A056537 Mapping from the column-by-column reading to the half-antidiagonal reading of the triangular tables. Inverse of A056536.

1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 11, 15, 20, 25, 10, 14, 19, 24, 30, 36, 13, 18, 23, 29, 35, 42, 49, 17, 22, 28, 34, 41, 48, 56, 64, 21, 27, 33, 40, 47, 55, 63, 72, 81, 26, 32, 39, 46, 54, 62, 71, 80, 90, 100, 31, 38, 45, 53, 61, 70, 79, 89, 99, 110, 121, 37, 44, 52, 60, 69

Offset: 1

Author

Antti Karttunen, Jun 20 2000

Keywords

• nonn
• tabl

Comments

Moves triangular numbers (A000217) to squares (A000290), i.e., A056537(A000217(i)) = A000290(i) for i >= 1.

As a square array, this is the dispersion of the complement of the squares; see A082152. - Clark Kimberling, Apr 05 2003

Examples

			As a square array, a northwest corner:
1 ... 2 ... 3 ... 5 ... 7 ... 10
4 ... 6 ... 8 ... 11 .. 14 .. 18
9 ... 12 .. 15 .. 19 .. 23 .. 28
16 .. 20 .. 24 .. 29 .. 34 .. 40
25 .. 30 .. 35 .. 41 .. 47 .. 54
36 .. 42 .. 48 .. 55 .. 62 .. 70
49 .. 56 .. 63 .. 71 .. 79 .. 88
64 .. 72 .. 80 .. 89 .. 98 .. 108
- _Clark Kimberling_, Aug 08 2013
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..4950
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A185787 (dispersion of complement of triangular numbers).

Cf. A082152 (dispersion of complement of pentagonal numbers).

Programs

Maple

# using Maple procedure nthmember given in A054426:
[seq(nthmember(j, A056536), j=1..105)];

Mathematica

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := n+Floor[1/2+Sqrt[n]] (* complement of column 1 *); mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]; rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]  (* A056537 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A056537 sequence *)
(* Clark Kimberling, Jun 06 2011 *)

Formula

Triangle T(n, k), 1<=k<=n, read by rows, defined by: T(n, k) = 0 for nA002620(n-k+1) + k*n + k - n if n>=k. T(n, n) = n^2; T(n, 1) = 1 + A002620(n) = A033638(n). - Philippe Deléham, Feb 16 2004

Square: t(n,k) = (n-1)(n+k) + k^2/4 + (1/8)(7+(-1)^k). - Clark Kimberling, Aug 08 2013