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 A322135 #23 Feb 22 2019 01:55:14
%S A322135 1,4,5,9,13,14,16,25,29,30,25,41,50,54,55,36,61,77,86,90,91,49,85,110,
%T A322135 126,135,139,140,64,113,149,174,190,199,203,204,81,145,194,230,255,
%U A322135 271,280,284,285,100,181,245,294,330,355,371,380,384,385,121,221,302
%N A322135 Table of truncated square pyramid numbers, read by antidiagonals.
%C A322135 The n-th row contains n numbers: n^2, n^2 + (n-1)^2, ..., n^2 + (n-1)^2 + ... + 1^2.
%C A322135 All numbers that appear in the table are listed in ascending order at A034705.
%C A322135 All numbers that appear twice or more are listed at A130052.
%C A322135 The left column is A000290 (the squares).
%C A322135 The top row is A000330 (the square pyramidal numbers).
%C A322135 The columns are A000290, A099776 (or a tail of A001844), a tail of A005918 or A120328, a tail of A027575, a tail of A027578, a tail of A027865, ...
%C A322135 The first two rows are A000330 and a tail of A168599, but subsequent rows are not currently in the OEIS, and are all tails of A000330 minus various constants.
%C A322135 The main diagonal is A050410.
%F A322135 T(n,k) = n^2 + (n+1)^2 + ... + (n+k-1)^2 = A000330(n + k - 1) - A000330(n - 1) = T(n, k) = k*n^2 + (k^2 - k)*n + (1/3*k^3 - 1/2*k^2 + 1/6*k)
%F A322135 G.f.: -y*(y*(1 + y) + x*(1 - 2*y - 3*y^2) + x^2*(1 - 3*y + 4*y^2))/((- 1 + x)^3*(- 1 + y)^4). - _Stefano Spezia_, Nov 28 2018
%e A322135 The 17th term is entry 2 on antidiagonal 6, so we sum two terms: 6^2 + 5^2 = 61.
%e A322135 Table begins:
%e A322135 1 5 14 30 55 91 140 204 ...
%e A322135 4 13 29 54 90 139 203 ...
%e A322135 9 25 50 86 135 199 ...
%e A322135 16 41 77 126 190 ...
%e A322135 25 61 110 174 ...
%e A322135 36 85 149 ...
%e A322135 49 113 ...
%e A322135 64 ...
%e A322135 ...
%t A322135 T[n_,k_] = Sum[(n+i)^2, {i,0,k-1}]; Table[T[n-k+1, k], {n,1,10}, {k,1,n}] // Flatten (* _Amiram Eldar_, Nov 28 2018 *)
%t A322135 f[n_] := Table[SeriesCoefficient[-((y (y (1 + y) + x (1 - 2 y - 3 y^2) + x^2 (1 - 3 y + 4 y^2)))/((-1 + x)^3 (-1 + y)^4)) , {x, 0,
%t A322135 i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f, 10]] (* _Stefano Spezia_, Nov 28 2018 *)
%Y A322135 See comments; also cf. A000330, A059255.
%K A322135 nonn,easy,tabl
%O A322135 1,2
%A A322135 _Allan C. Wechsler_, Nov 27 2018