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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. 45
 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, 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, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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. 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 LINKS Alois P. Heinz, Rows n = 0..140, flattened Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner. Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms. FORMULA T(n,k) = n*(k-1)*k/2+k. T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016 EXAMPLE The square array of nonnegatives together with polygonal numbers begins: ========================================================= ....................... A   A   .   .   A    A    A    A ....................... 0   0   .   .   0    0    1    1 ....................... 0   0   .   .   1    1    3    3 ....................... 0   0   .   .   6    7    9    9 ....................... 0   0   .   .   9    3    6    6 ....................... 0   1   .   .   5    2    0    0 ....................... 4   2   .   .   7    9    6    7 ========================================================= Nonnegatives . A001477: 0,  1,  2,  3,  4,   5,   6,   7, ... Triangulars .. A000217: 0,  1,  3,  6, 10,  15,  21,  28, ... Squares ...... A000290: 0,  1,  4,  9, 16,  25,  36,  49, ... Pentagonals .. A000326: 0,  1,  5, 12, 22,  35,  51,  70, ... Hexagonals ... A000384: 0,  1,  6, 15, 28,  45,  66,  91, ... Heptagonals .. A000566: 0,  1,  7, 18, 34,  55,  81, 112, ... Octagonals ... A000567: 0,  1,  8, 21, 40,  65,  96, 133, ... 9-gonals ..... A001106: 0,  1,  9, 24, 46,  75, 111, 154, ... 10-gonals .... A001107: 0,  1, 10, 27, 52,  85, 126, 175, ... 11-gonals .... A051682: 0,  1, 11, 30, 58,  95, 141, 196, ... 12-gonals .... A051624: 0,  1, 12, 33, 64, 105, 156, 217, ... ... ========================================================= 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. MAPLE T:= (n, k)-> n*(k-1)*k/2+k: seq(seq(T(d-k, k), k=0..d), d=0..14);  # Alois P. Heinz, Oct 14 2018 MATHEMATICA 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 *) PROG (Python 3) def A139600Row(n):     x, y = 1, 1     yield 0     while True:         yield x         x, y = x + y + n, y + n for n in range(8):         R = A139600Row(n) print([next(R) for _ in range(11)]) # Peter Luschny, Aug 04 2019 CROSSREFS Cf. A001477, A139601, A139617, A139618, A139620. A formal extension negative n is in A326728. 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 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). Sequence in context: A179329 A089112 A155584 * A198321 A325003 A166278 Adjacent sequences:  A139597 A139598 A139599 * A139601 A139602 A139603 KEYWORD nonn,tabl,easy AUTHOR Omar E. Pol, Apr 27 2008 EXTENSIONS Edited by Omar E. Pol, Jan 05 2009 STATUS approved

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Last modified December 11 07:41 EST 2019. Contains 329914 sequences. (Running on oeis4.)