A333822 Number of ways to write n as the difference of two k-gonal numbers for k >= 3.

1, 3, 3, 5, 4, 6, 4, 8, 5, 7, 6, 8, 5, 10, 7, 9, 6, 8, 6, 13, 8, 8, 7, 12, 6, 12, 8, 10, 9, 10, 7, 13, 8, 12, 10, 13, 6, 13, 9, 12, 8, 10, 8, 17, 11, 10, 10, 14, 8, 16, 9, 10, 9, 14, 10, 19, 9, 8, 10, 14, 7, 16, 12, 19, 12, 12, 7, 14, 12, 12, 11, 14, 8

Offset: 2

Author

Peter Kagey, Apr 06 2020

Keywords

• nonn

Comments

Records occur at indices 2, 3, 5, 7, 9, 15, 21, 45, 57, 81, 105, 145, 217, 225, 385, 435, 441, 495, 561, 651, 705, 945, 1105, ... - Peter Kagey, Nov 18 2020

Examples

			For n = 7, the a(7) = 6 ways to write 7 as the difference of k-gonal numbers are:
A000217(4) - A000217(2) = 10 -  3 (triangular),
A000217(7) - A000217(6) = 28 - 21 (triangular),
A000290(4) - A000290(3) = 16 -  9 (square),
A000326(3) - A000326(2) = 12 -  5 (pentagonal),
A000566(2) - A000566(0) =  7 -  0 (heptagonal), and
A000567(2) - A000567(1) =  8 -  1 (octagonal).
		

Links

• Peter Kagey, Table of n, a(n) for n = 2..5000
• OEIS Wiki, Figurate numbers: Polygonal numbers

Crossrefs

Cf. A000217, A000290, A000326, A000566, A000567, A139601.

Cf. A177025.

Analogous sequences for specific values of k: A001227 (k=3), A034178 (k=4), A333815 (k=5), A333816 (k=6), A333817 (k=7), A333818 (k=8).

Programs

Mathematica

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

Formula

G.f.: Sum_{m>=1} (-x + Sum_{k>=1} x^A139601(m-1,k)/(1 - x^(m*k))).