A241016 Triangle read by rows: T(n, k) = sum of k-th row of n X n square filled with the numbers 1 through n^2 reading across rows left-to-right.

1, 3, 7, 6, 15, 24, 10, 26, 42, 58, 15, 40, 65, 90, 115, 21, 57, 93, 129, 165, 201, 28, 77, 126, 175, 224, 273, 322, 36, 100, 164, 228, 292, 356, 420, 484, 45, 126, 207, 288, 369, 450, 531, 612, 693, 55, 155, 255, 355, 455, 555, 655, 755, 855, 955, 66, 187, 308, 429, 550

Offset: 1

Author

Kival Ngaokrajang, Aug 08 2014

Keywords

• nonn
• easy
• tabl

Comments

See illustration in links.

The corresponding triangle with column sums is found in A251630. - Wolfdieter Lang, Dec 09 2014

Examples

			The triangle T(n, k) begins:
n\k  1   2   3   4   5   6   7   8   9  10 ...
1:   1
2:   3   7
3:   6  15  24
4:  10  26  42  58
5:  15  40  65  90 115
6:  21  57  93 129 165 201
7:  28  77 126 175 224 273 322
8:  36 100 164 228 292 356 420 484
9:  45 126 207 288 369 450 531 612 693
10: 55 155 255 355 455 555 655 755 855 955
... reformatted - _Wolfdieter Lang_, Dec 08 2014
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Columns k=1..6: A000217, A005449, A005475, A022265, A022267, A022269.

Diagonals: A081436, A059270, ...

Row sums: A037270.

Programs

Mathematica

Table[Sum[n*(k - 1) + j, {j,1,n}], {n,1,10}, {k,1,n}] // Flatten (* G. C. Greubel, Aug 23 2017 *)

PARI

trg(nn) = {for (n=1, nn, mm = matrix(n, n, i, j, j + n*(i-1)); for (i=1, n, print1(sum(j=1, n, mm[i, j]), ", ");); print(););} \\ Michel Marcus, Sep 15 2014

Formula

T(n, k) = Sum_{j=1..n} (n*(k-1)+ j), for n >= k >= 1. See the Michel Marcus program. - Wolfdieter Lang, Dec 08 2014

T(n, k) = binomial(n+1, 2) + n^2*(k-1). - Wolfdieter Lang, Dec 09 2014

Extensions

Edited. - Wolfdieter Lang, Dec 08 2014