cp's OEIS Frontend

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.

A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m*((n - 2)*m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

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%I A303301 #65 Aug 29 2018 17:59:19
%S A303301 0,0,1,0,1,-3,0,1,-2,0,0,1,-1,1,-8,0,1,0,2,-5,-3,0,1,1,3,-2,0,-15,0,1,
%T A303301 2,4,1,3,-9,-8,0,1,3,5,4,6,-3,-2,-24,0,1,4,6,7,9,3,4,-14,-15,0,1,5,7,
%U A303301 10,12,9,10,-4,-5,-35,0,1,6,8,13,15,15,16,6,5,-20,-24,0,1,7,9,16,18,21,22,16,15,-5,-9,-48
%N A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m*((n - 2)*m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.
%C A303301 Note that the formula mentioned in the definition gives several kinds of numbers, for example:
%C A303301 Row 0 and row 1 give A317300 and A317301 respectively.
%C A303301 Row 2 gives A001057 (canonical enumeration of integers).
%C A303301 Row 3 gives 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
%C A303301 Row 4 gives A008794 (squares repeated) except the initial zero.
%C A303301 Finally, for n >= 5 row n gives the generalized k-gonal numbers (see Crossrefs section).
%H A303301 Alois P. Heinz, <a href="/A303301/b303301.txt">Antidiagonals n = 0..200</a> (first 45 antidiagonals from Robert G. Wilson v)
%F A303301 T(n,k) = A194801(n-3,k) if n >= 3.
%e A303301 Array begins:
%e A303301 ------------------------------------------------------------------
%e A303301 n\m  Seq. No.    0   1  -1   2  -2   3   -3    4   -4    5   -5
%e A303301 ------------------------------------------------------------------
%e A303301 0    A317300:    0,  1, -3,  0, -8, -3, -15,  -8, -24, -15, -35...
%e A303301 1    A317301:    0,  1, -2,  1, -5,  0,  -9,  -2, -14,  -5, -20...
%e A303301 2    A001057:    0,  1, -1,  2, -2,  3,  -3,   4,  -4,   5,  -5...
%e A303301 3   (A008795):   0,  1,  0,  3,  1,  6,   3,  10,   6,  15,  10...
%e A303301 4   (A008794):   0,  1,  1,  4,  4,  9,   9,  16,  16,  25,  25...
%e A303301 5    A001318:    0,  1,  2,  5,  7, 12,  15,  22,  26,  35,  40...
%e A303301 6    A000217:    0,  1,  3,  6, 10, 15,  21,  28,  36,  45,  55...
%e A303301 7    A085787:    0,  1,  4,  7, 13, 18,  27,  34,  46,  55,  70...
%e A303301 8    A001082:    0,  1,  5,  8, 16, 21,  33,  40,  56,  65,  85...
%e A303301 9    A118277:    0,  1,  6,  9, 19, 24,  39,  46,  66,  75, 100...
%e A303301 10   A074377:    0,  1,  7, 10, 22, 27,  45,  52,  76,  85, 115...
%e A303301 11   A195160:    0,  1,  8, 11, 25, 30,  51,  58,  86,  95, 130...
%e A303301 12   A195162:    0,  1,  9, 12, 28, 33,  57,  64,  96, 105, 145...
%e A303301 13   A195313:    0,  1, 10, 13, 31, 36,  63,  70, 106, 115, 160...
%e A303301 14   A195818:    0,  1, 11, 14, 34, 39,  69,  76, 116, 125, 175...
%e A303301 15   A277082:    0,  1, 12, 15, 37, 42,  75,  82, 126, 135, 190...
%e A303301 ...
%t A303301 t[n_, r_] := PolygonalNumber[n, If[OddQ@ r, Floor[(r + 1)/2], -r/2]]; Table[ t[n - r, r], {n, 0, 11}, {r, 0, n}] // Flatten (* also *)
%t A303301 (* to view the square array *)  Table[ t[n, r], {n, 0, 15}, {r, 0, 10}] // TableForm (* _Robert G. Wilson v_, Aug 08 2018 *)
%Y A303301 Columns 0..2 are A000004, A000012, A023445.
%Y A303301 Column 3 gives A001477 which coincides with the row numbers.
%Y A303301 Main diagonal gives A292551.
%Y A303301 Row 0-2 gives A317300, A317301, A001057.
%Y A303301 Row 3 gives 0 together with A008795.
%Y A303301 Row 4 gives A008794.
%Y A303301 For n >= 5, rows n gives the generalized n-gonal numbers: A001318 (n=5), A000217 (n=6), A085787 (n=7), A001082 (n=8), A118277 (n=9), A074377 (n=10), A195160 (n=11), A195162 (n=12), A195313 (n=13), A195818 (n=14), A277082 (n=15), A274978 (n=16), A303305 (n=17), A274979 (n=18), A303813 (n=19), A218864 (n=20), A303298 (n=21), A303299 (n=22), A303303 (n=23), A303814 (n=24), A303304 (n=25), A316724 (n=26), A316725 (n=27), A303812 (n=28), A303815 (n=29), A316729 (n=30).
%Y A303301 Cf. A194801, A195152.
%Y A303301 Cf. A317302 (a similar table but with polygonal numbers).
%K A303301 sign,tabl
%O A303301 0,6
%A A303301 _Omar E. Pol_, Jun 08 2018