This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A344235 #11 Jun 02 2021 17:43:54
%S A344235 0,1,1,5,5,4,14,14,13,9,30,30,29,25,16,55,55,54,50,41,25,91,91,90,86,
%T A344235 77,61,36,140,140,139,135,126,110,85,49,204,204,203,199,190,174,149,
%U A344235 113,64,285,285,284,280,271,255,230,194,145,81,385,385,384,380,371,355,330,294,245,181,100
%N A344235 Triangle T from the array A(k, n) giving the sums of k+1 consecutive squares starting with n^2, read as upwards antidiagonals, for k >= 0 and n >= 0.
%C A344235 Motivated by a proposal from _Charlie Marion_.
%D A344235 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd ed.; Addison-Wesley, 1994, pp. 283-290.
%F A344235 A(k, n) = Sum_{j=0..k} (n+j)^2, for k >= 0, n >= 0.
%F A344235 A(k, n) = Sum_{j=0..n+k} j^2 - (2*n-1)*n*(n-1)/3! = S(n+k) - (2*n-1)*n*(n-1)/3!, with S(n+k) = (1/3)*Sum_{j=0..2} binomial(3, j)*B_j*(n+k+1)^(3-j), with the Bernoulli numbers A027641 / A027642 (see Graham et al., pp. 283-290).
%F A344235 Recurrence for sequence of row k: A(k, n) = A(k, n-1) + (k+1)*(2*n + k - 1), n >= 1, with A(k, 0) = (2*k+1)*(k+1)*k/3!, for k >= 0.
%e A344235 The array A(k, n) begins:
%e A344235 k \ n 0 1 2 3 4 5 6 7 8 9 10 ...
%e A344235 -----------------------------------------------------------
%e A344235 0: 0 1 4 9 16 25 36 49 64 81 100 ...
%e A344235 1: 1 5 13 25 41 61 85 113 145 181 221 ...
%e A344235 2: 5 14 29 50 77 110 149 194 245 302 365 ...
%e A344235 3: 14 30 54 86 126 174 230 294 366 446 534 ...
%e A344235 4: 30 55 90 135 190 255 330 415 510 615 730 ...
%e A344235 5: 55 91 139 199 271 355 451 559 679 811 955 ...
%e A344235 6: 91 140 203 280 371 476 595 728 875 1036 1211 ...
%e A344235 7: 140 204 284 380 492 620 764 924 1100 1292 1500 ...
%e A344235 8: 204 285 384 501 636 789 960 1149 1356 1581 1824 ...
%e A344235 9: 285 385 505 645 805 985 1185 1405 1645 1905 2185 ...
%e A344235 ...
%e A344235 -----------------------------------------------------------
%e A344235 The triangle T(m, n) begins:
%e A344235 m \ n 0 1 2 3 4 5 6 7 8 9 ...
%e A344235 -----------------------------------------------------------
%e A344235 0: 0
%e A344235 1: 1 1
%e A344235 2: 5 5 4
%e A344235 3: 14 14 13 9
%e A344235 4: 30 30 29 25 16
%e A344235 5: 55 55 54 50 41 25
%e A344235 6: 91 91 90 86 77 61 36
%e A344235 7: 140 140 139 135 126 110 85 49
%e A344235 8: 204 204 203 199 190 174 149 113 64
%e A344235 9: 285 285 284 280 271 255 230 194 145 81
%e A344235 ...
%e A344235 ----------------------------------------------------------
%Y A344235 Rows of array A, diagonals of T: A000290, A001844, A005918(n+1), A027575, A027578, A027865, A260637, A276026, ...
%Y A344235 Columns of array A and T (without leading 0s): A000330, A000330(n+1), A168559(n+1), ...
%K A344235 nonn,tabl,easy
%O A344235 0,4
%A A344235 _Wolfdieter Lang_, May 27 2021