[seqfan] Truncated triangular numbers more than one way

Allan Wechsler acwacw at gmail.com
Wed Nov 14 00:50:47 CET 2018

Consider the numbers in A008912. These are numbers of spots that can be
arranged in a triangular lattice to form a hexagon whose sides alternate a
and b. Changing the variables in the formula at A008912 gives the form

F(a,b) = (4ab + a(a-3) + b(b-3) + 2) / 2

(Here a = k, b = (n-k) in the formula A008912.

F(a,b) = F(b,a), so for my purposes, assume a <= b.

36 = F(1,8) = F(3,5), and is the smallest integer that gives the size of
two different hexagons.

75 = F(2,10) = F(5,6) is the second such integer.

Unless I missed one, 91 = F(1,13) = F(6,6) is the third.

36, 75, 91 is not in OEIS. I must have missed one: double values of a
quadratic form are a really obvious idea. We definitely have the analogous
sequences for the Pythagorean form a^2 + b^2 and the Eisenstein form a^2 +
b^2 + ab.

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