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A130879
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An anti-diagonal triangular sequence based on sums of fractal self-similar level count totals of the sort:Sum[k^(2^n), {n, 0, m}].
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2, 3, 6, 4, 12, 22, 5, 20, 93, 278, 6, 30, 276, 6654, 65814, 7, 42, 655, 65812, 43053375, 4295033110, 8, 56, 1338, 391280, 4295033108, 1853020231905216, 18446744078004584726, 9, 72, 2457, 1680954, 152588281905
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| I used the "Reverse" in this anti-diagonal transform so that the first column is the low numbers of the fractal states themselves.
The row sum is given by:
Table[Apply[Plus, Table[a[[n, l - n]], {n, 1, l - 1}]], {l, 1, Dimensions[a][[1]] + 1}]
0,2,9,38,396,72780,4338153001,18448597102531915732...
This sort of statistics is basic to things like Zipf word frequency and other fractal dimension determinations.
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FORMULA
| a(m,k)=Sum[k^(2^n), {n, 0, m}] T(n,m)=anti-diagonal_transform[a(m,k)] ( see the Mathematica code below)
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EXAMPLE
| {2},
{3, 6},
{4, 12, 22},
{5, 20, 93, 278},
{6, 30, 276, 6654, 65814},
{7, 42, 655, 65812, 43053375, 4295033110},
{8, 56, 1338, 391280, 4295033108, 1853020231905216, 18446744078004584726}
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MATHEMATICA
| f[m_, k_] := Sum[k^(2^n), {n, 0, m}]; a = Table[f[m, k], {k, 2, 12}, {m, 0, 10}]; c = Table[Reverse[Table[a[[n, l - n]], {n, 1, l - 1}]], {l, 1, Dimensions[a][[1]] + 1}]; Flatten[c]
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CROSSREFS
| Sequence in context: A046202 A122866 A097275 * A119741 A126063 A137524
Adjacent sequences: A130876 A130877 A130878 * A130880 A130881 A130882
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KEYWORD
| nonn,tabl
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 21 2007
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