cp's OEIS Frontend

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

%I A333836 #12 Nov 19 2020 22:34:06
%S A333836 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,
%T A333836 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,
%U A333836 6,14,9,17,8,11,6,12,9,11,9,13,7
%N A333836 Number of ways to write n as the difference of two positive k-gonal numbers for k >= 3.
%C A333836 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
%H A333836 Peter Kagey, <a href="/A333836/b333836.txt">Table of n, a(n) for n = 1..5000</a>
%H A333836 OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers#Polygonal_numbers">Figurate numbers: Polygonal numbers</a>
%F A333836 a(n) = A333822(n) - A177025(n) for n > 2.
%e A333836 The a(9) = 6 ways of writing 9 as the difference of two k-gonal numbers are:
%e A333836 A000217(4) - A000217(1) = 10 -  1 (triangular),
%e A333836 A000217(5) - A000217(3) = 15 -  6 (triangular),
%e A333836 A000217(9) - A000217(8) = 45 - 36 (triangular),
%e A333836 A000290(5) - A000290(4) = 25 - 16 (square),
%e A333836 A000384(3) - A000384(2) = 15 -  6 (hexagonal), and
%e A333836 A001107(2) - A001107(1) = 10 -  1 (10-gonal).
%t A333836 b := 74
%t A333836 CoefficientList[
%t A333836 Series[Sum[
%t A333836    Sum[x^(k*(p*k - (p - 2))/2)*x^(p*k)/(1 - x^(p*k)), {k, 1, b}], {p,
%t A333836     1, b - 1}], {x, 0, b}], x]
%Y A333836 Cf. A177025, A333822.
%Y A333836 Cf. A000217, A000290, A000384, A001107.
%K A333836 nonn
%O A333836 1,3
%A A333836 _Peter Kagey_, Apr 07 2020