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Search: id:A101946
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| 1, 4, 13, 34, 79, 172, 361, 742, 1507, 3040, 6109, 12250, 24535, 49108, 98257, 196558, 393163, 786376, 1572805, 3145666, 6291391, 12582844, 25165753, 50331574, 100663219, 201326512, 402653101, 805306282, 1610612647, 3221225380
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Sequence generated from a 3 X 3 matrix, companion to A101945.
Characteristic polynomial of M = x^3 - 4x^2 + 5x - 2.
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FORMULA
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a(0)=1, a(1)=4, a(2)=13 and for n>2, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
a(n) = right term in M^n * [1 1 1], where M = the 3X3 matrix [1 0 0 / 2 2 0 / 1 2 1]. M^n * [1 1 1] = [1 A033484(n) a(n)].
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EXAMPLE
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a(4) = 79 = 4*34 - 5*13 + 2*4 = 4*a(3) - 5*a(2) + 2*a(1).
a(4) = right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 46 a(4)], where 46 = A033484(4).
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MATHEMATICA
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a[0] = 1; a[1] = 4; a[2] = 13; a[n_] := a[n] = 4a[n - 1] - 5a[n - 2] + 2a[n - 3]; Table[ a[n], {n, 0, 30}] (* Or *)
a[n_] := (MatrixPower[{{1, 0, 0}, {2, 2, 0}, {1, 2, 1}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 0, 30}] (from Robert G. Wilson v Jan 12 2005)
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CROSSREFS
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Cf. A101945, A033484.
Sequence in context: A036894 A135859 A161531 this_sequence A029860 A127981 A089453
Adjacent sequences: A101943 A101944 A101945 this_sequence A101947 A101948 A101949
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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 22 2004
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 12 2005
New definition from Ralf Stephan, May 17 2007
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