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A333836
Number of ways to write n as the difference of two positive k-gonal numbers for k >= 3.
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
1,3
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
FORMULA
a(n) = A333822(n) - A177025(n) for n > 2.
EXAMPLE
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).
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Kagey, Apr 07 2020
STATUS
approved