%I #33 Jun 08 2024 08:53:43

%S 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,

%T 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,

%U 18,46,29,81,91,29,81,46,2,1,12,3,43,75,6,112,12,54,1,57,3

%N 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).

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

%H Paolo Xausa, <a href="/A373169/b373169.txt">Table of n, a(n) for n = 2..11326</a> (first 150 antidiagonals, flattened).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>.

%e The array begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10 ...

%e ----------------------------------------------------

%e 2 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... = A177274

%e 3 | 1, 3, 6, 1, 6, 12, 19, 27, 36, 46, ... = A243658 (from n = 1)

%e 4 | 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, ... = A370812

%e 5 | 1, 5, 12, 22, 35, 51, 7, 29, 54, 82, ... = A373171

%e 6 | 1, 6, 15, 28, 45, 66, 91, 12, 45, 82, ... = A373172

%e 7 | 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, ...

%e 8 | 1, 8, 21, 4, 29, 6, 43, 86, 135, 19, ...

%e 9 | 1, 9, 24, 46, 75, 111, 154, 24, 81, 145, ...

%e 10 | 1, 1, 18, 43, 76, 117, 166, 223, 288, 361, ...

%e ... | \______ A373170 (main diagonal)

%e A004719 (from n = 2)

%t noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];

%t A373169[n_, k_] := A373169[n, k] = If[k == 1, 1, noz[A373169[n, k-1] + (k-1)*(n-2) + 1]];

%t Table[A373169[n - k + 1, k], {n, 2, 15}, {k, n - 1}]

%o (PARI) noz(n) = fromdigits(select(sign, digits(n)));

%o T(n,k) = if (k==1, 1, noz(T(n,k-1) + (k-1)*(n-2) + 1));

%o matrix(7,7,n,k,T(n+1,k)) \\ _Michel Marcus_, May 30 2024

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

%Y Cf. A373170 (main diagonal).

%Y Cf. A004719, A057145.

%K nonn,tabl,base,easy

%O 2,3

%A _Paolo Xausa_, May 27 2024