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Search: id:A100190
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| A100190 |
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The (4,1)-entry in the 4 X 4 matrix M^n, where M=[1,0,0,0 / 3,3,0,0 / 3,6,3,0 / 1,3,3,1]. |
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+0
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| 1, 20, 147, 760, 3317, 13164, 49255, 177200, 620073, 2125828, 7174523, 23914920, 78919069, 258280412, 839411151, 2711943520, 8716961105, 27894275316, 88913002339, 282429536600, 894360198981, 2824295364940, 8896530399287
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Suggested by "Mathematical Vistas", p. 178, Fig 14: The Pascal Tetrahedron. The first few levels are (Level 0): 1; (Level 1): 1; 1, 1; (Level 2): 1; 2, 2; 1, 2, 1; (Level 3): 1; 3, 3; 3, 6, 3; 1, 3, 3, 1.
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REFERENCES
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Peter Hilton, Derek Holton and Jean Pederson; "Mathematical Vistas, From a Room With Many Windows"; Springer, 2000; p. 178.
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FORMULA
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a(n)=8a(n-1)-22a(n-2)+24a(n-3)-9a(n-4) for n>=5 (derived from the minimal polynomial of the matrix M).
a(n) = (1/2) [(9n + 18) * 3^n + 11n + 18 ]. - Ralf Stephan, May 15 2007
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EXAMPLE
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a(6)=13164 because M^6 = [1,0,0,0 / 1092,729,0,0 / 10938,8748,729,0 / 13164,10938,1092,1]. Alternatively, a(6)=8a(5)-22a(4)+24a(3)-9a(2)=26536-16720+3528-180=13164.
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MAPLE
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with(linalg): M[1]:=matrix(4, 4, [1, 0, 0, 0, 3, 3, 0, 0, 3, 6, 3, 0, 1, 3, 3, 1]): for n from 2 to 27 do M[n]:=multiply(M[1], M[n-1]) od: seq(M[n][4, 1], n=1..27);
a[1]:=1:a[2]:=20:a[3]:=147:a[4]:=760: for n from 5 to 27 do a[n]:=8*a[n-1]-22*a[n-2]+24*a[n-3]-9*a[n-4] od: seq(a[n], n=1..27);
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CROSSREFS
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Sequence in context: A054389 A071816 A094171 this_sequence A022680 A108647 A164605
Adjacent sequences: A100187 A100188 A100189 this_sequence A100191 A100192 A100193
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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), Nov 07 2004
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 04 2006
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