt(3)]^(-1)*sqrt(3)-72*[-I*sqrt(3)]^n*[72-(72*I)*sqrt(3)]^(-1)-(24*I)*[-I *sqrt(3)]^n*[72-(72*I)*sqrt(3)]^(-1)*sqrt(3)+72*3^n*[72-(72*I)*sqrt(3)]^(-1)+72 *[72-(72*I)*sqrt(3)]^(-1)*[I*sqrt(3)]^n, with n>=0 and I=sqrt(-1) [From Paolo P. Lava (ppl(AT)spl.at), Aug 01 2008]
O.g.f.: x(1-2x+9x^2)/((1-3x)(1+3x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 22 2008]
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EXAMPLE
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a(5) = 81 since M^5 * [ 1 0 0] = [81 90 72].
a(5) = 81 = 99 - 27 + 9 = 3*33 - 3*9 + 9*1 = 3*a(4) - 3*a(3) + 9*a(2).
a(7) = 729 = 9^3. (let b = 3, then n = 2b+1 = 7; and a(2b+1) = 9^b.
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MATHEMATICA
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a[n_] := a[n] = 3a[n - 1] - 3a[n - 2] + 9a[n - 3]; a[1] = a[2] = 1; a[3] = 9; Table[ a[n], {n, 26}] (* Or *)
a[n_] := (MatrixPower[{{1, 0, 2}, {2, 1, 0}, {0, 2, 1}}, n].{{1}, {0}, {0}})[[1, 1]]; Table[ a[n], {n, 26}] (from Robert G. Wilson v Dec 23 2004)
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CROSSREFS
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Sequence in context: A092562 A103602 A081585 this_sequence A147170 A146823 A147027
Adjacent sequences: A101987 A101988 A101989 this_sequence A101991 A101992 A101993
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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), Dec 23 2004
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EXTENSIONS
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Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 23 2004
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