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%I #28 Aug 23 2017 16:06:21
%S 1,3,7,6,15,24,10,26,42,58,15,40,65,90,115,21,57,93,129,165,201,28,77,
%T 126,175,224,273,322,36,100,164,228,292,356,420,484,45,126,207,288,
%U 369,450,531,612,693,55,155,255,355,455,555,655,755,855,955,66,187,308,429,550
%N 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.
%C See illustration in links.
%C The corresponding triangle with column sums is found in A251630. - _Wolfdieter Lang_, Dec 09 2014
%H G. C. Greubel, <a href="/A241016/b241016.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Kival Ngaokrajang, <a href="/A241016/a241016.pdf">Illustration of initial terms</a>
%F 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
%F T(n, k) = binomial(n+1, 2) + n^2*(k-1). - _Wolfdieter Lang_, Dec 09 2014
%e The triangle T(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 ...
%e 1: 1
%e 2: 3 7
%e 3: 6 15 24
%e 4: 10 26 42 58
%e 5: 15 40 65 90 115
%e 6: 21 57 93 129 165 201
%e 7: 28 77 126 175 224 273 322
%e 8: 36 100 164 228 292 356 420 484
%e 9: 45 126 207 288 369 450 531 612 693
%e 10: 55 155 255 355 455 555 655 755 855 955
%e ... reformatted - _Wolfdieter Lang_, Dec 08 2014
%t Table[Sum[n*(k - 1) + j, {j,1,n}], {n,1,10}, {k,1,n}] // Flatten (* _G. C. Greubel_, Aug 23 2017 *)
%o (Small Basic)
%o For n=1 To 20
%o For k=1 To n*n-(n-1) Step n
%o c=0
%o For i=1 To n
%o If i=1 Then
%o a=k
%o Else
%o a=a+1
%o EndIf
%o c=c+a
%o EndFor
%o TextWindow.Write(c+", ")
%o EndFor
%o EndFor
%o (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
%Y Columns k=1..6: A000217, A005449, A005475, A022265, A022267, A022269.
%Y Diagonals: A081436, A059270, ...
%Y Row sums: A037270.
%K nonn,easy,tabl
%O 1,2
%A _Kival Ngaokrajang_, Aug 08 2014
%E Edited. - _Wolfdieter Lang_, Dec 08 2014
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