A377205 -id:A377205 - cp's OEIS frontend

A377229 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(F(k)*a(k)) < 1, F = Fibonacci.

2, 3, 4, 9, 44, 1486, 1357976, 1855074754595, 2975714380792664939835466, 46528348836004781630107949818181021469921360198769

Offset: 1

Alois P. Heinz, Oct 20 2024

nonn

			s(0), s(1), ... = 0, 1/2, 5/6, 23/24, 215/216, 11879/11880, 17653679/17653680, ... .

Alois P. Heinz, Table of n, a(n) for n = 1..14

Cf. A000045, A079586, A374663, A376058, A377205, A377230.

F:= combinat[fibonacci]:
s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+1/(F(n)*a(n))) end:
a:= proc(n) option remember; 1+floor(1/((1-s(n-1))*F(n))) end:
seq(a(n), n=1..11);

A377230 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(T(k)*a(k)) < 1, T = A000217.

Original entry on oeis.org

2, 1, 2, 2, 3, 5, 23, 806, 519065, 220441054222, 222723684271305542570701, 41974171914555858099300698444579076459265512901, 1510140949639448391630842209382251970116940997822995817347241840058937174456186756365141648201

Offset: 1

Alois P. Heinz, Oct 20 2024

nonn

			s(0), s(1), ... = 0, 1/2, 5/6, 11/12, 29/30, 89/90, 629/630, ... .

Alois P. Heinz, Table of n, a(n) for n = 1..16

Cf. A000217, A374663, A377205, A377229.

T:= n-> n*(n+1)/2:
s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+1/(T(n)*a(n))) end:
a:= proc(n) option remember; 1+floor(1/((1-s(n-1))*T(n))) end:
seq(a(n), n=1..13);

cp's OEIS Frontend

A377229 Lexicographically earliest sequence of positive integers a(1), a(2), ... such that for any n >= 0, s(n) = Sum_{k=1..n} 1/(F(k)*a(k)) < 1, F = Fibonacci.

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