A206532 -id:A206532 - cp's OEIS frontend

A206531 a(n) = (2(n+1)(2n+1)-1)a(n-1) + 2n(2n-1)a(n-2), a(0)=0, a(1)=2.

0, 2, 58, 3250, 292498, 38609738, 7026972314, 1686473355362, 516060846740770, 196103121761492602, 90599642253809582122, 50011002524102889331346, 32507151640666878065374898, 24575406640344159817423422890

Offset: 0

Seiichi Kirikami, Feb 11 2012

nonn

The numerators of the fractions limiting to the value of A206533.

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

G. C. Greubel, Table of n, a(n) for n = 0..220

Cf. A002939, A082108, A206532, A206533.

[n le 2 select 2*(n-1) else (2*n*(2*n-1)-1)*Self(n-1) + 2*(n-1)*(2*n-3)*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 21 2022

RecurrenceTable[{a[n]==(2(n+1)(2n+1)-1)a[n-1]+2n(2n-1)a[n-2],a[0]==0,a[1]==2},a,{n,15}]

@CachedFunction
def a(n): # a = A206531
    if (n<2): return 2*n
    else: return (2*(n+1)*(2*n+1)-1)*a(n-1) + 2*n*(2*n-1)*a(n-2)
[a(n) for n in range(31)] # G. C. Greubel, Dec 21 2022

a(n) = A082108(n)*a(n-1) + A002939(n)*a(n-2), a(0) = 0, a(1) = 2.

A206533 Decimal expansion of 1/(1-cos(1)).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Feb 11 2012

The value of the fractional limit of the numerators(A206531+2*A206532) and the denominators(A206532).

Abs(A206532/(1-cos(1)) - (A206531+2*A206532)) -> 0.

			2.17534264967002141077678675965606997584844...

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

Cf. A002939, A082108, A206531, A206532, A049470, A371936.

Mathematica
```
RealDigits[N[1/(1-Cos[1]), 150]][[1]]
```

1/(1 - cos(1)) \\ Stefano Spezia, Apr 21 2025

Equals 1/(1-A049470).
A206531/A206532+2 -> 1/(1-A049470).
Equals 1/A371936. - Hugo Pfoertner, Apr 21 2025

Incorrect a(86)=9 removed by Georg Fischer, Apr 04 2020

cp's OEIS Frontend

A206531 a(n) = (2(n+1)(2n+1)-1)a(n-1) + 2n(2n-1)a(n-2), a(0)=0, a(1)=2.

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A206533 Decimal expansion of 1/(1-cos(1)).

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cp's OEIS Frontend

A206531 a(n) = (2*(n+1)*(2*n+1)-1)*a(n-1) + 2*n*(2*n-1)*a(n-2), a(0)=0, a(1)=2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A206533 Decimal expansion of 1/(1-cos(1)).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A206531 a(n) = (2(n+1)(2n+1)-1)a(n-1) + 2n(2n-1)a(n-2), a(0)=0, a(1)=2.