cp's OEIS Frontend

A121002 Numerators of partial sums of Catalan numbers scaled by powers of 1/5.

%I A121002 #8 Aug 30 2019 03:47:44
%S A121002 1,6,32,33,839,4237,21317,107014,4292,2687362,13453606,67326816,
%T A121002 336842092,336990672,1685488248,8429380209,42153972579,210795791853,
%U A121002 210814897401,5270725887663,26354942262399
%N A121002 Numerators of partial sums of Catalan numbers scaled by powers of 1/5.
%C A121002 Denominators are given under A121003.
%C A121002 This is the first member (p=0) of the second p-family of partial sums of normalized scaled Catalan series CsnII(p):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..infinity) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi) = F(2*p+1)*sqrt(5)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
%C A121002 The partial sums of the above mentioned second p-family are rII(p;n):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..n), n>=0, for p=0,1,...
%C A121002 For more details on this p-family and the other three ones see the W. Lang link under A120996.
%C A121002 The limit lim_{n->infinity} r(n) = (3 - phi) = (2*phi-1)/phi = 1.38196601125010 (maple10, 15 digits). This is the square of the dimensionless pentagon side length.
%H A121002 W. Lang: <a href="/A121002/a121002.txt">Rationals r(n), limit.</a>
%F A121002 a(n)=numerator(r(n)) with r(n) := rII(p=0,n) = sum(C(k)/5^k,k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
%e A121002 Rationals r(n): [1, 6/5, 32/25, 33/25, 839/625, 4237/3125,
%e A121002 21317/15625, 107014/78125, 4292/3125, 2687362/1953125,...].
%Y A121002 Cf. A120786 (numerators, second member p=1).
%K A121002 nonn,frac,easy
%O A121002 0,2
%A A121002 _Wolfdieter Lang_, Aug 16 2006