cp's OEIS Frontend

A350887 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 Y sorted by X.

%I A350887 #49 Dec 23 2024 11:34:44
%S A350887 4,24,4,64,49,9,4,50,484,3249,81,361,49,4,289,64,16,3938,5041,4,36,
%T A350887 568,441,121,4761,33,1936,9,49,25872,4,64,8257,24,361,12024
%N A350887 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 Y sorted by X.
%C A350887 A generalization of the cannonball problem for pyramids with a slope of 1/A350888(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. a(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 A350887 This sequence contains only numbers which appear in A186699, too. A number which is in A186699, does not appear in this sequence if it is of the form: 2^m*p where p is a prime of the form 12*k+1 or 12*k-1. There are also some perfect squares like 25 excluded, as these would only provide valid solutions in the case of A350888(n) = 0, which is not part of the sequence definition.
%H A350887 Thomas Scheuerle, <a href="/A350887/a350887.txt">Some solutions to this problem sorted by a(n)</a>.
%H A350887 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 A350887 Thomas Scheuerle, <a href="/A350886/a350886.txt">Recursive solution formulas</a>.
%H A350887 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A350887 A350886(n)^2 = A350888(n)^2*binomial(2*a(n), 3)/4 + 2*A350888(n)*binomial(a(n), 2) + a(n).
%F A350887 A350886(n)^2 = c*((b*(4*b*c^2-(6*c-2)*b + 12*(c-1)) + 12)/12), with c = a(n) and b = A350888(n). Expanded to see factors more clearly.
%F A350887 A350886(n)^2 = c*b^2*( ((1/b) + (c-1)/2)^2 + (c^2-1)/12 ). Shorter form of above.
%e A350887 a(1) = 4 and A350886(1) = 54, A350888(1) = 14: 54^2 = 1^2 + 15^2 + 29^2 + 43^2.
%e A350887 a(2) = 24 and A350886(2) = 70, A350888(2) = 1: 70^2 = 1^2 + 2^2 + 3^2 + ... + 23^2 + 24^2. This is the classic solution for the cannonball problem.
%o A350887 (PARI)
%o A350887 sqtest(n, c)={q=1; r=1; for(t=2, c, t+=n; r+=1; q+=(t*t); if(issquare(q), break)); [q, r]}
%o A350887 z=500000; b=[;]; for(n=0,z,r=sqtest(n,z); if(issquare(r[1]), b=concat(b,[sqrtint(r[1]); r[2]]))); b=vecsort(b,1); vector(#b, k, b[2, k]) \\ Last valid value for z=500000 is 49.
%Y A350887 Cf. A350886 (squareroot of pyramid), A350888 (denominator of slope).
%Y A350887 Cf. A000330, A000447, A024215, A024381 (square pyramidal numbers for slope 1, 1/2, 1/3, 1/4).
%Y A350887 Cf. A076215, A001032, A134419, A106521. Some related problems.
%Y A350887 Cf. A186699.
%K A350887 nonn,more
%O A350887 1,1
%A A350887 _Thomas Scheuerle_, Feb 25 2022