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a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.
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%I #13 Feb 16 2025 08:34:00

%S 1,10,2180,10053736,13291443468940

%N a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%e Let S(k, m) denote the sum of m triangular numbers starting from k(k+1)/2. We have

%e a(1) = S(1, 1);

%e a(2) = S(4, 1) = S(1, 3);

%e a(3) = S(31, 4) = S(27, 5) = S(9, 15);

%e a(4) = S(945, 22) = S(571, 56) = S(968, 21) = S(131, 266);

%e a(5) = S(4109, 38947) = S(25213, 20540) = S(10296, 32943) = S(32801, 15834) = S(31654, 16472).

%Y Cf. A000217, A054859, A068314, A034706, A050943, A186337, A298467, A307666, A309783, A334008, A334010, A334011, A334012.

%K nonn,hard,more,changed

%O 1,2

%A _Ilya Gutkovskiy_, Apr 12 2020

%E a(5) from _Giovanni Resta_, Apr 13 2020