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.

A280443 a(n) = A280442(n)/A223067(n) = A067624(n)*A046161(n)/A223068(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 11, 17, 1, 23, 1, 11, 1, 1, 1, 17, 11, 1, 1, 1, 23, 11, 43, 17, 1, 1, 121, 1, 1, 1, 1, 4301, 1, 1, 1, 73, 11, 1, 1, 17, 1, 11, 23, 43, 1, 1, 11, 17, 1, 1, 1, 11, 101, 23, 89, 17, 11, 1, 1, 83, 1, 11, 1
Offset: 0

Views

Author

Johannes W. Meijer and Joseph Abate, Jan 03 2017

Keywords

Comments

This sequence is related in a peculiar way to A223067 and A223068, sequences related to the complete elliptic integral of the first kind K(k), and to A280442 and A046161, sequences related to the unsigned Euler numbers A000364.
In this sequence certain prime numbers appear on a regular basis, either by itself or as a factor of a composite number, i.e., a(n)=11 if n=7+5*k, a(n)=17 if n=13+8*k, a(n)=23 if n=15+11*k, a(n)=43 if n=28+21*k, a(n)=73 if n=41+36*k, a(n)=101 if n=58+50*k, a(n)=89 if n=60+44*k, a(n)=83 if n=65+41*k, in all cases k >= 0. We observe that the period T of each prime is apparently T = (prime-1)/2.
Conjecture: The sequence A280443 will not have a(n)=1 after some point.

Crossrefs

Cf. A000364 (Euler numbers), A046161, A067624, A223067, A223068, A280442.

Programs

  • Maple
    nmax:=68: A067624 := n -> 2^(2*n)*(2*n)!: f := series((exp(add((-1)^n*euler(2*n) * x^n/(2*n), n=1..nmax+1))), x=0, nmax+1): for n from 0 to nmax do b(n) := coeff(f, x, n); a(n) := numer(b(n))/numer(b(n)/A067624(n)) od: seq(a(n), n=0..nmax);
  • Sage
    def A280443_list(prec):
    P. = PowerSeriesRing(QQ, default_prec=2*prec)
    g = lambda x: exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
    R = P(g(x)).coefficients()
    d = lambda n: 2*n - sum(n.digits(2))
    return [(2^d(n)*R[n]/(numerator(R[n]/factorial(2*n)))) for n in (0..prec)]
print(A280443_list(68)) # Peter Luschny, Jan 05 2017

Formula

a(n) = A280442(n)/A223067(n).
a(n) = A067624(n)*A046161(n)/A223068(n).
a(n) = A280442(n)/numer((A280442(n)/A046161(n))/A067624(n)).

A067624 a(n) = 2^(2*n)*(2*n)!.

Original entry on oeis.org

1, 8, 384, 46080, 10321920, 3715891200, 1961990553600, 1428329123020800, 1371195958099968000, 1678343852714360832000, 2551082656125828464640000, 4714400748520531002654720000, 10409396852733332453861621760000
Offset: 0

Views

Author

Benoit Cloitre, Feb 02 2002

Keywords

Comments

For n >= 1, a(n) equals the absolute value of the determinant of the 4n X 4n matrix with i's along the superdiagonal (where i is the imaginary unit), and 2, 3, 4, ... 4*n along the subdiagonal, and 0's everywhere else. (See Mathematica code below.) - John M. Campbell, Jun 04 2011

Links

Crossrefs

Cf. A000165.
Appears in A162445, A061549 and A120738. - Johannes W. Meijer, Jul 06 2009

Programs

  • Magma
    [2^(2*n)*Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Feb 18 2018
  • Maple
    for n from 0 to 30 by 2 do printf(`%d,`,2^(n)*(n)!) od: # James Sellers, Feb 11 2002
A067624 := n -> 2^(2*n)*(2*n)!: seq(A067624(n), n=0..12); # Johannes W. Meijer, Jan 05 2017
  • Mathematica
    Table[Abs[Det[Array[KroneckerDelta[#1 + 1, #2]*I &, {4*n, 4*n}] + Array[KroneckerDelta[#1 - 1, #2]*#1 &, {4*n, 4*n}]]], {n, 1, 20}] (* John M. Campbell, Jun 04 2011 *)
Table[2^(2 n) (2 n)!, {n, 0, 30}] (* Vincenzo Librandi, Feb 18 2018 *)

Formula

a(n) = A000165(2*n) where A000165(k) are the double factorial numbers 2^k*k!=(2k)!!. - Corrected by Johannes W. Meijer, Jul 05 2009
a(n) = (4*n)!! = 2^(2*n)*(2*n)!. - Johannes W. Meijer, Jul 06 2009
sqrt((1+cos(x))/2) = Sum_{n>=0} (-1)^n * x^(2*n) / a(n).
a(n) = (A280442(n)/A046161(n))/(A223067(n)/A223068(n)). - Johannes W. Meijer, Jan 05 2017
From Amiram Eldar, Jul 12 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(1/2).
Sum_{n>=0} (-1)^n/a(n) = cos(1/2). (End)

Extensions

More terms from James Sellers, Feb 11 2002

A223067 A sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes.

Original entry on oeis.org

1, 1, 11, 173, 22931, 1319183, 233526463, 2673857519, 39959591850371, 8797116290975003, 4872532317019728133, 1657631603843299234219, 247098748783812523360613, 77729277912104164732573547, 1503342018433974345747514544039
Offset: 0

Views

Author

Johannes W. Meijer, Mar 14 2013

Keywords

Comments

For small angles the period T of a simple gravity pendulum obeys Christiaan Huygens’s law, i.e. T = 2*Pi*sqrt(L/g) with L the length of the pendulum and g the acceleration due to gravity. For arbitrary amplitudes the period T is given below, see Wikipedia. The Taylor series expansion of T as a function of the angular displacement phi leads for the numerators of the even powers of phi to the sequence given above and for the denominators to A223068.

Examples

			T = 2*Pi*sqrt(L/g) * (1 + (1/16)*phi^2 + (11/3072)*phi^4 + (173/737280)*phi^6 + … ).

References

  • C. D. Andriesse and Sally Miedema, Huygens: The Man Behind the Principle, Ch. 8, 2005.

Links

Crossrefs

Cf. A223068 (denominators), A019692 (2*Pi).
Cf. A280442, A280443

Programs

  • Maple
    nmax:=14: f := series((2/Pi)*EllipticK(sin(phi/2)), phi, 2*nmax+1): for n from 0 to nmax do a(n):= numer(coeff(f, phi, 2*n)) od: seq(a(n), n=0..nmax); # End first program.
nmax:=14: f := series(1/((Pi/4)*(1+cos(phi/2))/EllipticK((1-cos(phi/2))/(1+cos(phi/2)))), phi, 2*nmax+1): for n from 0 to nmax do a(n):= numer(coeff(f, phi, 2*n)) od: seq(a(n), n=0..nmax); # End second program. - Johannes W. Meijer, Dec 28 2016
  • Mathematica
    s = Series[EllipticK[Sin[t/2]^2 ], {t, 0, 60}]; CoefficientList[s/Pi, t^2] // Numerator (* Jean-François Alcover, Oct 07 2014 *)
  • Sage
    def A223067_list(prec):
    P. = PowerSeriesRing(QQ, default_prec=2*prec)
    g = lambda x: exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
    R = P(g(x)).coefficients()
    return [numerator(R[n]/factorial(2*n)) for n in (0..prec)]
print(A223067_list(14)) # Peter Luschny, Jan 05 2017

Formula

T = 2*Pi*sqrt(L/g)*(2/Pi)*K(sin(phi/2)) with K(k) the complete elliptic integral of the first kind.
T = 2*Pi*sqrt(L/g)/M(1,cos(phi/2)) where M(x,y) = (Pi/4)*((x+y)/(K((x-y)/(x+y)) is the arithmetic-geometric mean of x and y. - Johannes W. Meijer, Dec 28 2016
Let S = Sum_{n>=0} (-1)^n*euler(2*n)*x^n/(2*n) then a(n) = numerator(1/(2*n)! * [x^n] exp(S)). - Peter Luschny, Jan 05 2017
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.