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


%S 0,0,0,0,1,2,7,10,22,29,51,61,99,115,163,192,262,287,385,428,528,600,

%T 730,780,963,1054,1202,1337,1545,1646,1908,2059,2269,2516,2770,2933,

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

%C Related to "Louisville's Last Theorem".

%H Robert Israel, <a href="/A191832/b191832.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 with(numtheory);

%p D00:=n->add(tau(j)*tau(n-j),j=1..n-1);

%p D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);

%p D000:=proc(n) local t1,i,j;

%p t1:=0;

%p for i from 1 to n-1 do

%p for j from 1 to n-1 do

%p if (i+j < n) then t1 := t1+numtheory:-tau(i)*numtheory:-tau(j)*numtheory:-tau(n-i-j); fi;

%p od; od;

%p t1;

%p end;

%p L5:=n->D000(n)/6+D00(n)+D01(n)/2+(2*n-1/6)*tau(n)-11*sigma[2](n)/6;

%p [seq(L5(n),n=1..60)];

%p # Alternate:

%p g:= proc(n,k,j) option remember;

%p if n < k-1 then 0

%p elif k = 2 then

%p if n mod j = 0 then 1 else 0 fi

%p else

%p add(procname(n-j*x,k-1,x), x=1 .. floor((n-k+2)/j))

%p fi

%p end proc:

%p f:= n -> add(g(n,6,j),j=1..n-4);

%p seq(f(n),n=1..100); # _Robert Israel_, Dec 02 2015

%Y Cf. A000005, A000203, A002133, A055507, A191822, A191829, A191831.

%K nonn

%O 1,6

%A _N. J. A. Sloane_, Jun 17 2011

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Last modified December 12 07:00 EST 2019. Contains 329948 sequences. (Running on oeis4.)