A053988 -id:A053988 - cp's OEIS frontend

A053987 Numerators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-...))))))).

1, 6, 59, 820, 14701, 322602, 8372951, 250865928, 8521068601, 323549740910, 13580568049619, 624382580541564, 31205548459028581, 1684475234207001810, 97668358035547076399, 6053753722969711734928, 399450077357965427428849, 27955451661334610208284502

Offset: 1

Vladeta Jovovic, Apr 03 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..200
S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

Cf. A001517, A053988, A058798.

A053987:= func< n| &+[(-1)^k*Factorial(2*n-2*k-1)/(Factorial(n-2*k-1)* Factorial(2*k+1)): k in [0..Floor((n-1)/2)]] >;
[A053987(n) : n in [1..20]]; // G. C. Greubel, May 17 2020

A053987 := n -> local k; add((-1)^k*(2*n-2*k-1)!/((n-2*k-1)!*(2*k+1)!), k = 0..floor((n-1)/2)); seq(A053987(n), n = 1..20); # G. C. Greubel, May 17 2020

Rest[CoefficientList[Series[Sin[(1-Sqrt[1-4*x])/2]/Sqrt[1-4*x], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 25 2014 *)

a(n)=sum(k=0,floor((n-1)/2),(-1)^k*(2*n-2*k-1)!/(n-2*k-1)!/(2*k+1)!) \\ Benoit Cloitre, Jan 03 2006

def A053987(n):
    return 4^n*gamma(n+1/2)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], -1/4)/sqrt(4*pi)
[round(A053987(n).n(100)) for n in (1..18)] # Peter Luschny, Sep 10 2014

a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k*(2*n-2*k-1)!/((n-2*k-1)! * (2*k+1)!). - Benoit Cloitre, Jan 03 2006
E.g.f.: 1-cos(x*C(x)), C(x)=(1-sqrt(1-4*x))/(2*x) (A000108). - Vladimir Kruchinin, Aug 10 2010
From Peter Bala, Aug 01 2013, (Start)
a(n+1) = (4*n+2)*a(n) - a(n-1) with a(0) = 0 and a(1) = 1.
a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*4^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1, k)*binomial(n-k-1/2, k+1/2), see A058798. (End)
a(n) ~ sin(1/2) * 2^(2*n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Feb 25 2014
a(n) = 4^n*Gamma(n+1/2)*hypergeometric([1/2-n/2,1-n/2], [3/2,1/2-n,1-n], -1/4)/sqrt(4*Pi). - Peter Luschny, Sep 10 2014

a(16)-a(17) from Wesley Ivan Hurt, Feb 28 2014

A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).

Original entry on oeis.org

1, 1, 0, 3, 0, -1, 15, 0, -6, 0, 105, 0, -45, 0, 1, 945, 0, -420, 0, 15, 0, 10395, 0, -4725, 0, 210, 0, -1, 135135, 0, -62370, 0, 3150, 0, -28, 0, 2027025, 0, -945945, 0, 51975, 0, -630, 0, 1, 34459425, 0, -16216200, 0, 945945, 0, -13860, 0, 45, 0, 654729075, 0, -310134825, 0, 18918900, 0, -315315, 0, 1485, 0, -1

Offset: 0

G. C. Greubel, May 12 2020, following a suggestion from Michel Marcus

Lambert's numerator polynomials related to convergents of tan(x), g(n, x), are given in A334824.

			Polynomials:
f(0, x) = 1;
f(1, x) = x;
f(2, x) = 3*x^2 - 1;
f(3, x) = 15*x^3 - 6*x;
f(4, x) = 105*x^4 - 45*x^2 + 1;
f(5, x) = 945*x^5 - 420*x^3 + 15*x;
f(6, x) = 10395*x^6 - 4725*x^4 + 210*x^2 - 1;
f(7, x) = 135135*x^7 - 62370*x^5 + 3150*x^3 - 28*x;
f(8, x) = 2027025*x^8 - 945945*x^6 + 51975*x^4 - 630*x^2 + 1.
Triangle of coefficients begins as:
        1;
        1, 0;
        3, 0,      -1;
       15, 0,      -6, 0;
      105, 0,     -45, 0,     1;
      945, 0,    -420, 0,    15, 0;
    10395, 0,   -4725, 0,   210, 0,   -1;
   135135, 0,  -62370, 0,  3150, 0,  -28, 0;
  2027025, 0, -945945, 0, 51975, 0, -630, 0, 1.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
J.-H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See also.

Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094674, A334824.

Columns k: A001147 (k=0), A001879 (k=2), A001880 (k=4), A038121 (k=6).

C := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k)*Factorial(n-k)) ) >;
[T(n,k): k in [0..n], n in [0..10]];

Maple

T:= (n, k) -> I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!); seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10);

Mathematica

(* First program *) y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x]; f[n_, k_]:= Coefficient[((-I)^n/2)*(y[n, I*x] + (-1)^n*y[n, -I*x]), x, k]; Table[f[n, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Second program *) Table[ I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten

Sage

[[ i^k*factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*factorial(k)*factorial(n-k)) for k in (0..n)] for n in (0..10)]

Formula

Equals the coefficients of the polynomials, f(n, x), defined by: (Start)

f(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k)!/((2*k)!*(n-2*k)!))*(x/2)^(n-2*k).

f(n, x) = ((2*n)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 1/2, -n, -n+1/2; -1/x^2).

f(n, x) = ((-i)^n/2)*(y(n, i*x) + (-1)^n*y(n, -i*x)), where y(n, x) are the Bessel Polynomials.

f(n, x) = (2*n-1)*x*f(n-1, x) - f(n-2, x).

E.g.f. of f(n, x): cos((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).

f(n, 1) = (-1)^n*f(n, -1) = A053983(n) = (-1)^(n+1)*A053984(-n-1) = (-1)^(n+1) * g(-n-1, 1).

f(n, 2) = (-1)^n*f(n, -2) = A053988(n+1). (End)

As a number triangle:

T(n, k) = i^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), where i = sqrt(-1).

T(n, 0) = A001147(n).

A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).

Original entry on oeis.org

1, 3, 0, 15, 0, -1, 105, 0, -10, 0, 945, 0, -105, 0, 1, 10395, 0, -1260, 0, 21, 0, 135135, 0, -17325, 0, 378, 0, -1, 2027025, 0, -270270, 0, 6930, 0, -36, 0, 34459425, 0, -4729725, 0, 135135, 0, -990, 0, 1, 654729075, 0, -91891800, 0, 2837835, 0, -25740, 0, 55, 0, 13749310575, 0, -1964187225, 0, 64324260, 0, -675675, 0, 2145, 0, -1

Offset: 0

G. C. Greubel, May 13 2020, following a suggestion from Michel Marcus

Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.

			Polynomials:
g(0, x) = 1;
g(1, x) = 3*x;
g(2, x) = 15*x^2 - 1;
g(3, x) = 105*x^3 - 10*x;
g(4, x) = 945*x^4 - 105*x^2 + 1;
g(5, x) = 10395*x^5 - 1260*x^3 + 21*x;
g(6, x) = 135135*x^6 - 17325*x^4 + 378*x^2 - 1;
g(7, x) = 2027025*x^7 - 270270*x^5 + 6930*x^3 - 36*x.
Triangle of coefficients begins as:
        1;
        3, 0;
       15, 0,      -1;
      105, 0,     -10, 0;
      945, 0,    -105, 0,    1;
    10395, 0,   -1260, 0,   21, 0;
   135135, 0,  -17325, 0,  378, 0,  -1;
  2027025, 0, -270270, 0, 6930, 0, -36, 0.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
J.-H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See also.

Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094675, A334823.

Columns k: A001147 (k=0), A000457 (k=2), A001881 (k=4), A130563 (k=6).

C := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k+1)*Factorial(n-k)) ) >;
[T(n,k): k in [0..n], n in [0..10]];

Maple

T:= (n, k) -> I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!); seq(seq(T(n, k), k = 0..n), n = 0..10);

Mathematica

(* First program *) y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x]; g[n_, k_]:= Coefficient[((-I)^n/2)*(y[n+1, I*x] + (-1)^n*y[n+1, -I*x]), x, k]; Table[g[n, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Second program *) Table[I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten

Sage

[[ i^k*factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*factorial(k+1)*factorial(n-k)) for k in (0..n)] for n in (0..10)]

Formula

Equals the coefficients of the polynomials, g(n, x), defined by: (Start)

g(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k+1)!/((2*k+1)!*(n-2*k)!))*(x/2)^(n-2*k).

g(n, x) = ((2*n+1)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 3/2, -n, -n-1/2; -1/x^2).

g(n, x) = ((-i)^n/2)*(y(n+1, i*x) + (-1)^n*y(n+1, -i*x)), where y(n, x) are the Bessel Polynomials.

g(n, x) = (2*n-1)*x*g(n-1, x) - g(n-2, x).

E.g.f. of g(n, x): sin((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).

g(n, 1) = (-1)^n*g(n, -1) = A053984(n) = (-1)^n*A053983(-n-1) = (-1)^n*f(-n-1, 1).

g(n, 2) = (-1)^n*g(n, -2) = A053987(n+1). (End)

As a number triangle:

T(n, k) = i^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), where i = sqrt(-1).

T(n, 0) = A001147(n+1).

A201505 Decimal expansion of cos(1/2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 02 2011

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.877582561890372716... = sqrt((1+A049470)/2) = sqrt(1-A201504^2).

Index entries for transcendental numbers

Cf. A053988.

Maple
```
evalf(cos(1/2)) ;
```

RealDigits[Cos[1/2],10,120][[1]] (* Harvey P. Dale, Sep 22 2016 *)

cos(1/2) \\ Charles R Greathouse IV, May 13 2019

Taylor series representation: 1 -1/(2^2*2!) +1/(2^4*4!) -1/(2^6*6!) -...

cp's OEIS Frontend

A053987 Numerators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-...))))))).

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A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).

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Magma

Maple

Mathematica

Sage

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A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).

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Magma

Maple

Mathematica

Sage

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A201505 Decimal expansion of cos(1/2).

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Maple

Mathematica

PARI

Formula