A025547 -id:A025547 - cp's OEIS frontend

A215746 Numerator of Sum_{i=0..n} (-1)^i4/(2i + 1).

4, 8, 52, 304, 1052, 10312, 147916, 135904, 2490548, 44257352, 47028692, 1023461776, 5385020324, 15411418072, 467009482388, 13895021563328, 14442004718228, 13926277743608, 533322720625196, 516197940314096, 21831981985010836, 911392701638017048, 937558224301357108

Offset: 0

Alonso del Arte, Aug 22 2012

Denominator of the sum divides A025547(n+1), but is not always equal to it: the first exception is n = 32.

x(n) = Sum_{i=0..n} (-1)^i*4/(2*i+1) very slowly converges to Pi, with x(n) > Pi when n is even and x(n) < Pi when n is odd.

			a(2) = 52 because 4 - 4/3 + 4/5 = 60/15 - 20/15 + 12/15 = 52/15.

Robert Israel, Table of n, a(n) for n = 0..229
Eric W. Weisstein, MathWorld: Pi formulas

Cf. A007509.

N:= 100; # to get terms up to a[N]
T[0]:= 4;
A215746[0]:= 4;
for i from 1 to N do
  T[i]:= T[i-1] + (-1)^i*4/(2*i+1);
  A215746[i]:= numer(T[i])
od:
[seq](A215746[i],i=0..N); # Robert Israel, Apr 27 2014

Table[Numerator[Sum[(-1)^i 4/(2i + 1), {i, 0, n}]], {n, 0, 39}]

Definition and comments corrected by Robert Israel, Apr 27 2014

A275790 Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2cos(v), with v = u/((2/Pi)K(k)).

Original entry on oeis.org

1, 8, 1, -32, 11, 3, -736, -92, 9, 15, 2816, -593, -249, -65, 35, 48976, 6122, 1581, -970, -1295, 315, -951424, 61252, 67791, 46030, 18515, -21735, 3465, -1045952, -130744, -92082, -30445, 14455, 53928, -25179, 3003, 26933248, 1069361, -1666047, -634255, -1167740, -1258236, 1562253, -471471, 45045, 634836808, 79354601, 24881793, 17914550, 30289840, 12635028, -71064609, 42480438, -9594585, 765765

Offset: 0

Wolfdieter Lang, Aug 09 2016

The dimensionless scaled phase space coordinates of the plane pendulum are (qtilde(tau, k), ptilde(tau, k)) with tau = omega_0*t, omega^2 = g/L (L is the  length of the pendulum, g the acceleration), and the energy variable E = 2*k^2 = 2*sin^2(Theta_0/2), with the maximal deflection angle Theta_0 (from [0, Pi/2]). qtilde = Theta/(2*k)) with the deflection angle Theta. Similarly ptilde = (d(Theta)/d(tau))/(2*k).
The exact solution is qtilde(tau, k) = (1/k)*arcsin(k*sn(tau, k)) with Jacobi's elliptic sn function, and ptilde(tau,k) = cn(tau, k) with the elliptic cn function.
Here the expansion in new variables v and q is used where v = tau/((2/Pi)*K(k)) and q = exp(-Pi*K'(k)/K(k)) with the real and imaginary quarter periods K and K'. This leads to qhat(v, q) = qtilde(tau(v, q), k(q)) with tau(v, q) = theta_3^2(0, q)*v. (For theta_3^2(0, q) see A004018.) Because k is actually a function of k^2 one uses the q expansion of (k/4)^2 given in A005798.
Using the result for the sn expansion in q and v from A274662 one obtains qhat(v, q) = sin(v)*Sum_{n >= 0} q^n/L(n)*Sum_{m=0..n} T(n, m)*(2*cos(v))^(2*m) with L(n) = A025547(n+1) = lcm{1, 3, ..., (2*n+1)}.
This entry is inspired by a paper of Bradley Klee giving an approximation to the phase space solution of the plane pendulum (see A273506). Thanks for discussions via e-mail go to him.

			The triangle T(n, m) begins:
n\m    0    1    2    3     4   5 ...
0:     1
1:     8    1
2:   -32   11    3
3:  -736  -92    9   15
4:  2816 -593 -249  -65    35
5: 48976 6122 1581 -970 -1295 315
...
row n=6: -951424 61252 67791 46030 18515 -21735 3465,
row n=7: -1045952 -130744 -92082 -30445 14455 53928 -25179 3003,
row n=8: 26933248 1069361 -1666047 -634255 -1167740 -1258236 1562253 -471471 45045,
row n=9: 634836808 79354601 24881793 17914550 30289840 12635028 -71064609 42480438 -9594585 765765.
...
The corresponding L(n) = A025547(n+1) numbers are 1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535,...
n=4: the contribution to qhat(v, q) of order q^4 is (q^4/315)*(2816 - 593*(2*cos(v))^2 - 249*(2*cos(v))^4 - 65*(2*cos(v))^6 + 35*(2*cos(v))^8).

Cf. A004018, A005798, A025547, A274662.

T(n, m)*(2*cos(v))^(2*m)), n >= 0, m = 0, 1, ..., n, gives the contribution to q^n/L(n) (L(n) = A025547(n+1)) in the rescaled phase space coordinate qhat(v, q) expansion of the plane pendulum. See a comment above for details.

A346781 a(n) is the numerator of the sum of the first n terms of 1 - 1/3 - 1/5 + 1/7 + 1/9 - 1/11 - 1/13 + ... .

Original entry on oeis.org

1, 2, 7, 64, 227, 2182, 24901, 27904, 519413, 9103082, 8410247, 207985216, 1106853941, 3134651098, 85885292267, 2808012157952, 2944757946677, 402260886146, 14238994069127, 14850593365952, 632726700580207, 26229300849325726, 25294817897063581, 1230908174013784832

Offset: 1

Hugo Pfoertner, Aug 03 2021

The limit for n->oo of the sum 1 - 1/3 - 1/5 + 1/7 + ... is log(1+sqrt(2))/sqrt(2) (A196525). See there for more information.

			1, 2/3, 7/15, 64/105, 227/315, 2182/3465, 24901/45045, 27904/45045, ...

Barry Mazur, Chapter IV.1 Algebraic Numbers, page 316, in The Princeton Companion to Mathematics, ed. Timothy Gowers, Princeton University Press, Princeton and Oxford, 2008.

The corresponding denominators are A025547.

Cf. A091337, A196525.

a346781(limit)={my(s=0,b(n)=1/(n*sign(4-(n+2)%8)));forstep(k=1,limit,2,print1(numerator(s+=b(k)),", "))};
a346781(47)

A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 15, 9, 1, 1, 105, 225, 27, 1, 765765, 405810405, 91398648466125, 48049812916875, 1033788065625, 89339709375, 3796875, 729, 1, 1, 1, 315, 11025, 3375, 27, 1, 1, 3465, 99225, 1157625, 16875, 81, 1, 1, 45045, 12006225, 31255875, 40516875, 253125, 243, 1, 1, 45045, 2029052025

Offset: 0

Thomas Scheuerle, Apr 21 2024

			array begins:
1,    1,        1,           1,              1,                  1
1,    1,        1,           1,              1,                  1
3,    9,        27,          27,             81,                 243
15,   225,      3375,        16875,          253125,             759375
105,  11025,    1157625,     40516875,       4254271875,         89339709375
315,  99225,    31255875,    3281866875,     1033788065625,      65128648134375
3465, 12006225, 41601569625, 48049812916875, 166492601756971875, 115379373017581509375

Cf. A370692 (numerators),
Cf. A025547 (first column), A128492 (second column).
Cf. A128507.
Cf. A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
Cf. A255009 (numerators polygamma(n, 1) - polygamma(n, k)).

A := (n, k) -> Psi(k, n + 1/2) - Psi(k, 1/2):
seq(lprint(seq(denom(A(n, k)), k = 0..4)), n=0..6);

T(n, k) = denominator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)

T(n, k) = denominator( polygamma(k, n + 1/2) - polygamma(k, 1/2) ).
T(n, k) = denominator( k!*(-1)^(k+1)*(zeta((k+1), 1/2 + n) - zeta((k+1), 1/2)) ), where zeta is the Hurwitz zeta function.
T(n, 0) = A025547(n).
T(n, 1) = A128492(n).
Conjectured: T(n, 2) = A128507(n).

cp's OEIS Frontend

A215746 Numerator of Sum_{i=0..n} (-1)^i*4/(2*i + 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A275790 Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2*cos(v), with v = u/((2/Pi)*K(k)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A346781 a(n) is the numerator of the sum of the first n terms of 1 - 1/3 - 1/5 + 1/7 + 1/9 - 1/11 - 1/13 + ... .

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2*k!*(-2)^k*Sum_{m=1..n}( 1/(2*m-1)^(k+1) ) ).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

A215746 Numerator of Sum_{i=0..n} (-1)^i4/(2i + 1).

A275790 Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2cos(v), with v = u/((2/Pi)K(k)).

A370691 Square array read by upward antidiagonals: T(n, k) = denominator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).