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.

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A254795 Numerators of the convergents of the generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))).

Original entry on oeis.org

2, 9, 54, 441, 4410, 53361, 747054, 12006225, 216112050, 4334247225, 95353438950, 2292816782025, 59613236332650, 1671463434096225, 50143903022886750, 1606276360166472225, 54613396245660055650, 1967688541203928475625, 74772164565749282073750
Offset: 0

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Author

Peter Bala, Feb 23 2015

Keywords

Comments

Brouncker gave the generalized continued fraction expansion 4/Pi = 1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... ))). The sequence of convergents begins [1/1, 3/2, 15/13, 105/76, ... ]. The numerators of the convergents are in A001147, the denominators in A024199.
In extending Brouckner's result, Osler showed that 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) = L^2/Pi, where L is the lemniscate constant A062539. The sequence of convergents to Osler's continued fraction begins [2/1, 9/4, 54/25, 441/200, 4410/2025, ...]. Here we list the (unreduced) numerators of these convergents. See A254796 for the sequence of denominators. See A254794 for the decimal expansion of L^2/Pi.

Links

Crossrefs

Cf. A008545, A001147, A024199, A142970, A254794, A254796.

Programs

  • Maple
    a[0] := 2: a[1] := 9:
for n from 2 to 18 do a[n] := 4*a[n-1] + (2*n-1)^2*a[n-2] end do:
seq(a[n], n = 0 .. 18);

Formula

a(2*n-1) = ( A008545(n) )^2 = ( Product {k = 0..n-1} 4*k + 3 )^2.
a(2*n) = (4*n + 2)*( A008545(n) )^2 = (4*n + 2)*( Product {k = 0..n-1} 4*k + 3 )^2.
a(n) = 4*a(n-1) + (2*n - 1)^2*a(n-2) with a(0) = 2, a(1) = 9.
a(2*n) = (4*n + 2)*a(2*n-1); a(2*n+1) = (4*n + 4)*a(2*n) + a(2*n-1).

A254796 Denominators of the convergents of the generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))).

Original entry on oeis.org

1, 4, 25, 200, 2025, 24300, 342225, 5475600, 98903025, 1978060500, 43616234025, 1046789616600, 27260146265625, 763284095437500, 22925783009390625, 733625056300500000, 24966177697226390625, 898782397100150062500, 34178697267502928765625
Offset: 0

Views

Author

Peter Bala, Feb 23 2015

Keywords

Comments

The generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) represents the constant L^2/Pi = 2.188439... = A254794, where L is the lemniscate constant A062539. See A254795 for the numerators of the convergents of the continued fraction.

Examples

			54/25 = 2.16, 441/200 = 2.205 etc approach 2.188..

Links

Crossrefs

Cf. A007696, A024199, A062539, A254794, A254795.

Programs

  • Magma
    I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1)+(2*n-3)^2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 24 2015
  • Maple
    a[0] := 1: a[1] := 4:
for n from 2 to 18 do a[n] := 4*a[n-1] + (2*n-1)^2*a[n-2] end do:
seq(a[n], n = 0 .. 18);
  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 4, a[n] == 4 a[n - 1] + (2 n - 1)^2 a[n - 2]}, a, {n, 20}] (* Vincenzo Librandi, Feb 24 2015 *)

Formula

a(2*n) = A007696(n+1)^2 = ( Product {k = 0..n} 4*k + 1 )^2.
a(2*n-1) = 4*n*A007696(n)^2 = 4*n * ( Product {k = 0..n-1} 4*k + 1 )^2.
a(n) = 4*a(n-1) + (2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 4.
a(2*n+1) = 4*(n + 1)*a(2*n); a(2*n) = (4*n + 2)*a(2*n-1) + a(2*n-2).
Empirical e.g.f.: ((-Q(1/2, -3)-Q(-1/2, -3))*P(1/2, (2*x+3)/(2*x-1))+Q(1/2, (2*x+3)/(2*x-1))*(P(1/2, -3)+P(-1/2, -3)))/((1-2*x)^(3/2)*(-Q(-1/2, -3)*P(1/2, -3)+Q(1/2, -3)*P(-1/2, -3))) where P and Q are Legendre functions of the first and second kinds. - Robert Israel, Feb 24 2015

A377542 Decimal expansion of Gamma(1/4)^4/(16*Pi^2).

Original entry on oeis.org

1, 0, 9, 4, 2, 1, 9, 8, 0, 7, 6, 1, 3, 2, 3, 8, 3, 1, 9, 4, 1, 8, 3, 8, 4, 9, 7, 0, 3, 5, 2, 2, 3, 2, 2, 7, 1, 6, 2, 9, 6, 8, 6, 3, 6, 1, 4, 1, 2, 7, 8, 3, 3, 6, 1, 0, 5, 9, 6, 4, 3, 1, 0, 5, 2, 8, 9, 7, 2, 5, 9, 6, 9, 2, 2, 2, 9, 6, 6, 4, 7, 3, 8, 8, 5, 5, 1, 6, 5, 7, 4, 8, 3, 8, 7, 8, 0, 4, 3, 1
Offset: 1

Views

Author

Stefano Spezia, Oct 31 2024

Keywords

Examples

			1.09421980761323831941838497035223227162968636141...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

Links

Crossrefs

Cf. A002388, A068465, A068466, A068467, A254794.

Programs

  • Mathematica
    RealDigits[Gamma[1/4]^4/(16Pi^2),10,100][[1]]

Formula

Equals Product_{n>=1} (1 - 1/(2*n + 1)^2)^(-1)^n (see Finch).
Equals Product_{n>=1} (4*n - 1)^2*((4*n + 1)^2 - 1)/(((4*n - 1)^2 - 1)*(4*n + 1)^2) (see Shamos).
Equals 2*(Gamma(5/4)/Gamma(3/4))^2.
Equals A254794/2. - Hugo Pfoertner, Oct 31 2024
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