A195310 Triangle read by rows with T(n,k) = n - A001318(k), n >= 1, k >= 1, if (n - A001318(k)) >= 0.

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

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

Table of n, a(n) for n=1..78.

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium

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