cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A333089 a(n) is the denominator of Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)).

Original entry on oeis.org

1, 18, 960, 76440, 37437400, 157024707840, 2777798704721040, 4169982785629476816, 331259342780844858796416, 743322803326470921519628462800, 163037651356772148158514292729628880, 187555796967791569325602741834073910082560, 3838658658324493911932517275499048601188128008800
Offset: 1

Views

Author

A.H.M. Smeets, Mar 07 2020

Keywords

Comments

The numerators are given in A333088.
Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)) is a fraction for n > 0.
Sum_{i > 0} 1/Fibonacci(i)^2, i.e., the n = 0 case, is known to be transcendental. See A105393.
Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n-1)) is an irrational number for n > 0. See for instance A290565 (n = 1).

Examples

			These infinite sums begin: 1, 7/18, 143/960, ...

Links

Crossrefs

Cf. A105393, A290565, A333088 (numerator).

Programs

  • Mathematica
    a[n_] := Denominator[Sum[1/(Fibonacci[2i-1]*Fibonacci[2i]),{i,1,n}] / Fibonacci[2n]]; Array[a, 13] (* Amiram Eldar, Mar 10 2020 *)
  • PARI
    a(n) = denominator(sum(i=1, n, 1/(fibonacci(2*i-1)*fibonacci(2*i)))/ fibonacci(2*n)); \\ Michel Marcus, Mar 10 2020
  • Python
    from math import gcd
f0, f1, snum, sden, n = 1, 1, 0, 1, 0
while n < 13:
    snum, sden, n = f0*f1*snum+sden, sden*f0*f1, n+1
    d = gcd(snum, sden*f0)
    print(n, sden*f0//d)
    f0, f1 = 2*f0+f1, f0+f1 # A.H.M. Smeets, May 16 2020

Formula

a(n) = denominator of (1/Fibonacci(2n)) * Sum_{0 < i <= n} 1/(Fibonacci(2i-1)*Fibonacci(2i)).

A333209 a(n) is the denominator of Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)), with Lucas(i) as defined in A000032.

Original entry on oeis.org

2, 36, 7392, 1688148, 197412831, 21085413226416, 101768454084335346, 60343478516053297339236, 73240105330540144095414793632, 1956470757376233684880813258936380492, 32802418997525523144166495047229414174839, 202042966989952174292936124782341088713724476716231
Offset: 1

Views

Author

A.H.M. Smeets, Mar 11 2020

Keywords

Comments

The numerators are given in A333208.
See A333088 and A333089 for similar fractions for infinite sums of Fibonacci numbers.
Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)) is a fraction for n > 0.
Sum_{i >= 0} 1/Lucas(i)^2 = 1/4 + A105394, i.e., the n = 0 case, is believed to be transcendental.

Examples

			These infinite sums begin: 1/2, 7/36, 551/7392, ...

Links

Crossrefs

Cf. A000032, A105394, A132318, A333088, A333089, A333208.

Programs

  • Mathematica
    a[n_] := Denominator[Sum[1/(LucasL[2 i - 2]*LucasL[2 i - 1]), {i, 1, n}]/Fibonacci[2 n]]; Array[a, 12] (* Amiram Eldar, Mar 11 2020 *)
  • Python
    from math import gcd
f0, f1, g0, g1, snum, sden, n = 1, 1, 1, 2, 0, 1, 0
while n < 12:
    n = n+1
    snum, sden = g0*g1*snum+sden, sden*g0*g1
    d = gcd(snum,sden*f0)
    print(n,sden*f0//d)
f0, f1, g0, g1 = 2*f0+f1, f0+f1, 2*g0+g1, g0+g1 # A.H.M. Smeets, Nov 30 2020
  • Python
    from math import gcd
f0, f1, g0, g1, snum, sden, n = 1, 1, 1, 2, 0, 1, 0
while n < 12:
    n = n+1
    snum, sden = g0*g1*snum+sden, sden*g0*g1
    d = gcd(snum,sden*f0)
    print(n,sden*f0//d)
    f0, f1, g0, g1 = 2*f0+f1, f0+f1, 2*g0+g1, g0+g1 # A.H.M. Smeets, Nov 30 2020

Formula

a(n) = denominator of (1/Fibonacci(2n)) * Sum_{i = 1..n} 1/(Lucas(2i-2)*Lucas(2i-1)).
Lim_{n -> inf} (A333208(n)/a(n)) / (A333208(n-1)/a(n-1)) = 1 - 1/phi = 1/phi^2 = A132318.
The following generalization holds: (Start)
Let H_(a,b) (n) be defined by H_(a,b) (0) = a, H_(a,b) (1) = b and H_(a,b) (n) = H_(a,b) (n-1) + H_(a,b) (n-2) for n > 1, then
Sum_{i >= 0} 1/(H_(a,b) (i)*H_(a,b) (i+2n)) = (1/Fibonacci(2n)) * Sum_{i=1..n} 1/(H_(a,b) (2i-2)*H_(a,b) (2i-1)) for n > 0, and are thus all fractions. Specially, H_(0,1) are the Fibonacci numbers A000045, H_(2,1) as here, are the Lucas numbers A000032, and H_(3,1) are the Pibonacci numbers A104449. (End)

A333208 a(n) is the numerator of Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)), with Lucas(i) as defined in A000032.

Original entry on oeis.org

1, 7, 551, 48091, 2148268, 87644575267, 161577754532123, 36595152483523582367, 16965509829762630129638831, 173107561150078104051618631740949, 1108595900580419409151086339986148307, 2608169750203411467722731179728125652086612772
Offset: 1

Views

Author

A.H.M. Smeets, Mar 11 2020

Keywords

Comments

The denominators are given in A333209.
See A333088 and A333089 for similar fractions for infinite sums of Fibonacci numbers.
Sum_{i >= 0} 1/(Lucas(i)*Lucas(i+2n)) is a fraction for n > 0.
Sum_{i >= 0} 1/Lucas(i)^2 = 1/4 + A105394, i.e., the n = 0 case, is believed to be transcendental.

Examples

			These infinite sums begin: 1/2, 7/36, 551/7392, ...

Crossrefs

Cf. A000032, A105394, A132318, A333088, A333089, A333209.

Programs

  • Mathematica
    a[n_] := Numerator[Sum[1/(LucasL[2 i - 2]*LucasL[2 i - 1]), {i, 1, n}]/Fibonacci[2 n]]; Array[a, 12] (* Amiram Eldar, Mar 11 2020 *)

Formula

a(n) = numerator of (1/Fibonacci(2n)) * Sum_{i=1..n} 1/(Lucas(2i-2)*Lucas(2i-1)).
Lim_{n -> inf} (a(n)/A333209(n)) / (a(n-1)/A333209(n-1)) = 1 - 1/phi = 1/phi^2 = A132318.
The following generalization holds: (Start)
Let H_(a,b) (n) be defined by H_(a,b) (0) = a, H_(a,b) (1) = b and H_(a,b) (n) = H_(a,b) (n-1) + H_(a,b) (n-2) for n > 1, then
Sum_{i >= 0} 1/(H_(a,b) (i)*H_(a,b) (i+2n)) = (1/Fibonacci(2n)) * Sum_{i=1..n} 1/(H_(a,b) (2i-2)*H_(a,b) (2i-1)) for n > 0, and are thus all fractions. Specially, H_(0,1) are the Fibonacci numbers A000045, H_(2,1) as here, are the Lucas numbers A000032, and H_(3,1) are the Pibonacci numbers A104449. (End)
Showing 1-3 of 3 results.

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