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A242443
Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an even square (A016742) and a generalized pentagonal number (A001318).
2
1, 1, 2, 1, 2, 3, 3, 4, 1, 4, 3, 4, 2, 2, 5, 3, 5, 3, 5, 4, 5, 7, 3, 4, 4, 6, 6, 4, 6, 3, 5, 7, 6, 4, 1, 7, 7, 6, 5, 6, 9, 5, 7, 7, 8, 6, 8, 4, 6, 6, 7, 9, 4, 10, 3, 6, 9, 7, 8, 5, 9, 7, 6, 7, 5, 11, 9, 7, 3, 7, 12, 13, 7, 7, 6, 9, 11, 6, 11, 8, 7, 10, 10, 8, 8, 8, 11, 5, 8, 5, 8, 11, 10, 10, 6, 14, 10, 6, 7, 7
OFFSET
1,3
COMMENTS
It is conjectured (1.1) and then proved by theorem 1.2 that all positive integers can be so represented [Sun, pp. 4-5].
LINKS
Zhi-Wei Sun, On Universal Sums Of Polygonal Numbers, arXiv:0905.0635v18 [math.NT] 26 Oct 2011
MATHEMATICA
planeFigurative[n_, r_] := pf[n, r] = (n - 2) Binomial[r, 2] + r; s = Sort@ Table[ planeFigurative[3, i] + planeFigurative[3, j] + planeFigurative[3, k], {i, 0, 14}, {j, 0, 10, 2}, {k, -8, 8}]; Table[ Count[s, n], {n, 0, 50}]
KEYWORD
nonn,easy
AUTHOR
Robert G. Wilson v, May 14 2014
STATUS
approved