login
a(1)=1; a(n) is the smallest integer > a(n-1) such that the sum of elements of the simple continued fraction for S(n)=1/a(1)+1/a(2)+...+1/a(n) equals n^2.
2

%I #8 Mar 30 2012 18:39:00

%S 1,3,27,51,135,138,186,187,352,479,525,923,932,1286,1578,1807,1886,

%T 1926,2816,3358,3438,5727,50152,156816,204512,213094,221368,255348,

%U 257350,329046,403004,434010,442520,449599,450069,452430,457524,459665

%N a(1)=1; a(n) is the smallest integer > a(n-1) such that the sum of elements of the simple continued fraction for S(n)=1/a(1)+1/a(2)+...+1/a(n) equals n^2.

%e 1/a(1)+1/a(2)+1/a(3)+1/a(4) = (1+1/3+1/27+1/51) which continued fraction is {1, 2, 1, 1, 3, 2, 1, 1, 4} and 1+2+1+1+3+2+1+1+4 = 16 = 4^2.

%t a[1] = 1; a[n_] := a[n] = (s = Sum[1/a[i], {i, 1, n - 1}]; While[Plus @@ ContinuedFraction[s + 1/k] != n^2, k++ ]; k); k = 1; Do[ Print[ a[n]], {n, 1, 40}]

%o (PARI) s=1; t=1; for(n=2,40,s=s+1/t; while(abs(n^2+1-sum(i=1,length(contfrac(s+1/t)), component(contfrac(s+1/t), i)))>0,t++); print1(t,","))

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Jun 10 2002

%E Edited by _Robert G. Wilson v_, Jun 17 2002