A206308 -id:A206308 - cp's OEIS frontend

A206530 Decimal expansion of 1/(1-sin(1)).

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

Offset: 1

Seiichi Kirikami, Feb 11 2012

The value of the limit of (A206307+6*A206308) / (A206308).

			6.3079935164437400275135217398...

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

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002943, A049469, A125202, A206307, A206308.

SetDefaultRealField(RealField(150)); 1/(1-Sin(1)); // G. C. Greubel, Dec 20 2022

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

numerical_approx(1/(1-sin(1)), digits=150) # G. C. Greubel, Dec 20 2022

Equals 1/(1-A049469).
A206307/A206308 + 6 -> 1/(1-A049469).
Abs(A206308/(1-sin(1)) - (A206307 + 6*A206308)) -> 0.

A206307 a(n) = ((2n+2)(2n+3) - 1)a(n-1) + 2n(2n+1)a(n-2), a(0)=0, a(1)=6.

Original entry on oeis.org

0, 6, 246, 17718, 1948974, 304039950, 63848389494, 17366761942374, 5939432584291902, 2494561685402598846, 1262248212813715016070, 757348927688229009642006, 531658947237136764768688206, 431707065156555052992174823278

Offset: 0

Seiichi Kirikami, Feb 11 2012

nonn

The numerators of the fractions limiting to the value of A206530.

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

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

Cf. A002943, A125202, A206308, A206530.

[n le 2 select 6*(n-1) else (4*n^2+2*n-1)*Self(n-1) + 2*(n-1)*(2*n-1)*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 20 2022

RecurrenceTable[{a[n]==((2n+3)(2n+2)-1)a[n-1]+2n(2n+1)a[n-2],a[0]==0,a[1]==6},a,{n,15}]

@CachedFunction
def a(n): return 6*n if (n<2) else (4*n^2+10*n+5)*a(n-1) + 2*n*(2*n+1)*a(n-2)
[a(n) for n in range(31)] # G. C. Greubel, Dec 20 2022

a(n) = A125202(n+2)*a(n-1) + A002943(n)*a(n-2), a(0) = 0, a(1) = 6.

cp's OEIS Frontend

A206530 Decimal expansion of 1/(1-sin(1)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A206307 a(n) = ((2n+2)(2n+3) - 1)a(n-1) + 2n(2n+1)a(n-2), a(0)=0, a(1)=6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

cp's OEIS Frontend

A206530 Decimal expansion of 1/(1-sin(1)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A206307 a(n) = ((2*n+2)*(2*n+3) - 1)*a(n-1) + 2*n*(2*n+1)*a(n-2), a(0)=0, a(1)=6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A206307 a(n) = ((2n+2)(2n+3) - 1)a(n-1) + 2n(2n+1)a(n-2), a(0)=0, a(1)=6.