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A346163 Numbers k such that there exist equal sums of k and 2k consecutive positive squares.

%I A346163 #29 Nov 05 2024 18:53:53
%S A346163 1,17,23,25,49,55,71,73,79,89,95,103,113,127,143,161,167,175,185,191,
%T A346163 193,199,215,217,233,239,241,265,271,287,289,305,361,377,391,409,415,
%U A346163 431,433,457,473,481,505,511,521,535,545,553,569,593,599,617,631,647
%N A346163 Numbers k such that there exist equal sums of k and 2k consecutive positive squares.
%C A346163 a(n) is congruent to 1 or 5 (mod 6).
%C A346163 a(n) is not congruent to 3, 4 or 5 (mod 8) or to 7, 11, 16 or 20 (mod 27), see Alder and Alfred.
%C A346163 k is in the sequence if the quadratic Diophantine equation 6*(k*x^2 - 2*k*y^2 + k*(k-1)*x + 2*k*(1-2*k)*y) - 14*k^3 + 9*k^2 - k = 0 has solutions x, y in the positive integers.
%H A346163 H. L. Alder and U. Alfred, <a href="https://www.jstor.org/stable/2310889">n and n+k Consecutive Integers with Equal Sums of Squares</a>, The American Mathematical Monthly, 71:7 (1964), 749-754.
%H A346163 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A346163 a(1): 5^2 = 3^2 + 4^2. Here the left-hand side has k = 1 term, and the right-hand side has 2k = 2 terms. Hence k = 1 is in the sequence.
%e A346163 a(2): 29^2 + 30^2 + ... + 44^2 + 45^2 = 8^2 + 9^2 + ... + 40^2 + 41^2 = 23681. Here the left and right sums have k = 17 and 2k = 34 terms, respectively. Hence k = 17 is in the sequence.
%o A346163 (Python)
%o A346163 import sympy # Version 1.8
%o A346163 xx, yy = sympy.symbols("x y")
%o A346163 def pyramidal(n):
%o A346163     return n*(n+1)*(2*n+1)/6 # A000330(n)
%o A346163 def expanded_diophantine(k,n):
%o A346163     left_hand_side =  pyramidal(xx+n-1) - pyramidal(xx-1)
%o A346163     right_hand_side =  pyramidal(yy+n+k-1) - pyramidal(yy-1)
%o A346163     return sympy.expand(right_hand_side-left_hand_side)
%o A346163 def has_solutions(k,n):
%o A346163     return len(sympy.solvers.diophantine(expanded_diophantine(k,n))) != 0
%o A346163 def k_in_a346163(k):
%o A346163     return has_solutions(k,k)
%Y A346163 Cf. A000330, A001032, A130052.
%K A346163 nonn
%O A346163 1,2
%A A346163 _Johan Westin_, Jul 08 2021