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A328343 Numbers k such that it is possible to find k consecutive squares whose sum is equal to the sum of two consecutive squares.

%I A328343 #22 Oct 24 2019 14:33:52
%S A328343 1,2,3,10,17,25,26,34,41,50,51,65,73,82,89,97,106,113,122,123,145,146,
%T A328343 169,170,178,185,194,218,219,241,250,257,267,274,281,291,298,305,314,
%U A328343 338,339,353,362,370,377,386,394,401,409,410,411,433,449,457,505,530,545
%N A328343 Numbers k such that it is possible to find k consecutive squares whose sum is equal to the sum of two consecutive squares.
%C A328343 The program generates solutions to the problem, but not necessarily all solutions. To exclude the existence of further solution you have to apply coset arithmetics (modulo operations).
%e A328343 k = 1: 3^2 + 4^2 = 5^2.
%e A328343 k = 2: 3^2 + 4^2 = 3^2 + 4^2.
%e A328343 k = 3: 13^2 + 14^2 = 10^2 + 11^2 + 12^2.
%e A328343 k = 10: 26^2 + 27^2 = 7^2 + ... + 16^2.
%e A328343 k = 17: 40^2 + 41^2 = 5^2 + ... + 21^2.
%e A328343 k = 25: 78^2 + 79^2 = 9^2 + ... + 33^2.
%e A328343 k = 26: 205^2 + 206^2 = 44^2 + ... + 49^2.
%e A328343 k = 34: 856^2 + 857^2 = 191^2 + ... + 224^2.
%e A328343 k = 41: 3029^2 + 3030^2 = 649^2 + ... + 689^2.
%e A328343 k = 50: 146^2 + 147^2 = 1^2 + ... + 50^2.
%e A328343 k = 51: 210^2 + 211^2 = 14^2 + ... + 64^2.
%e A328343 k = 65: 236^2 + 237^2 = 5^2 + ... + 69^2.
%e A328343 k = 73: 278^2 + 279^2 = 5^2 + ... + 76^2.
%e A328343 k = 82: 1070^2 + 1071^2 = 125^2 + ... + 206^2.
%e A328343 k = 89: 147445^2 + 147446^2 = 22059^2 + ... + 22147^2.
%e A328343 k = 97: 544^2 + 545^2 = 25^2 + ... + 121^2.
%t A328343 Select[Range[60], {} != FindInstance[ Sum[t^2, {t, x, x+#-1}] == y^2 + (y + 1)^2, {x, y}, Integers] &] (* _Giovanni Resta_, Oct 23 2019 *)
%o A328343 (Python)
%o A328343 import math
%o A328343 for n in range(1, 100):
%o A328343    for b in range(1, 10000000):
%o A328343       d = (6*b*b*(n+1)+6*b*n*(n+1)+2*n*n*n+3*n*n+n)
%o A328343       w = int((math.sqrt(d/6)))
%o A328343       a = w
%o A328343       if 6*a*a-6*b*b*(n+1)-6*b*n*(n+1)-2*n*n*n-3*n*n-n == 0:
%o A328343          print(a,b,n+1)
%o A328343       a = w+1
%o A328343       if 6*a*a-6*b*b*(n+1)-6*b*n*(n+1)-2*n*n*n-3*n*n-n == 0:
%o A328343          print(a,b,n+1)
%o A328343       a = w-1
%o A328343       if 6*a*a-6*b*b*(n+1)-6*b*n*(n+1)-2*n*n*n-3*n*n-n == 0:
%o A328343          print(a,b,n+1)
%Y A328343 Cf. A001032, A185545.
%K A328343 nonn
%O A328343 1,2
%A A328343 _Reiner Moewald_, Oct 13 2019
%E A328343 a(17)-a(38) from _Jon E. Schoenfield_, Oct 22 2019
%E A328343 a(39)-a(57) from _Jon E. Schoenfield_ and _Giovanni Resta_, Oct 23 2019