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%I #65 Jul 13 2024 02:14:30
%S 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 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 24,40,55,66,70,64,45,10,0,1,11,27,46,65,81,91,92,81,55,11
%N Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.
%C 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 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 Alois P. Heinz, <a href="/A139600/b139600.txt">Rows n = 0..140, flattened</a>
%H 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 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polnum01.jpg">Polygonal numbers</a>, An alternative illustration of initial terms.
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F T(n,k) = n*(k-1)*k/2+k.
%F T(n,k) = A057145(n+2,k). - _R. J. Mathar_, Jul 28 2016
%F From _Stefano Spezia_, Apr 12 2024: (Start)
%F G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
%F E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)
%e The square array of nonnegatives together with polygonal numbers begins:
%e =========================================================
%e ....................... A A . . A A A A
%e ....................... 0 0 . . 0 0 1 1
%e ....................... 0 0 . . 1 1 3 3
%e ....................... 0 0 . . 6 7 9 9
%e ....................... 0 0 . . 9 3 6 6
%e ....................... 0 1 . . 5 2 0 0
%e ....................... 4 2 . . 7 9 6 7
%e =========================================================
%e Nonnegatives . A001477: 0, 1, 2, 3, 4, 5, 6, 7, ...
%e Triangulars .. A000217: 0, 1, 3, 6, 10, 15, 21, 28, ...
%e Squares ...... A000290: 0, 1, 4, 9, 16, 25, 36, 49, ...
%e Pentagonals .. A000326: 0, 1, 5, 12, 22, 35, 51, 70, ...
%e Hexagonals ... A000384: 0, 1, 6, 15, 28, 45, 66, 91, ...
%e Heptagonals .. A000566: 0, 1, 7, 18, 34, 55, 81, 112, ...
%e Octagonals ... A000567: 0, 1, 8, 21, 40, 65, 96, 133, ...
%e 9-gonals ..... A001106: 0, 1, 9, 24, 46, 75, 111, 154, ...
%e 10-gonals .... A001107: 0, 1, 10, 27, 52, 85, 126, 175, ...
%e 11-gonals .... A051682: 0, 1, 11, 30, 58, 95, 141, 196, ...
%e 12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217, ...
%e ...
%e =========================================================
%e 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 T:= (n, k)-> n*(k-1)*k/2+k:
%p seq(seq(T(d-k, k), k=0..d), d=0..14); # _Alois P. Heinz_, Oct 14 2018
%t 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 (Python)
%o def A139600Row(n):
%o x, y = 1, 1
%o yield 0
%o while True:
%o yield x
%o x, y = x + y + n, y + n
%o for n in range(8):
%o R = A139600Row(n)
%o print([next(R) for _ in range(11)]) # _Peter Luschny_, Aug 04 2019
%o (Magma)
%o T:= func< n,k | k*(n*(k-1)+2)/2 >;
%o A139600:= func< n,k | T(n-k, k) >;
%o [A139600(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 12 2024
%o (SageMath)
%o def T(n,k): return k*(n*(k-1)+2)/2
%o def A139600(n,k): return T(n-k, k)
%o flatten([[A139600(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 12 2024
%Y Cf. A001477, A057145, A139601, A139617, A139618, A139620, A180662.
%Y A formal extension negative n is in A326728.
%Y 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 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 nonn,tabl,easy
%O 0,6
%A _Omar E. Pol_, Apr 27 2008
%E Edited by _Omar E. Pol_, Jan 05 2009
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