This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227847 #38 Mar 05 2025 02:05:19
%S A227847 0,1,1,1,2,2,2,6,10,31,77,206,568,1704,5037,15554
%N A227847 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-1} x_i + 2x_n)^2.
%C A227847 An n-tuple meeting the first condition is called an n-SCESS ("sum of cubes equals square of sum").
%C A227847 In other words, a(n) is the number of tuples (x_1, x_2, ..., x_n) satisfying SCESS such that (x_1, x_2, ..., x_{n-1}, x_n - 1, x_n + 1) also satisfies SCESS. - _Max Alekseyev_, Mar 04 2025
%C A227847 x_1 + x_2 + ... + x_{n-1} = A152948(x_n). - _Balarka Sen_, Aug 01 2013
%H A227847 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 A227847 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 A227847 A001055(n) <= a(n) <= A158649(n). - _Balarka Sen_, Aug 01 2013
%e A227847 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 A227847 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 A227847 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 A227847 (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 A227847 Cf. A001055, A152948, A158649, A225819.
%K A227847 more,nonn
%O A227847 1,5
%A A227847 _Jimmy Zotos_, Aug 01 2013
%E A227847 a(11)-a(15) from _Balarka Sen_, Aug 01 2013
%E A227847 a(16) from _Balarka Sen_, Aug 11 2013
%E A227847 Definition corrected by _Max Alekseyev_, Mar 04 2025