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Number of tuples (x_1, x_2, ..., x_n) with 1 <= x_1 <= x_2 <= ... <= x_n such that Sum_{i=1..n} x_i^3 = (Sum_{i=1..n} x_i)^2 and Sum_{i=1..n-1} x_i^3 + (x_n-1)^3 + (x_n+1)^3 = (Sum_{i=1..n} x_i + 2x_n)^2.
1

%I #30 Oct 21 2017 16:22:12

%S 0,1,1,1,2,2,2,6,10,31,77,206,568,1704,5037,15554

%N Number of tuples (x_1, x_2, ..., x_n) with 1 <= x_1 <= x_2 <= ... <= x_n such that Sum_{i=1..n} x_i^3 = (Sum_{i=1..n} x_i)^2 and Sum_{i=1..n-1} x_i^3 + (x_n-1)^3 + (x_n+1)^3 = (Sum_{i=1..n} x_i + 2x_n)^2.

%C An n-tuple meeting the first condition is called an n-SCESS ("sum of cubes equals square of sum").

%C x_1 + x_2 + ... + x_{n-1} = A152948(x_n). - _Balarka Sen_, Aug 01 2013

%H Edward Barbeau and Samer Seraj, <a href="http://arxiv.org/abs/1306.5257">Sum of cubes is square of sum</a>, arXiv:1306.5257 [math.NT], 2013.

%H John Mason, <a href="http://www.jstor.org/stable/3620469">Generalising 'sums of cubes equal to squares of sums'</a>, The Mathematical Gazette 85:502 (2001), pp. 50-58.

%F A001055(n) <= a(n) <= A158649(n). - _Balarka Sen_, Aug 01 2013

%e a(3) = 1 since the only 3-SCESS is (1, 2, 3) for which the corresponding ordered tuple (1, 2, 2, 4) satisfy the SCESS property. (See Mason et al.)

%e a(5) = 2 since the only 5-SCESS are (1, 2, 2, 3, 5) and (3, 3, 3, 3, 6) for which the corresponding ordered tuples (1, 2, 2, 3, 4, 6) and (3, 3, 3, 3, 5, 7) satisfy the SCESS property.

%e a(8) = 6 since the only 8-SCESS are (1, 1, 2, 4, 5, 5, 5, 8), (1, 2, 2, 3, 4, 5, 6, 8), (2, 2, 4, 4, 6, 6, 6, 9), (2, 6, 6, 6, 6, 6, 6, 10), (3, 3, 3, 3, 5, 6, 7, 9) and (3, 5, 5, 5, 6, 7, 7, 10) for which the corresponding ordered tuples (1, 1, 2, 4, 5, 5, 5, 7, 9), (1, 2, 2, 3, 4, 5, 6, 7, 9), (2, 2, 4, 4, 6, 6, 6, 8, 10), (2, 6, 6, 6, 6, 6, 6, 9, 11), (3, 3, 3, 3, 5, 6, 7, 8, 10) and (3, 5, 5, 5, 6, 7, 7, 9, 11) satisfy the SCESS property.

%o (PARI) a(n)=my(v=vector(n, i, 1), N=n^(4/3), k); while(v[#v]<N, v[1]++; if(v[1]>N, for(i=2, N, if(v[i]<N, v[i]++; for(j=1, i-1, v[j]=v[i]); break))); if(sum(i=1, n, v[i]^3)==sum(i=1, n, v[i])^2 && sum(i=2,n,v[i]^3)+(v[1]-1)^3+(v[1]+1)^3==(sum(i=2,n,v[i])+2*v[1])^2, k++));k /* _Balarka Sen_, Aug 01 2013 */

%Y Cf. A158649, A225819.

%K more,nonn

%O 1,5

%A _Jimmy Zotos_, Aug 01 2013

%E a(11)-a(15) from _Balarka Sen_, Aug 01 2013

%E a(16) from _Balarka Sen_, Aug 11 2013