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 A195310 Triangle read by rows with T(n,k) = n - A001318(k), n >= 1, k >= 1, if (n - A001318(k)) >= 0. 24
 0, 1, 0, 2, 1, 3, 2, 4, 3, 0, 5, 4, 1, 6, 5, 2, 0, 7, 6, 3, 1, 8, 7, 4, 2, 9, 8, 5, 3, 10, 9, 6, 4, 11, 10, 7, 5, 0, 12, 11, 8, 6, 1, 13, 12, 9, 7, 2, 14, 13, 10, 8, 3, 0, 15, 14, 11, 9, 4, 1, 16, 15, 12, 10, 5, 2, 17, 16, 13, 11, 6, 3, 18, 17, 14, 12, 7, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A001318(k). This sequence is related to Euler's Pentagonal Number Theorem. A000041(a(n)) gives the absolute value of A175003(n). To get the number of partitions of n see the example. LINKS L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005. Eric Weisstein's World of Mathematics, Pentagonal Number Theorem Wikipedia, Pentagonal number theorem FORMULA A175003(n,k) = A057077(k-1)*A000041(T(n,k)), n >= 1, k >= 1. EXAMPLE Written as a triangle:    0;    1,  0;    2,  1;    3,  2;    4,  3,  0;    5,  4,  1;    6,  5,  2,  0;    7,  6,  3,  1;    8,  7,  4,  2;    9,  8,  5,  3;   10,  9,  6,  4;   11, 10,  7,  5,  0;   12, 11,  8,  6,  1;   13, 12,  9,  7,  2;   14, 13, 10,  8,  3,  0; . For n = 15, consider row 15 which lists the numbers 14, 13, 10, 8, 3, 0. From Euler's Pentagonal Number Theorem we have that the number of partitions of 15 is p(15) = p(14) + p(13) - p(10) - p(8) + p(3) + p(0) = 135 + 101 - 42 - 22 + 3 + 1 = 176. MATHEMATICA rows = 20; a1318[n_] := If[EvenQ[n], n(3n/2+1)/4, (n+1)(3n+1)/8]; T[n_, k_] := n - a1318[k]; Table[DeleteCases[Table[T[n, k], {k, 1, n}], _?Negative], {n, 1, rows}] // Flatten (* Jean-François Alcover, Sep 22 2018 *) CROSSREFS Row sums give A195311. Cf. A000041, A001318, A010815, A026741, A057077, A175003. Sequence in context: A185314 A285120 A282744 * A051282 A274121 A052306 Adjacent sequences:  A195307 A195308 A195309 * A195311 A195312 A195313 KEYWORD nonn,tabf AUTHOR Omar E. Pol, Sep 21 2011 EXTENSIONS Name essentially suggested by Franklin T. Adams-Watters (see history), Sep 21 2011 STATUS approved

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Last modified February 23 11:52 EST 2020. Contains 332159 sequences. (Running on oeis4.)