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%I #39 Sep 30 2017 04:40:32
%S 5,13,14,25,26,27,41,42,43,44,61,62,63,64,65,85,86,87,88,89,90,113,
%T 114,115,116,117,118,119,145,146,147,148,149,150,151,152,181,182,183,
%U 184,185,186,187,188,189,221,222,223,224,225,226,227,228,229,230,265
%N Take 0, skip 3 * 1 + 1, take 1, skip 3 * 2 + 1, take 2, skip 3 * 3 + 1, ...
%H Michael Boardman, <a href="http://www.jstor.org/stable/2691496">Proof Without Words: Pythagorean Runs</a>, Math. Mag., 73 (2000), 59.
%F Sum_{n = (k-1)*k/2+1 .. k*(k+1)/2} a(n)^2 = Sum_{n = k*(k+1)/2 .. (k+1)*(k+2)/2-1} A292564(n)^2 = A059255(k) for k > 0.
%F a(n) = n + 4 + (3k^2 + 11k)/2 where k = floor((sqrt(2*n) - 1/2)). - _Jon E. Schoenfield_, Sep 30 2017
%e k| A292564(n)^2 | a(n)^2 | Sum
%e --------------------------------------------------------------------------------
%e 0| 0^2 (= 0)
%e 1| 3^2 + 4^2 = 5^2 (= 25)
%e 2| 10^2 + 11^2 + 12^2 = 13^2 + 14^2 (= 365)
%e 3| 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2 (= 2030)
%e 4| 36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2 (= 7230)
%e | ...
%e Row 3 is proved by the following:
%e (25^2 - 24^2) + (26^2 - 23^2) + (27^2 - 22^2) = 49*1 + 49*3 + 49*5 = 7^2*3^2 = 21^2.
%e Row k is proved by the same way.
%t Block[{s = Array[{# - 1, 3 # + 1} &, 12], r}, r = Range@ Total@ Flatten@ s; Map[Function[{a, b}, {First@ #, Set[r, Drop[Last@ #, b]]} &@ TakeDrop[r, a]] @@ # &, s][[All, 1]] // Flatten] (* _Michael De Vlieger_, Sep 25 2017 *)
%Y Cf. A000217, A059255, A063657, A292564.
%K nonn,easy
%O 1,1
%A _Seiichi Manyama_, Sep 19 2017
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