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A144434
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A triangle of rounded combinations made from golden mean like generalized factorials: Phi=-(1 - Sqrt[5])/2; b(n)=b(n-1)+phi; a(n)=b(n)*a(n-1); t(n,m)=Round((1+Phi)*a(n)/(a(m)*a(n-m))).
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1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 4, 7, 9, 7, 4, 1, 1, 4, 9, 14, 14, 9, 4, 1, 1, 4, 12, 21, 25, 21, 12, 4, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 6, 18, 41, 66, 77, 66, 41, 18, 6, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Row sums are:
{1, 2, 4, 6, 9, 18, 33, 56, 101, 186, 341}.
The actual results are plain rational numbers:
{1},
{1, 1},
{1, 3/2, 1},
{1, 2, 2, 1},
{1, 5/2, 10/3, 5/2, 1},
{1, 3, 5, 5, 3, 1},
{1, 7/2, 7, 35/4, 7, 7/2, 1},
{1, 4, 28/3, 14, 14, 28/3, 4, 1},
{1, 9/2, 12, 21, 126/5, 21, 12, 9/2, 1},
{1, 5, 15, 30, 42, 42, 30, 15, 5, 1},
{1, 11/2, 55/3, 165/4, 66, 77, 66, 165/4, 55/3, 11/2, 1}
such that Round() seems to give the major sense of them.
The rationalization/ reasoning behind these is that in natural terms
the ratios like the golden mean behave very much like integers,
so that a naturally smaller set of results is possible
in terms of scaled probabilities.
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FORMULA
| Phi=-(1 - Sqrt[5])/2; b(n)=b(n-1)+phi; a(n)=b(n)*a(n-1); t(n,m)=Round((1+Phi)*a(n)/(a(m)*a(n-m))).
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EXAMPLE
| {1},
{1, 1},
{1, 2, 1},
{1, 2, 2, 1},
{1, 2, 3, 2, 1},
{1, 3, 5, 5, 3, 1},
{1, 4, 7, 9, 7, 4, 1},
{1, 4, 9, 14, 14, 9, 4, 1},
{1, 4, 12, 21, 25, 21, 12, 4, 1},
{1, 5, 15, 30, 42, 42, 30, 15, 5, 1},
{1, 6, 18, 41, 66, 77, 66, 41, 18, 6, 1}
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MATHEMATICA
| Clear[a, n, b, c] (* generalized Beta integer factorial of the golden mean*) b[0] = -(1 - Sqrt[5])/2; b[n_] := b[n] = b[n - 1] - (1 - Sqrt[5])/2; a[0] = -(1 - Sqrt[5])/2; a[n_] := a[n] = b[n]*a[n - 1]; (* combinations based on the gen-beta -factorials*) c = Table[Table[FullSimplify[ExpandAll[((-1 + Sqrt[5])/2)*a[n]/(a[m]*a[n - m])]], {m, 0, n}], {n, 0, 10}] Round[c] Flatten[%]
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CROSSREFS
| Sequence in context: A087062 A204026 A110537 * A159936 A144464 A138015
Adjacent sequences: A144431 A144432 A144433 * A144435 A144436 A144437
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 04 2008
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