A130879 An antidiagonal triangular sequence based on sums of fractal self-similar level count totals of the sort: Sum_{n=0..m} k^(2^n).

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

Offset

1,1

Comments

I used the "Reverse" in this antidiagonal 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.

Links

Table of n, a(n) for n=1..33.

Formula

a(m,k) = Sum_{n=0..m} k^(2^n); T(n,m) = antidiagonal_transform(a(m,k)) (see the Mathematica code below).

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}

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]

Crossrefs

Sequence in context: A328443 A122866 A097275 * A119741 A268216 A346928

Adjacent sequences: A130876 A130877 A130878 * A130880 A130881 A130882

Keyword

nonn,tabl

Author

Roger L. Bagula, Aug 21 2007

Status

approved