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Mapping from the column-by-column reading to the half-antidiagonal reading of the triangular tables. Inverse of A056536.
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%I #32 Aug 30 2015 07:22:34

%S 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,

%T 42,49,17,22,28,34,41,48,56,64,21,27,33,40,47,55,63,72,81,26,32,39,46,

%U 54,62,71,80,90,100,31,38,45,53,61,70,79,89,99,110,121,37,44,52,60,69

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

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

%C As a square array, this is the dispersion of the complement of the squares; see A082152. - _Clark Kimberling_, Apr 05 2003

%H Ivan Neretin, <a href="/A056537/b056537.txt">Table of n, a(n) for n = 1..4950</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F Triangle T(n, k), 1<=k<=n, read by rows, defined by: T(n, k) = 0 for n<k and T(n, k) = A002620(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

%F Square: t(n,k) = (n-1)(n+k) + k^2/4 + (1/8)(7+(-1)^k). - _Clark Kimberling_, Aug 08 2013

%e As a square array, a northwest corner:

%e 1 ... 2 ... 3 ... 5 ... 7 ... 10

%e 4 ... 6 ... 8 ... 11 .. 14 .. 18

%e 9 ... 12 .. 15 .. 19 .. 23 .. 28

%e 16 .. 20 .. 24 .. 29 .. 34 .. 40

%e 25 .. 30 .. 35 .. 41 .. 47 .. 54

%e 36 .. 42 .. 48 .. 55 .. 62 .. 70

%e 49 .. 56 .. 63 .. 71 .. 79 .. 88

%e 64 .. 72 .. 80 .. 89 .. 98 .. 108

%e - _Clark Kimberling_, Aug 08 2013

%p # using Maple procedure nthmember given in A054426:

%p [seq(nthmember(j, A056536), j=1..105)];

%t (* Program generates the dispersion array T of the increasing sequence f[n] *)

%t 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 *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A056537 sequence *)

%t (* _Clark Kimberling_, Jun 06 2011 *)

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

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

%K nonn,tabl

%O 1,2

%A _Antti Karttunen_, Jun 20 2000