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A350888 Let X,Y,Z be positive integer solutions to X^2 = Sum_{j=0..Y-1} (1+Z*j)^2, where solutions for Y or Z < 1 are excluded. This sequence lists the values for Z sorted by X.

%I A350888 #45 Dec 23 2024 11:34:36
%S A350888 14,1,432,8,13,1932,12958,367,30,3,1554,194,5082,388320,1349,15254,
%T A350888 178542,163,181,11636654,418782,6791,11928,192638,1086,2209447,5166,
%U A350888 19317900,1981979,262,348711312,4799102,7379,60240793
%N A350888 Let X,Y,Z be positive integer solutions to X^2 = Sum_{j=0..Y-1} (1+Z*j)^2, where solutions for Y or Z < 1 are excluded. This sequence lists the values for Z sorted by X.
%C A350888 This is a generalization of the cannonball problem for pyramids with a slope of 1/a(n). In the cannonball problem, a square pyramid of stacked balls shall contain a square number of balls in total. Each layer of such a pyramid consists of a square number of balls, in the classic case the top layer has one ball, the layers below contain adjacent square numbers of balls. For this sequence we ignore the fact that if adjacent layers do not alternate between odd and even squares the stacking will become difficult at least for sphere-like objects. We start in the top layer always with one ball, but will skip a constant count of square numbers between each layer. This results in pyramids which slope <= 1. A350887(n) is the number of layers needed to stack such a pyramid. As this sequence is only a list of solutions, n has no known relation to its definition.
%C A350888 For each slope 1/a(n) there exists exactly one or no such pyramid with a square number of balls.
%H A350888 Thomas Scheuerle, <a href="/A350887/a350887.txt">Some solutions to this problem sorted by A350887</a>.
%H A350888 Anji Dong, Katerina Saettone, Kendra Song, and Alexandru Zaharescu, <a href="https://arxiv.org/abs/2412.10097">An Equidistribution Result for Differences Associated with Square Pyramidal Numbers</a>, arXiv:2412.10097 [math.NT], 2024. See p. 1.
%H A350888 Thomas Scheuerle, <a href="/A350886/a350886.txt">Recursive solution formulas</a>.
%H A350888 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>.
%F A350888 A350886(n)^2 = a(n)^2*binomial(2*A350887(n), 3)/4 + 2*a(n)*binomial(A350887(n), 2) + A350887(n).
%F A350888 A350886(n)^2 = c*((b*(4*b*c^2-(6*c-2)*b + 12*(c-1)) + 12)/12), with c = A350887(n) and b = a(n). Expanded to see factors more clearly.
%F A350888 A350886(n)^2 = c*b^2*( ((1/b) + (c-1)/2)^2 + (c^2-1)/12 ). Shorter form of above.
%F A350888 a(n) != a(m) if n != m.
%F A350888 Let s(n) be the sequence of numbers such that A350887(s(n)) = 4 and s(n) is sorted in ascending order, then a(s(n)) has the ordinary generating function (2*(-7 + z))/(-1 + 31*z - 31*z^2 + z^3).
%F A350888 Let s(n) be the sequence of numbers such that A350887(s(n)) = 49 and s(n) is sorted in ascending order, then a(s(n)) has the ordinary generating function (-13 + z)/(-1 + 391*z - 391*z^2 + z^3).
%e A350888 a(1) = 14 and A350886(1) = 54, A350887(1) = 4:
%e A350888   54^2 = 1^2 + 15^2 + 29^2 + 43^2.
%e A350888 a(2) = 1 and A350886(2) = 70, A350887(2) = 24:
%e A350888   70^2 = 1^2 + 2^2 + 3^2 + ... + 23^2 + 24^2. This is the classic solution for the cannonball problem.
%o A350888 (PARI)
%o A350888 sqtest(n, c)={q=1; for(t=2, c, t+=n; q+=(t*t); if(issquare(q), break)); q}
%o A350888 z=500000; b=[; ]; for(n=0, z, r=sqtest(n, z); if(issquare(r), b=concat(b, [sqrtint(r); n+1]))); b=vecsort(b, 1); vector(#b, k, b[2,k]) \\ Last valid value for z=500000 is 5082.
%Y A350888 Cf. A350886 (square root of pyramid), A350887 (number of layers).
%Y A350888 Cf. A000330, A000447, A024215, A024381 (square pyramidal numbers for slope 1, 1/2, 1/3, 1/4).
%Y A350888 Cf. A076215, A001032, A134419, A106521. Some related problems.
%Y A350888 Cf. A186699.
%K A350888 nonn,more
%O A350888 1,1
%A A350888 _Thomas Scheuerle_, Feb 25 2022