A350669 -id:A350669 - cp's OEIS frontend

A074599 Numerator of 2 * H(n,2,1), a generalized harmonic number. See A075135. Also 2 * A350669.

2, 8, 46, 352, 1126, 13016, 176138, 182144, 3186538, 62075752, 63461422, 1488711776, 7577414602, 23104065256, 680057071574, 21372905414144, 21646396991594, 21904260478904, 819482859775298, 828045249930848

Offset: 1

Robert G. Wilson v, Aug 27 2002

2*(1 + 1/3 + ... + 1/(2*n-1))/Pi = a(n)/(A350670(n)*Pi) is the equivalent resistance between the points (0,0) and (n,n) on a 2-dimension infinite square grid of unit resistors. - Jianing Song, Apr 28 2025

MathPages, The Algebra of an Infinite Grid of Resistors
Physics Stack Exchange, On this infinite grid of resistors, what's the equivalent resistance?

Cf. A350669. The denominators are in A350670.
Cf. A025547, A025550, A075135.
Not always equal to the second left hand column of A161198 triangle divided by A025549. - Johannes W. Meijer, Jun 08 2009

Table[ Numerator[ Sum[1/i, {i, 1/2, n}]], {n, 1, 20}]

A025550 a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2n-1) )LCM(1, 3, 5, ..., 2*n-1).

Original entry on oeis.org

1, 4, 23, 176, 563, 6508, 88069, 91072, 1593269, 31037876, 31730711, 744355888, 3788707301, 11552032628, 340028535787, 10686452707072, 10823198495797, 10952130239452, 409741429887649, 414022624965424, 17141894231615609, 743947082888833412, 750488463554668427

Offset: 1

Clark Kimberling

Or, numerator of 1/1 + 1/3 + ... + 1/(2n-1) up to a(38).
Following similar remark by T. D. Noe in A025547, this coincides with f(n) = numerator of 1 + 1/3 + 1/5 + 1/7 + ... + 1/(2n-1) iff n <= 38. But a(39) = 18048708369314455836683437302413, f(39) = 1640791669937677803334857936583. Note that f(n) = numerator(digamma(n+1/2)/2 + log(2) + euler_gamma/2). - Paul Barry, Aug 19 2005 [See A350669(n-1).]
2*(1 + 1/3 + ... + 1/(2*n-1))/Pi = 2*a(n)/(A025547(n)*Pi) is the equivalent resistance between the points (0,0) and (n,n) on a 2-dimension infinite square grid of unit resistors. - Jianing Song, Apr 28 2025

Georg Fischer, Table of n, a(n) for n = 1..200 (first 39 terms from Jean-François Alcover)
MathPages, The Algebra of an Infinite Grid of Resistors
Physics Stack Exchange, On this infinite grid of resistors, what's the equivalent resistance?
Eric Weisstein's World of Mathematics, Jeep Problem

Cf. A025547, A075135, A002428, A350669.

a025550 n = numerator $ sum  $ map (1 %) $ take n [1, 3 ..]
-- Reinhard Zumkeller, Jan 22 2012

[&+[1/d: d in i]*Lcm(i) where i is [1..2*n-1 by 2]: n in [1..21]]; // Bruno Berselli, Apr 16 2015

a:= n-> (f-> add(1/p, p=f)*ilcm(f[]))([2*i-1$i=1..n]):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 16 2015

Table[(Total[1/Range[1,2n-1,2]])LCM@@Range[1,2n-1,2],{n,30}] (* Harvey P. Dale, Sep 09 2020 *)

a(n)=my(v=vector(n,i,2*i-1));sum(i=1,#v,1/v[i])*lcm(v) \\ Charles R Greathouse IV, Feb 28 2013

1 + 1/3 + ... + 1/(2*n-1) = a(n)/A025547(n) = A350669(n-1)/A350670(n-1). - Jianing Song, Apr 28 2025

Value of a(39) corrected by Jean-François Alcover, Apr 16 2015

A350670 Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

Original entry on oeis.org

1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725, 96845140757687397075, 96845140757687397075, 5132792460157432044975

Offset: 0

Wolfdieter Lang, Mar 16 2022

For the numerators see A350669.
This sequence coincides with A025547(n+1), for n = 0, 1, ..., 37. See the comments there.
Thanks to Ralf Steiner for sending me a paper where this and similar sums appear.

Hugo Pfoertner, Table of n, a(n) for n = 0..100
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions. p. 258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 258.
Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
Comparison to A025547 using Plot 2.

Cf. A001620, A025547, A025550, A350669 (numerators).

[Denominator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1))): n in [0..40]]; // G. C. Greubel, Jul 24 2023

With[{H=HarmonicNumber}, Table[Denominator[2*H[2n+2] -H[n+1]], {n,0,50}]] (* G. C. Greubel, Jul 24 2023 *)

a(n) = denominator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022

[denominator(2*harmonic_number(2*n+2,1) - harmonic_number(n+1,1)) for n in range(41)] # G. C. Greubel, Jul 24 2023

a(n) = denominator((Psi(n+3/2) + gamma + 2*log(2))/2), with the Digamma function Psi(z), and the Euler-Mascheroni constant gamma = A001620. See Abramowitz-Stegun, p. 258, 6.3.4.

a(n) = denominator of ( 2*H_{2*n+2} - H_{n+1} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023

A355566 T(j,k) are the numerators u in the representation R = s/t + (2/Pi)*u/v of the resistance between two nodes separated by the distance vector (j,k) in an infinite square lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= j, is a triangle read by rows.

Original entry on oeis.org

0, 0, 1, -2, 2, 4, -12, 23, 2, 23, -184, 40, -118, 12, 176, -940, 3323, -1118, 499, 20, 563, -24526, 1234, -18412, 13462, -626, 118, 6508, -130424, 721937, -71230, 327143, -1312, 14369, 262, 88069, -4924064, 191776, -6601046, 2395676, -888568, 131972, -300766, 1624, 91072

Offset: 0

Hugo Pfoertner, Jul 07 2022

See A355565 for more information.

On the diagonal we have T(0,0) = 0 and T(n,n) = A350669(n-1) for n > 0. - Rainer Rosenthal, Aug 01 2022

			The triangle begins:
       0;
       0,    1;
      -2,    2,      4;
     -12,   23,      2,    23;
    -184,   40,   -118,    12,  176;
    -940, 3323,  -1118,   499,   20, 563;
  -24526, 1234, -18412, 13462, -626, 118, 6508;

See A211074 for references and links.

Rainer Rosenthal, Table of n, a(n) for n = 0..135, rows 0..15 of triangle, flattened.

A355567 are the corresponding denominators v.
A355565 and A131406 (with changed offset) are s and t.
Cf. A350669.

\\ uses function R(m, p, x) given in A355565
for (j=0, 8, for (k=0, j, my(q=(pi/2)*R(j,k)); print1(numerator(polcoef(q,0,pi)),", ")); print())

A111877 a(n) = denominator of 3Sum_{j=0..n+1} 1/(2j+1).

Original entry on oeis.org

1, 5, 35, 105, 1155, 15015, 15015, 255255, 4849845, 4849845, 111546435, 557732175, 1673196525, 48522699225, 1504203675975, 1504203675975, 1504203675975, 55655536011075, 55655536011075, 2281876976454075, 98120709987525225

Offset: 0

Paul Barry, Aug 19 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001620, A025547, A350669 (numerators).

[Denominator((2*HarmonicNumber(2*n+4) - HarmonicNumber(n+2)))/3: n in [0..40]]; // G. C. Greubel, Jul 24 2023

f[x_]:= 2*x+1; a[1]= f[1]; a[n_]:= LCM[f[n], a[n-1]]; Array[a, 21]/3 (* Robert G. Wilson v, Jan 04 2013 *)

[denominator(2*harmonic_number(2*n+4,1) - harmonic_number(n+2,1))/3 for n in range(41)] # G. C. Greubel, Jul 24 2023

a(n) = denominator of (3/2)*(digamma(n+5/2) + 2*log(2) + euler_gamma).
a(n) = denominator of ( 3*Sum_{j=0..n+1} 1/(2*j+1) ).
a(n) = (1/3) * denominator of ( 2*H_{2*n+4} - H_{n+2} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023

Name edited by G. C. Greubel, Jul 24 2023

A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).

Original entry on oeis.org

1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725

Offset: 0

Wolfdieter Lang, Apr 06 2022

This is not the sequence A025547(n+1)_{n>=0}, because a(32) = 1420993851085122917681925 but A025547(33) = 18472920064106597929865025. Hence it is also not the sequence A350670.
The alternating sum Sum_{k=0..n} (-1)^k/(2*k+1) = (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2, with the Digamma function Psi(z).
Proof by subtracting twice the negative fractions from Sum_{k=0..n} 1/(2*k+1) = A350669(n)/A350670(n) (Abramowitz-Stegun, p. 258, eq. 6.3.4.), using Sum_{j=0..k} 1/(4*j + 3) = A074637((k+1)/4)/A074638(k+1) (Abramowitz-Stegun, p. 258, eqs. 6.3.6. with 6.3.5.) and, finally, replacing in the results for the even and odd n cases the formula for Psi(3/4) = -A200134.

Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. p.258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. p. 258.

Cf. A007509 (numerators).

Cf. A025547, A074637, A074638, A200134, A350669, A350670.

Denominator @ Accumulate @ Table[(-1)^k/(2*k + 1), {k, 0, 25}] (* Amiram Eldar, Apr 08 2022 *)

a(n) = denominator(sum(k=0, n, (-1)^k / (2*k+1))); \\ Michel Marcus, Apr 07 2022

from fractions import Fraction
def A352395(n): return sum(Fraction(-1 if k % 2 else 1,2*k+1) for k in range(n+1)).denominator # Chai Wah Wu, May 18 2022

a(n) = denominator( (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2), for n >= 0, with the Digamma function. See the above comment.

a(n) = denominator(Pi/4 + (-1)^n * (Psi((n + 5/2)/2) - Psi((n + 3/2)/2))/4). - Vaclav Kotesovec, May 16 2022

cp's OEIS Frontend

A074599 Numerator of 2 * H(n,2,1), a generalized harmonic number. See A075135. Also 2 * A350669.

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A025550 a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2*n-1) )*LCM(1, 3, 5, ..., 2*n-1).

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A350670 Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

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A355566 T(j,k) are the numerators u in the representation R = s/t + (2/Pi)*u/v of the resistance between two nodes separated by the distance vector (j,k) in an infinite square lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= j, is a triangle read by rows.

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A111877 a(n) = denominator of 3*Sum_{j=0..n+1} 1/(2*j+1).

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Magma

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A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).

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Python

Formula

A025550 a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2n-1) )LCM(1, 3, 5, ..., 2*n-1).

A111877 a(n) = denominator of 3Sum_{j=0..n+1} 1/(2j+1).