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A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

%I A139600 #65 Jul 13 2024 02:14:30
%S A139600 0,0,1,0,1,2,0,1,3,3,0,1,4,6,4,0,1,5,9,10,5,0,1,6,12,16,15,6,0,1,7,15,
%T A139600 22,25,21,7,0,1,8,18,28,35,36,28,8,0,1,9,21,34,45,51,49,36,9,0,1,10,
%U A139600 24,40,55,66,70,64,45,10,0,1,11,27,46,65,81,91,92,81,55,11
%N A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.
%C A139600 A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.
%C A139600 The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - _Johannes W. Meijer_, Apr 29 2011
%H A139600 Alois P. Heinz, <a href="/A139600/b139600.txt">Rows n = 0..140, flattened</a>
%H A139600 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FigurateNumber">Figurate number — a very short introduction</a>. With plots from Stefan Friedrich Birkner.
%H A139600 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polnum01.jpg">Polygonal numbers</a>, An alternative illustration of initial terms.
%H A139600 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A139600 T(n,k) = n*(k-1)*k/2+k.
%F A139600 T(n,k) = A057145(n+2,k). - _R. J. Mathar_, Jul 28 2016
%F A139600 From _Stefano Spezia_, Apr 12 2024: (Start)
%F A139600 G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
%F A139600 E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)
%e A139600 The square array of nonnegatives together with polygonal numbers begins:
%e A139600 =========================================================
%e A139600 ....................... A   A   .   .   A    A    A    A
%e A139600 ....................... 0   0   .   .   0    0    1    1
%e A139600 ....................... 0   0   .   .   1    1    3    3
%e A139600 ....................... 0   0   .   .   6    7    9    9
%e A139600 ....................... 0   0   .   .   9    3    6    6
%e A139600 ....................... 0   1   .   .   5    2    0    0
%e A139600 ....................... 4   2   .   .   7    9    6    7
%e A139600 =========================================================
%e A139600 Nonnegatives . A001477: 0,  1,  2,  3,  4,   5,   6,   7, ...
%e A139600 Triangulars .. A000217: 0,  1,  3,  6, 10,  15,  21,  28, ...
%e A139600 Squares ...... A000290: 0,  1,  4,  9, 16,  25,  36,  49, ...
%e A139600 Pentagonals .. A000326: 0,  1,  5, 12, 22,  35,  51,  70, ...
%e A139600 Hexagonals ... A000384: 0,  1,  6, 15, 28,  45,  66,  91, ...
%e A139600 Heptagonals .. A000566: 0,  1,  7, 18, 34,  55,  81, 112, ...
%e A139600 Octagonals ... A000567: 0,  1,  8, 21, 40,  65,  96, 133, ...
%e A139600 9-gonals ..... A001106: 0,  1,  9, 24, 46,  75, 111, 154, ...
%e A139600 10-gonals .... A001107: 0,  1, 10, 27, 52,  85, 126, 175, ...
%e A139600 11-gonals .... A051682: 0,  1, 11, 30, 58,  95, 141, 196, ...
%e A139600 12-gonals .... A051624: 0,  1, 12, 33, 64, 105, 156, 217, ...
%e A139600 ...
%e A139600 =========================================================
%e A139600 The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.
%p A139600 T:= (n, k)-> n*(k-1)*k/2+k:
%p A139600 seq(seq(T(d-k, k), k=0..d), d=0..14);  # _Alois P. Heinz_, Oct 14 2018
%t A139600 T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Robert G. Wilson v_, Jul 12 2009 *)
%o A139600 (Python)
%o A139600 def A139600Row(n):
%o A139600     x, y = 1, 1
%o A139600     yield 0
%o A139600     while True:
%o A139600         yield x
%o A139600         x, y = x + y + n, y + n
%o A139600 for n in range(8):
%o A139600     R = A139600Row(n)
%o A139600     print([next(R) for _ in range(11)]) # _Peter Luschny_, Aug 04 2019
%o A139600 (Magma)
%o A139600 T:= func< n,k | k*(n*(k-1)+2)/2 >;
%o A139600 A139600:= func< n,k | T(n-k, k) >;
%o A139600 [A139600(n,k): k in  [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 12 2024
%o A139600 (SageMath)
%o A139600 def T(n,k): return k*(n*(k-1)+2)/2
%o A139600 def A139600(n,k): return T(n-k, k)
%o A139600 flatten([[A139600(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 12 2024
%Y A139600 Cf. A001477, A057145, A139601, A139617, A139618, A139620, A180662.
%Y A139600 A formal extension negative n is in A326728.
%Y A139600 Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - _Johannes W. Meijer_, Apr 29 2011
%Y A139600 Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
%K A139600 nonn,tabl,easy
%O A139600 0,6
%A A139600 _Omar E. Pol_, Apr 27 2008
%E A139600 Edited by _Omar E. Pol_, Jan 05 2009