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.

A068379 Engel expansion of sinh(1/2).

Original entry on oeis.org

2, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648, 25920, 27224, 28560
Offset: 1

Views

Author

Benoit Cloitre, Mar 03 2002

Keywords

Comments

Cf. A006784 for Engel expansion definition.
The MathWorld link mentions the closed form of the Engel expansion of sinh(1) = A068377. - Georg Fischer, Nov 22 2020

Examples

			sinh(1/2) = 1/2 + 1/(2*24) + 1/(2*24*80) + 1/(2*24*80*168) + 1/(2*24*80*168*288) + ... = 0.52109530549374736162242562641... = A334367.

Links

Crossrefs

Cf. A002943, A006784, A033586, A068377, A068380, A334367.

Programs

Formula

a(n) = 8*(n*(2*n-3)+1) for n > 1, a(1)=2.
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: 2*x*(1+9*x+7*x^2-x^3)/(1-x)^3. (End)
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = (3-log(2))/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/4 - Pi/16 - log(2)/8. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: 2*(4*exp(x)*(1 - x + 2*x^2) + (x - 4)).
a(n) = 2*A033586(n-1) for n >= 2.
a(n) = 4*A002943(n-1) for n >= 2. (End)

Extensions

Edited, offset 1 and a(1)=2 in programs and b-file by Georg Fischer, Nov 22 2020

A334366 Decimal expansion of Sum_{k>=0} 1/(4*k)!!.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 24 2020

Keywords

Comments

This constant is transcendental.

Examples

			1/(2^0*0!) + 1/(2^2*2!) + 1/(2^4*4!) + ... = 1.1276259652063807852...

Links

Crossrefs

Cf. A019774, A067624, A073743, A143280, A160327, A201505, A334367.

Programs

  • Mathematica
    RealDigits[Cosh[1/2], 10, 110] [[1]]
  • PARI
    cosh(1/2) \\ Michel Marcus, Apr 25 2020

Formula

Equals cosh(1/2).
Equals Product_{k>=0} 1 + 1/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020

A352526 a(n) = Product_{k=0..n} Nimsum (2*k + 2), with Nimsum (2 + 2) = 0 replaced by 1.

Original entry on oeis.org

2, 2, 12, 48, 480, 3840, 53760, 645120, 11612160, 185794560, 4087480320, 81749606400, 2125489766400, 51011754393600, 1530352631808000, 42849873690624000, 1456895705481216000, 46620662575398912000, 1771585177865158656000, 63777066403145711616000, 2678636788932119887872000
Offset: 0

Views

Author

Peter McNair, Mar 19 2022

Keywords

Comments

Nimsum 2*k + 2 = A004443(2*k).
Sum_{n>0} 1/a(n) = 1/sqrt(e) = A092605.
Sum_{n>0} 1/a(2*n-1) = sinh(1/2) = A334367.
Sum_{n>0} 1/a(2*n) = cosh(1/2) - 2*sinh(1/2).
a(n)/2^n = abs(A265376(n+1)) = Product_{k=0..n} Nimsum k + 1, with Nimsum 1 + 1 = 0 replaced by 1, n > 0.

Crossrefs

Cf. A004443, A092605, A265376, A334367.

Programs

  • Mathematica
    a[n_] := Product[If[k == 1, 1, BitXor[2*k, 2]], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Mar 19 2022 *)
  • PARI
    a(n) = 2*prod(k=2,n,bitxor(2*k, 2))

Formula

a(n) = 2*Product_{k=2..n} A004443(2*k).
a(n) = 2^(n-1)*(n+1)!/floor((n+1)/2), n > 0.
a(n) = 2^(n-1)*(1+(-1)^n)*((n-1)!+n!)-((-1)^n-1)*(2*n)!!/2, n > 0.
a(n) = 2*a(n-1)*(n+(-1)^n), n > 1, with a(1) = 2.
Showing 1-3 of 3 results.

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