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

2,2

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

Links

Peter Kagey, Table of n, a(n) for n = 2..5000

OEIS Wiki, Figurate numbers: Polygonal numbers

Formula

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

Example

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).

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]

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).

Sequence in context: A131950 A116192 A090104 * A075795 A058268 A341729

Adjacent sequences: A333819 A333820 A333821 * A333823 A333824 A333825

Keyword

nonn

Author

Peter Kagey, Apr 06 2020

Status

approved