A138744 - OEIS

A138744

Let r_1 = 1. Let r_{m+1} = r_1 + 1/(r_2 + 1/(r_3 +...(r_{m-1} + 1/r_m)...)), a continued fraction of rational terms. Then a(n) is the sum of the (positive integer) terms in the simple continued fraction of r_n.

%I #16 Oct 10 2022 07:55:33

%S 1,1,2,4,8,33,128,109,344,3760,1829,18367,11168,35246,41103,79356,

%T 151643,344725,1249071,1678788,5385320,19780986,17348076,30966961,

%U 85647848,160394455,451333739,623813606

%N Let r_1 = 1. Let r_{m+1} = r_1 + 1/(r_2 + 1/(r_3 +...(r_{m-1} + 1/r_m)...)), a continued fraction of rational terms. Then a(n) is the sum of the (positive integer) terms in the simple continued fraction of r_n.

%C This sequence is the sum of terms in the n-th row of irregular array A138742.

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A138742%2B3%2B4.py">A138742+3+4.py</a>

%e r_5 = 31/18, for instance, equals the simple continued fraction 1+ 1/(1 + 1/(2 + 1/(1 + 1/(1 +1/2)))). The integer terms in this continued fraction are (1,1,2,1,1,2); so a(5) = 1+1+2+1+1+2 = 8.

%Y Cf. A138742, A138743.

%K nonn,more

%O 1,3

%A _Leroy Quet_, Mar 27 2008

%E a(7)-a(28) from _Lucas A. Brown_, Apr 12 2021