cp's OEIS Frontend

A191832 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 = n, with all xi >= 1.

%I A191832 #27 Sep 25 2020 08:40:53
%S A191832 0,0,0,0,1,2,7,10,22,29,51,61,99,115,163,192,262,287,385,428,528,600,
%T A191832 730,780,963,1054,1202,1337,1545,1646,1908,2059,2269,2516,2770,2933,
%U A191832 3298,3568,3792,4142,4493,4786,5183,5562,5831,6423,6745,7140,7639,8231,8479,9216,9603,10260,10663,11488,11752,12838,13100,13887
%N A191832 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 = n, with all xi >= 1.
%C A191832 Related to "Liouville's Last Theorem".
%H A191832 Robert Israel, <a href="/A191832/b191832.txt">Table of n, a(n) for n = 1..1000</a>
%H A191832 George E. Andrews, <a href="http://dx.doi.org/10.1007/BF01608779">Stacked lattice boxes</a>, Ann. Comb. 3 (1999), 115-130. See L_5(n).
%p A191832 with(numtheory);
%p A191832 D00:=n->add(tau(j)*tau(n-j),j=1..n-1);
%p A191832 D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);
%p A191832 D000:=proc(n) local t1,i,j;
%p A191832 t1:=0;
%p A191832 for i from 1 to n-1 do
%p A191832 for j from 1 to n-1 do
%p A191832 if (i+j < n) then t1 := t1+numtheory:-tau(i)*numtheory:-tau(j)*numtheory:-tau(n-i-j); fi;
%p A191832 od; od;
%p A191832 t1;
%p A191832 end;
%p A191832 L5:=n->D000(n)/6+D00(n)+D01(n)/2+(2*n-1/6)*tau(n)-11*sigma[2](n)/6;
%p A191832 [seq(L5(n),n=1..60)];
%p A191832 # Alternate:
%p A191832 g:= proc(n,k,j) option remember;
%p A191832      if n < k-1 then 0
%p A191832      elif k = 2 then
%p A191832         if n mod j = 0 then 1 else 0 fi
%p A191832      else
%p A191832         add(procname(n-j*x,k-1,x), x=1 .. floor((n-k+2)/j))
%p A191832      fi
%p A191832 end proc:
%p A191832 f:= n -> add(g(n,6,j),j=1..n-4);
%p A191832 seq(f(n),n=1..100); # _Robert Israel_, Dec 02 2015
%t A191832 g[n_, k_, j_] := g[n, k, j] = If[n < k - 1, 0, If[k == 2, If[ Mod[n, j] == 0, 1, 0], Sum[g[n - j x, k - 1, x], {x, 1, Floor[(n - k + 2)/j]}]]];
%t A191832 f[n_] := Sum[g[n, 6, j], {j, 1, n - 4}];
%t A191832 Array[f, 100] (* _Jean-François Alcover_, Sep 25 2020, after _Robert Israel_ *)
%Y A191832 Cf. A000005, A000203, A002133, A055507, A191822, A191829, A191831.
%K A191832 nonn
%O A191832 1,6
%A A191832 _N. J. A. Sloane_, Jun 17 2011