A373169 Square array read by ascending antidiagonals: T(n,k) = noz(T(n,k-1) + (k-1)*(n-2) + 1), with T(n,1) = 1, n >= 2, k >= 1, where noz(n) = A004719(n).

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 9, 1, 5, 1, 6, 12, 16, 6, 6, 1, 7, 15, 22, 25, 12, 7, 1, 8, 18, 28, 35, 36, 19, 8, 1, 9, 21, 34, 45, 51, 49, 27, 9, 1, 1, 24, 4, 55, 66, 7, 64, 36, 1, 1, 11, 18, 46, 29, 81, 91, 29, 81, 46, 2, 1, 12, 3, 43, 75, 6, 112, 12, 54, 1, 57, 3

Offset: 2

Author

Paolo Xausa, May 27 2024

Keywords

• nonn
• tabl
• base
• easy

Comments

Row n is the zeroless analog of the positive n-gonal numbers.

Examples

			The array begins:
  n\k|  1  2   3   4   5    6    7    8    9   10  ...
  ----------------------------------------------------
   2 |  1, 2,  3,  4,  5,   6,   7,   8,   9,   1, ... = A177274
   3 |  1, 3,  6,  1,  6,  12,  19,  27,  36,  46, ... = A243658 (from n = 1)
   4 |  1, 4,  9, 16, 25,  36,  49,  64,  81,   1, ... = A370812
   5 |  1, 5, 12, 22, 35,  51,   7,  29,  54,  82, ... = A373171
   6 |  1, 6, 15, 28, 45,  66,  91,  12,  45,  82, ... = A373172
   7 |  1, 7, 18, 34, 55,  81, 112, 148, 189, 235, ...
   8 |  1, 8, 21,  4, 29,   6,  43,  86, 135,  19, ...
   9 |  1, 9, 24, 46, 75, 111, 154,  24,  81, 145, ...
  10 |  1, 1, 18, 43, 76, 117, 166, 223, 288, 361, ...
  ...      |                                     \______ A373170 (main diagonal)
        A004719 (from n = 2)
		

Links

• Paolo Xausa, Table of n, a(n) for n = 2..11326 (first 150 antidiagonals, flattened).
• Wikipedia, Polygonal number.

Crossrefs

Cf. rows 2..6: A177274, A243658, A370812, A373171, A373172.

Cf. A373170 (main diagonal).

Cf. A004719, A057145.

Programs

Mathematica

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A373169[n_, k_] := A373169[n, k] = If[k == 1, 1, noz[A373169[n, k-1] + (k-1)*(n-2) + 1]];
Table[A373169[n - k + 1, k], {n, 2, 15}, {k, n - 1}]

PARI

noz(n) = fromdigits(select(sign, digits(n)));
T(n,k) = if (k==1, 1, noz(T(n,k-1) + (k-1)*(n-2) + 1));
matrix(7,7,n,k,T(n+1,k)) \\ Michel Marcus, May 30 2024