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%I #20 Jun 25 2013 22:05:54
%S 5,18,53,102,197,306,491,684,989,1290,1745,2178,2813,3402,4247,5016,
%T 6101,7074,8429,9630,11285,12738,14723,16452,18797,20826,23561,25914,
%U 29069,31770,35375,38448,42533,46002,50597,54486,59621,63954,69659,74460,80765,86058
%N Shiraishi numbers: a parametrized family of solutions c to the Diophantine equation a^3 + b^3 + c^3 = d^3 with d = c+1.
%C Shiraishi's solutions to a^3 + b^3 + c^3 = d^3 are a = 3n^2; b = 6n^2 - 3n + 1 or 6n^2 + 3n + 1; c = 9n^3 - 6n^2 + 3n - 1 or 9n^3 + 6n^2 + 3n, respectively, for n > 0; and d = c+1. See Smith and Mikami for a derivation.
%C Shiraishi's formulas show that the sequence is infinite. Hence the sequences A023042 (solutions to x^3 + y^3 + z^3 = w^3), A225908 (solutions to a^3 + b^3 = c^3 - d^3), A225909 (solutions to a^3 + b^3 = (c+1)^3 - c^3) and A226902 (numbers c in A225909) are also infinite.
%C Shiraishi's solution b = 6n^2 +/- 3n + 1 is the centered triangular numbers A005448 except 1.
%D Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.
%H David Eugene Smith and Yoshio Mikami, <a href="http://archive.org/details/historyofjapanes00smitiala">A History of Japanese Mathematics</a>, Open Court, Chicago, 1914; Dover reprint, 2004; pp. 233-235.
%H Wikipedia (French), <a href="http://fr.wikipedia.org/wiki/Shiraishi_Nagatada">Shiraishi Nagatada</a>
%H Wikipedia (German), <a href="http://de.wikipedia.org/wiki/Shiraishi_Nagatada">Shiraishi Nagatada</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F a(2n-1) = 9n^3 - 6n^2 + 3n - 1.
%F a(2n) = 9n^3 + 6n^2 + 3n.
%F G.f.: x*(5 + 13*x + 20*x^2 + 10*x^3 + 5*x^4 + x^5) / ((1 + x)^3*(1 - x)^4). [_Bruno Berselli_, Jun 22 2013]
%F a(n) = (18*n^3 + 27*n^2 + 27*n + 1 - (3*n^2 + 3*n + 1)*(-1)^n)/16. [_Bruno Berselli_, Jun 22 2013]
%e The first two terms are a(1) = 9 - 6 + 3 - 1 = 5 and a(2) = 9 + 6 + 3 = 18. Then Shiraishi's formulas give 3^3 + 4^3 + 5^3 = 6^3 and 3^3 + 10^3 + 18^3 = 19^3.
%Y Cf. A003325, A005448, A023042, A181123, A225908, A225909, A226902.
%K nonn,easy
%O 1,1
%A _Jonathan Sondow_, Jun 22 2013
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