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Search: id:A095265
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| A095265 |
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A sequence generated from a 4th degree Pascal's Triangle polynomial. |
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+0
1
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| 1, 22, 103, 284, 605, 1106, 1827, 2808, 4089, 5710, 7711, 10132, 13013, 16394, 20315, 24816, 29937, 35718, 42199, 49420, 57421, 66242, 75923, 86504, 98025, 110526, 124047, 138628, 154309, 171130, 189131, 208352, 228833, 250614, 273735, 298236
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The characteristic polynomial of M = x^4 - 4x^3 + 6x^2 - 4x + 1. (the recursive multipliers are seen in the polynomial with changed signs: (4), (-6), (4), (-1).
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FORMULA
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a(n+4) = 4*a(n+3) - 6*a(n+2) + 4*a(n+1) - a(n), (multipliers which are present with changed signs in the characteristic polynomial, x^4 - 4x^3 + 6x^2 - 4x + 1. Given the 4 X 4 matrix derived from an A056939 triangle (fill in with zeros): M = [1 0 0 0 / 1 1 0 0 / 1 4 1 0 / 1 10 10 1], then M^n * [1 0 0 0] = [1 n A000384(n) a(n)] where A000384 is the hexagonal series 1, 6, 15, 28... 3. a(n) = (20/3)n^3 - 10n^2 + (13/3)n.
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EXAMPLE
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a(13) = 13013 = 4*a(12) - 6*a(11) + 4*a(10) - a(9) = 4*10132 - 6*7711 + 4*5710 - 4089.
a(6) = 1106 since M^6 * [1 0 0 0] = [ 1 6 66 1106].
a(6) = 1106 = f(n) = (20/3)(6)^3 -10*(6^2) +(13/3)*6 = 1440 - 360 + 26.
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MATHEMATICA
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a[n_] := (MatrixPower[{{1, 0, 0, 0}, {1, 1, 0, 0}, {1, 4, 1, 0}, {1, 10, 10, 1}}, n].{{1}, {0}, {0}, {0}})[[4, 1]]; Table[ a[n], {n, 36}] (from Robert G. Wilson v Jun 05 2004)
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CROSSREFS
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Cf. A056939, A000384.
Sequence in context: A060382 A044273 A044654 this_sequence A066450 A124950 A126409
Adjacent sequences: A095262 A095263 A095264 this_sequence A095266 A095267 A095268
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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), May 31 2004
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EXTENSIONS
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Edited and corrected by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 05 2004
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