A206533 -id:A206533 - 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.

A206532 a(n) = (2(n+1)(2n+1)-1) * a(n-1) + 2n(2n-1) * a(n-2), a(0) = 1, a(1) = 11.

Original entry on oeis.org

1, 11, 331, 18535, 1668151, 220195931, 40075659443, 9618158266319, 2943156429493615, 1118399443207573699, 516700542761899048939, 285218699604568275014327, 185392154742969378759312551, 140156468985684850342040288555

Offset: 0

Seiichi Kirikami, Feb 11 2012

nonn

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

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

Robert Israel, Table of n, a(n) for n = 0..223

Cf. A002939, A082108, A206531, A206533.

f:= gfun:-rectoproc({a(n) = (2*(n+1)*(2*n+1)-1) * a(n-1) + 2*n*(2*n-1) * a(n-2), a(0) = 1, a(1) = 11},a(n),remember):
map(f, [$0..40]); # Robert Israel, Sep 16 2018

RecurrenceTable[{a[n]==(2(n+1)(2n+1)-1)a[n-1]+2n(2n-1)a[n-2],a[0]==1,a[1]==11},a,{n,15}]

a(n) = A082108*a(n-1) + A002939*a(n-2), a(0) = 1, a(1) = 11.

a(n) = -4*n*(-1)^n*(n+1)*LommelS1(2*n+1/2, 3/2, 1)-2*(-1)^n*(n+1)*LommelS1(2*n+3/2, 1/2, 1)+(1-cos(1))*(2*n+2)!+(-1)^n. - Robert Israel, Sep 16 2018

A371936 Decimal expansion of Sum_{k>=0} (-1)^k / (2(k+1)(2*k+1)!).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Apr 24 2024

			0.459697694131860282599063392557023396267689579382077772...

Cf. A049470, A206533, A371935.

s = N[Sum[(-1)^k/(2(k + 1) (2 k + 1)!), {k, 0, Infinity}], 120]
First[RealDigits[s]]

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

Equals 1 - cos(1) = A371935/2 = 1 - A049470.

Equals 1/A206533. - Hugo Pfoertner, Apr 26 2024

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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A206532 a(n) = (2(n+1)(2n+1)-1) * a(n-1) + 2n(2n-1) * a(n-2), a(0) = 1, a(1) = 11.

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A371936 Decimal expansion of Sum_{k>=0} (-1)^k / (2(k+1)(2*k+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

A206532 a(n) = (2(n+1)(2n+1)-1) * a(n-1) + 2n(2n-1) * a(n-2), a(0) = 1, a(1) = 11.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A371936 Decimal expansion of Sum_{k>=0} (-1)^k / (2*(k+1)*(2*k+1)!).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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

A371936 Decimal expansion of Sum_{k>=0} (-1)^k / (2(k+1)(2*k+1)!).