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A379979 Number of pairs (m,k), 1 <= m < k <= N such that there exists 1 <= x < y < k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2, N = A355812(n).

%I A379979 #11 Jan 08 2025 09:27:19
%S A379979 1,3,5,7,8,10,12,14,16,17,19,21,23,25,27,28,30,32,34,36,38,40,42,44,
%T A379979 45,47,49,51,53,55,57,60,62,64,65,67,69,71,75,76,78,82,84,86,89,91,93,
%U A379979 95,97,99,101,105,108,110,112,116,118,119,121,123,129,131,133,137,139
%N A379979 Number of pairs (m,k), 1 <= m < k <= N such that there exists 1 <= x < y < k-1 such that 1/x^2 - 1/y^2 = 1/m^2 - 1/k^2, N = A355812(n).
%C A379979 Partial sums of A379895.
%C A379979 Let S(N) = {1/x^2 - 1/y^2 : 1 <= x < y <= N}, then binomial(N,2) - a(n) is the size of |S(N)|, N = A355812(n). Note that S_N is the number of distinct energy differences within the first N energy levels of a hydrogen atom.
%H A379979 Jianing Song, <a href="/A379979/b379979.txt">Table of n, a(n) for n = 1..307</a> (corresponding to A355812(n) <= 1500)
%e A379979 a(3) = 5 since A355812(3) = 56, and there are 5 such pairs (m,k), 1 <= m < k <= 56:
%e A379979 (m,k) = (7,35): 1/5^2 - 1/7^2 = 1/7^2 - 1/35^2;
%e A379979 (m,k) = (11,55): 1/10^2 - 1/22^2 = 1/11^2 - 1/55^2;
%e A379979 (m,k) = (22,55): 1/10^2 - 1/11^2 = 1/22^2 - 1/55^2;
%e A379979 (m,k) = (8,56): 1/7^2 - 1/14^2 = 1/8^2 - 1/56^2;
%e A379979 (m,k) = (14,56): 1/7^2 - 1/8^2 = 1/14^2 - 1/56^2.
%e A379979 Correspondingly, the set {1/x^2 - 1/y^2 : 1 <= x < y <= 56} is of size binomial(56,2) - 5.
%o A379979 (PARI) b(n) = my(v=[], m2); for(y=1, n-1, for(x=1, y-1, m2=1/(1/x^2-1/y^2+1/n^2); if(m2==m2\1 && issquare(m2), v=concat(v, [m2])))); #Set(v) \\ #v gives A355813
%o A379979 my(s=0); for(n=1, 1500, if(b(n)>0, s+=b(n); print1(s, ", ")))
%Y A379979 Cf. A355812, A379895, A355813.
%K A379979 nonn
%O A379979 1,2
%A A379979 _Jianing Song_, Jan 07 2025