A056537 -id:A056537 - cp's OEIS frontend

A056536 Mapping from half-antidiagonal reading of the triangle (as used in A028297) to the column-by-column reading of the triangular tables.

1, 2, 4, 3, 7, 5, 11, 8, 6, 16, 12, 9, 22, 17, 13, 10, 29, 23, 18, 14, 37, 30, 24, 19, 15, 46, 38, 31, 25, 20, 56, 47, 39, 32, 26, 21, 67, 57, 48, 40, 33, 27, 79, 68, 58, 49, 41, 34, 28, 92, 80, 69, 59, 50, 42, 35, 106, 93, 81, 70, 60, 51, 43, 36, 121, 107, 94, 82, 71, 61, 52

Offset: 1

Antti Karttunen, Jun 20 2000

nonn

Moves squares (A000290) to triangular numbers (A000217). See 1st formula.

This sequence may be regarded as a triangular array read by rows: 1; 2; 4, 3; 7, 5; 11, 8, 6; 16, 12, 9; 22, 17, 13, 10; .... with row sums: A079824 = [1, 2, 7, 12, 25, 37, 62, 84, ...]. - Philippe Deléham, Feb 16 2004

			As a triangular array read by rows:
    1;
    2;
    4,  3;
    7,  5;
   11,  8, 6;
   16, 12, 9;
   22, 17, 13, 10;
   29, 23, 18, 14;
   37, 30, 24, 19, 15;
   46, 38, 31, 25, 20;
   56, 47, 39, 32, 26, 21;
   67, 57, 48, 40, 33, 27;
   79, 68, 58, 49, 41, 34, 28;
   92, 80, 69, 59, 50, 42, 35;
  106, 93, 81, 70, 60, 51, 43, 36;
  ...

Index entries for sequences that are permutations of the natural numbers

Cf. A000217, A000290, A056537 (inverse), A079824, A091018.

triang_perm := proc(upto_d) local a,i,j; a := []; for i from 1 to upto_d do for j from 1 to floor((i+1)/2) do a := [op(a),binomial((i-j)+1,2)+j]; od; od; RETURN(a); end;

a(A000290(i)) = A000217(i) for all i >= 1.

a(n) = A091018(n-1) + 1.

A082156 Dispersion of the complement of row 1 of A056536.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 11, 7, 36, 30, 24, 19, 14, 10, 49, 42, 35, 29, 23, 18, 13, 64, 56, 48, 41, 34, 28, 22, 17, 81, 72, 63, 55, 47, 40, 33, 27, 21, 100, 90, 80, 71, 62, 54, 46, 39, 32, 26, 121, 110, 99, 89, 79, 70, 61, 53, 45, 38, 31, 144, 132, 120, 109

Offset: 1

Clark Kimberling, Apr 05 2003

Rectangular array read by antidiagonals; a permutation of the natural numbers. (Row 1) = squares = A000290(n) = n^2. (Dispersion of complement of column 1 of A082156) = (Transpose of A056537). The associated fractal sequence is A122196.

			Northwest corner:
1 4 9 16 25
2 6 12 20 30
3 8 15 24 35
5 11 19 29 41
7 14 23 34 47

C. Kimberling, Interspersions and Dispersions.
C. Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A056536, A000290.

The transpose of A056536.

A361975 (2,1)-block array, B(2,1), of the natural number array (A000027), read by descending antidiagonals.

Original entry on oeis.org

4, 7, 16, 12, 23, 36, 19, 32, 47, 64, 28, 43, 60, 79, 100, 39, 56, 75, 96, 119, 144, 52, 71, 92, 115, 140, 167, 196, 67, 88, 111, 136, 163, 192, 223, 256, 84, 107, 132, 159, 188, 219, 252, 287, 324, 103, 128, 155, 184, 215, 248, 283, 320, 359, 400, 124, 151

Offset: 1

Clark Kimberling, Apr 01 2023

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(2,1) is a row-splitting array. The rows and columns of B(2,1) are linearly recurrent with signature (3,-3,1). It appears that the order array (as defined in A333029) of B(2,1) is given by A056537 (the dispersion of the nonsquares).

			Corner of B(2,1):
    4    7   12   19   28   39   52
   16   23   32   43   56   71   88
   36   47   60   75   92  111  132
   64   79   96  115  136  159  184
  100  119  140  163  188  215  244
  144  167  192  219  238  279  312
(column 1 of A000027) = (1,3,6,10,15,21,...), so (column 1 of B(2,1)) = (4,16,64,...);
(column 2 of A000027) = (2,5,9,14,20,27,...), so (column 2 of B(2,1)) = (7,23,47,...).

Cf. A000027, A056537, A333029, A361974 (array B(1,2)), A361976 (array B(2,2)).

zz = 10; z = 13;
w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
t[h_, k_] := w[2 h - 1, k] + w[2 h, k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* this sequence as an array *)

B(2,1) = (b(i,j)), where b(i,j) = w(2i-1, j) + w(2i, j) for i >= 1, j >= 1, where (w(i,j)) is the natural number array (A000027).

b(i,j) = 4i - 1 + (2i + j - 2)^2.

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0

Offset: 0

Ross La Haye, Mar 02 2004

			Table begins
   0;
   1,  0;
   4,  2,  0;
   9,  6,  3,  0;
  16, 12,  8,  4,  0;
  25, 20, 15, 10,  5,  0;
  36, 30, 24, 18, 12,  6,  0;
  ...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.

Muniru A Asiru, Table of n, a(n) for n = 0..5150 (rows n = 0..100,flattened)
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.

Cf. A056536, A056537, A082156.

Flat(List([0..11],j->List([0..j],i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018

Maple
```
seq(seq((j-i)*j,i=0..j),j=0..14);
```

Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)

G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic, Mar 05 2004

More terms from Emeric Deutsch, Mar 15 2004

cp's OEIS Frontend

A056536 Mapping from half-antidiagonal reading of the triangle (as used in A028297) to the column-by-column reading of the triangular tables.

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A082156 Dispersion of the complement of row 1 of A056536.

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A361975 (2,1)-block array, B(2,1), of the natural number array (A000027), read by descending antidiagonals.

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A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

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