A373172 -id:A373172

Search: a373172 -id:a373172

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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).

+10
6

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

2,3

COMMENTS

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

LINKS

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

Wikipedia, Polygonal number.

EXAMPLE

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)

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}]

PROG

(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

CROSSREFS

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

Cf. A373170 (main diagonal).

Cf. A004719, A057145.

KEYWORD

nonn,tabl,base,easy

AUTHOR

Paolo Xausa, May 27 2024

STATUS

approved

A370812

a(1) = 1; for n >= 2, a(n) = noz(a(n-1) + 2*n - 1), where noz(n) = A004719(n).

+10
4

1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 22, 45, 7, 34, 63, 94, 127, 162, 199, 238, 279, 322, 367, 414, 463, 514, 567, 622, 679, 738, 799, 862, 927, 994, 163, 234, 37, 112, 189, 268, 349, 432, 517, 64, 153, 244, 337, 432, 529, 628, 729, 832, 937, 144, 253, 364, 477

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Zeroless analog of the positive squares.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

MATHEMATICA

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

Block[{n = 1}, NestList[noz[++n*2 - 1 + #] &, 1, 100]]

CROSSREFS

Cf. A000290, A004719, A243657, A243658, A373171, A373172.

Row n = 4 of A373169.

KEYWORD

nonn,base,easy

AUTHOR

Paolo Xausa, May 24 2024

STATUS

approved

A373171

a(1) = 1; for n >= 2, a(n) = noz(a(n-1) + 3*n - 2), where noz(n) = A004719(n).

+10
4

1, 5, 12, 22, 35, 51, 7, 29, 54, 82, 113, 147, 184, 224, 267, 313, 362, 414, 469, 527, 588, 652, 719, 789, 862, 938, 117, 199, 284, 372, 463, 557, 654, 754, 857, 963, 172, 284, 399, 517, 638, 762, 889, 119, 252, 388, 527, 669, 814, 962, 1113, 1267, 1424, 1584

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Zeroless analog of the positive pentagonal numbers.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

MATHEMATICA

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

Block[{n = 1}, NestList[noz[++n*3 - 2 + #] &, 1, 100]]

PROG

(PARI) noz(n) = fromdigits(select(sign, digits(n))); \\ A004719

lista(nn) = my(va=vector(nn)); for (n=1, nn, va[n] = if (n==1, 1, noz(va[n-1] + 3*n - 2))); va; \\ Michel Marcus, Jun 03 2024

CROSSREFS

Row n = 5 of A373169.

Cf. A000326, A004719, A243658, A370812, A373172.

KEYWORD

nonn,base,easy

AUTHOR

Paolo Xausa, May 28 2024

STATUS

approved

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