login
A333822
Number of ways to write n as the difference of two k-gonal numbers for k >= 3.
5
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
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. 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
KEYWORD
nonn
AUTHOR
Peter Kagey, Apr 06 2020
STATUS
approved