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 A125764 #16 Feb 16 2025 08:33:04
%S A125764 1,3,2,6,7,3,10,15,12,4,15,26,27,18,5,21,40,48,42,25,6,28,57,75,76,60,
%T A125764 33,7,36,77,108,120,110,81,42,8,45,100,147,174,175,150,105,52,9,55,
%U A125764 126,192,238,255,240,196,132,63,10,66,155,243,312,350,351,315,248,162,75,11
%N A125764 Array of partial sums of rows of array in A086271, read by antidiagonals.
%C A125764 Row 3 is = 3rd triangular number + 3rd square + 3rd pentagonal number + 3rd hexagonal number + ... + 3rd k-gonal number. First column is triangular numbers. A086271 Rectangular array T(n,k) of polygonal numbers, by diagonals.
%H A125764 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number.</a> See equation (4), our partial sums are on this as array element values.
%F A125764 a(k,n) = (k*(k-1)/2)n^2 + (k*(k+3)/4)n. a(k,n) = row k of array of partial sums = k-th triangular number + k-th square + k-th pentagonal number + k-th hexagonal number + ... = A000217(k) + A000290(k) + A000326(k) + A000384(k) + ... a(1,n) = n. a(2,n) = (n(n+1)/2)-3 = A000217(n) - 3. a(3,n) = 3*n(n+3)/2 = A000096 with offset 3.
%e A125764 Partial row sum array begins:
%e A125764 1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... n.
%e A125764 2 | 3, 7, 12, 18, 25, 33, 42, 52, (n(n+1)/2)-3.
%e A125764 3 | 6, 15, 27, 42, 60, 81, 105, ... (3/2)n^2 + (9/2) n.
%e A125764 4 | 10, 26, 48, 76, 110, 150, ... 3n^2 + 7n.
%e A125764 5 | 15, 40, 75, ... 5n^2 + 10n.
%e A125764 6 | 21, 57, 108, ... (15/2)n^2 + (27/2)n.
%p A125764 A086271 := proc(n,k) k*binomial(n,2)+n ; end: A125764 := proc(n,k) add(A086271(n,i),i=1..k) ; end: for d from 1 to 15 do for k from 1 to d do printf("%d, ",A125764(d-k+1,k)) ; od: od: # _R. J. Mathar_, Nov 02 2007
%Y A125764 Cf. A000096, A000217, A000290, A000326, A000384, A086270, A086271, A086272, A086273.
%K A125764 easy,nonn,tabl
%O A125764 1,2
%A A125764 _Jonathan Vos Post_ and _Joshua Zucker_, Feb 03 2007
%E A125764 More terms from _R. J. Mathar_, Nov 02 2007
%E A125764 Keyword tabl added by _Michel Marcus_, Apr 08 2013