A376059
a(n) is the denominator of the sum S(n) defined in A376058.
Original entry on oeis.org
1, 2, 6, 78, 18330, 1679962830, 22578200883132834030, 6627077016548303724729207245056971365730, 922281145448518091883798423085535218757314338662318933097843039655721026758456630
Offset: 0
The first few values of S(n) are 0, 1/2, 5/6, 77/78, 18329/18330, 1679962829/1679962830, 22578200883132834029/22578200883132834030, ...
-
RecurrenceTable[{a[n+1] == Fibonacci[n+1]*a[n]^2 + a[n], a[0] == 1}, a, {n, 0, 8}] (* Amiram Eldar, Sep 15 2024 *)
A376056
Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} (2*k-1)/a(k) < 1.
Original entry on oeis.org
2, 7, 71, 6959, 62255215, 4736981006316791, 26518805245879857416837904442871, 811438882694890436523185183518581584358651922339197834228784351
Offset: 1
-
# Given a sequence b(1), b(2), b(3), ... of nonnegative real numbers, this program computes the first M terms of the lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... with the property that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1.
# For the present sequence we set b(k) = 2*k - 1.
b := Array(0..100,-1); a := Array(0..100,-1); S := Array(0..100,-1); d := Array(0..100,-1);
for k from 1 to 100 do b[k]:=2*k-1; od:
M:=8;
S[0] := 0; d[0] := 1;
for n from 1 to M do
a[n] := floor(b[n]/d[n-1])+1;
S[n] := S[n-1] + b[n]/a[n];
d[n] := 1 - S[n];
od:
La:=[seq(a[n],n=1..M)]; # the present sequence
Ls:=[seq(S[n],n=1..M)]; # the sums S(n)
Lsn:=[seq(numer(S[n]),n=1..M)];
Lsd:=[seq(denom(S[n]),n=1..M)]; # A376057
Lsd-Lsn; # As a check, by the above theorem, this should (and does) produce the all-1's sequence
# Some small changes to the program are needed if the starting sequence {b(n)} has offset 0, as for example in the case of the Fibonacci or Catalan numbers (see A376058-A376061).
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.
Original entry on oeis.org
2, 3, 4, 9, 44, 1486, 1357976, 1855074754595, 2975714380792664939835466, 46528348836004781630107949818181021469921360198769
Offset: 1
s(0), s(1), ... = 0, 1/2, 5/6, 23/24, 215/216, 11879/11880, 17653679/17653680, ... .
-
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);
Showing 1-3 of 3 results.
Comments