A333836 Number of ways to write n as the difference of two positive k-gonal numbers for k >= 3.

0, 1, 2, 2, 4, 2, 5, 3, 6, 3, 6, 4, 7, 4, 7, 5, 8, 4, 7, 5, 10, 6, 7, 5, 10, 5, 10, 5, 9, 7, 9, 6, 11, 6, 10, 6, 12, 5, 11, 7, 11, 6, 9, 7, 13, 9, 9, 8, 12, 7, 13, 7, 9, 7, 11, 9, 17, 7, 7, 8, 13, 6, 14, 9, 17, 8, 11, 6, 12, 9, 11, 9, 13, 7

Offset: 1

Author

Peter Kagey, Apr 07 2020

Keywords

• nonn

Comments

Records occur at indices 1, 2, 3, 5, 7, 9, 13, 17, 21, 33, 37, 45, 57, 105, 145, 217, 225, 273, 385, 495, 561, 651, 705, 945, 1105, ... - Peter Kagey, Nov 18 2020

Examples

			The a(9) = 6 ways of writing 9 as the difference of two k-gonal numbers are:
A000217(4) - A000217(1) = 10 -  1 (triangular),
A000217(5) - A000217(3) = 15 -  6 (triangular),
A000217(9) - A000217(8) = 45 - 36 (triangular),
A000290(5) - A000290(4) = 25 - 16 (square),
A000384(3) - A000384(2) = 15 -  6 (hexagonal), and
A001107(2) - A001107(1) = 10 -  1 (10-gonal).
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..5000
• OEIS Wiki, Figurate numbers: Polygonal numbers

Crossrefs

Cf. A177025, A333822.

Cf. A000217, A000290, A000384, A001107.

Programs

Mathematica

b := 74
CoefficientList[
Series[Sum[
   Sum[x^(k*(p*k - (p - 2))/2)*x^(p*k)/(1 - x^(p*k)), {k, 1, b}], {p,
    1, b - 1}], {x, 0, b}], x]

Formula

a(n) = A333822(n) - A177025(n) for n > 2.