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A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.

%I A377729 #27 Feb 16 2025 08:34:07
%S A377729 19,90,162,299,509,816,1248,1837,2619,3634,4926,6543,8537,10964,13884,
%T A377729 17361,21463,26262,31834,38259,45621,54008,63512,74229,86259,99706,
%U A377729 114678,131287,149649,169884,192116,216473,243087,272094,303634,337851,374893,414912,458064,504509
%N A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.
%C A377729 From _David A. Corneth_, Nov 06 2024: (Start)
%C A377729 a(n) <= (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. Proof:
%C A377729 A polygonal number is of the form P(m, n) = m/2 * ((n - 2) * m - n + 4).
%C A377729 We have P(n - 5, n) + P(n - 4, n) + P(n, n) = P(n - 6, n) + P(n - 2, n) + P(n - 1, n) = (3*n^3 - 18*n^2 + 21*n) / 2.
%C A377729 This lets us find the upper bound on a(n) by making two lists from 1 through n + 3. From one of them we remove n-2, n-1 and n + 3 and from the other we remove n-3, n+1 and n+2. The sum for remaining polygonal numbers is the same giving an upper bound on a(n) which turns out to be (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 (End)
%H A377729 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>
%F A377729 From _David A. Corneth_, Nov 06 2024: (Start)
%F A377729 a(n) >= A006484(n).
%F A377729 Conjecture: a(n) = (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. (End)
%F A377729 Conjectured g.f.: x^3*(19 - 5*x - 98*x^2 + 199*x^3 - 171*x^4 + 72*x^5 - 12*x^6) / (1 - x)^5.
%e A377729 a(3) = 19 = 1 + 3 + 15 = 3 + 6 + 10.
%e A377729 a(4) = 90 = 1^2 + 2^2 + 6^2 + 7^2 = 1^2 + 3^2 + 4^2 + 8^2.
%Y A377729 Cf. A006484, A025377, A057145, A350405, A374144, A374256, A374287.
%K A377729 nonn
%O A377729 3,1
%A A377729 _Ilya Gutkovskiy_, Nov 05 2024
%E A377729 a(12)-a(36) from _Michael S. Branicky_, Nov 06 2024
%E A377729 More terms from _David A. Corneth_, Nov 10 2024