A244811 -id:A244811 - cp's OEIS frontend

A244818 The hexagonal spiral of Champernowne, read along the 120-degree ray.

1, 1, 6, 7, 1, 8, 1, 4, 1, 3, 6, 6, 6, 4, 3, 9, 3, 7, 2, 1, 2, 3, 7, 1, 7, 9, 3, 2, 8, 5, 0, 3, 5, 0, 8, 4, 8, 4, 6, 4, 7, 9, 6, 5, 2, 3, 7, 6, 3, 6, 8, 8, 0, 9, 1, 9, 3, 9, 1, 0, 0, 1, 3, 1, 2, 2, 1, 8, 1, 3, 4, 1, 6, 1, 5, 5, 1, 5, 1, 7, 7, 1, 7, 1, 9, 9, 2, 0, 2, 1, 1, 2, 6, 2, 3, 4, 2, 3, 2, 5, 6, 2, 3, 2, 8

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A244807 example section for its diagram.

Cf. A007376, A033577, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817.

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 12n^2 - 17n + 6 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2-17n+6)th almost natural number (A033307), also see formula section of A056105.

A255812 Rectangular array: row n gives the denominators in the positive convolutory n-th root of (1,1,1,...).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 8, 3, 1, 1, 16, 9, 4, 1, 1, 128, 81, 32, 5, 1, 1, 256, 243, 128, 25, 6, 1, 1, 1024, 729, 2048, 125, 72, 7, 1, 1, 2048, 6561, 8192, 625, 1296, 49, 8, 1, 1, 32768, 19683, 65536, 15625, 31104, 343, 128, 9, 1, 1, 65536, 59049, 262144, 78125

Offset: 1

Clark Kimberling, Mar 11 2015

(See Comments at A255811.)

			First, regarding the numbers numerator/denominator, we have
row 1:  1,1,1,1,1,1,1,1,1,1,1,1,..., the 0th self-convolution of (1,1,1,...);
row 2:  1,1/2,3/8,5/16,35/128,63/256, ..., convolutory sqrt of (1,1,1,...);
row 3:  1,1/3,2/9,14/81,35/243,91/729,..., convolutory 3rd root
row 4:  1,1/4,5/32,15/128,195/2048,663/8192,..., convolutory 4th root.
Taking only denominators:
row 1:  1,1,1,1,1,1,1,...
row 2:  1,2,8,16,128,...
row 3:  1,3,9,81,243,729,...
row 4:  1,4,32,128,2048,8192,...

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A244811, A000012.

z = 15; t[n_] := CoefficientList[Normal[Series[(1 - t)^(-1/n), {t, 0, z}]], t];
u = Table[Numerator[t[n]], {n, 1, z}]
TableForm[Table[u[[n, k]], {n, 1, z}, {k, 1, z}]]     (*A255811 array*)
Table[u[[n - k + 1, k]], {n, z}, {k, n, 1, -1}] // Flatten (*A255811 sequence*)
v = Table[Denominator[t[n]], {n, 1, z}]
TableForm[Table[v[[n, k]], {n, 1, z}, {k, 1, z}]]     (*A255812 array*)
Table[v[[n - k + 1, k]], {n, z}, {k, n, 1, -1}] // Flatten  (*A255812 sequence*)

G.f. of s: (1 - t)^(-1/n).

cp's OEIS Frontend

A244818 The hexagonal spiral of Champernowne, read along the 120-degree ray.

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