A005728 -id:A005728 - cp's OEIS frontend

A359654 Number of vertices formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

4, 9, 77, 593, 6749, 15569, 93281, 222933, 623409, 1087393, 3453289, 5011009, 13271517

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 10 2023

The number of points internal to each edge is given by A005728(n) - 2.

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.

Cf. A359653 (regions), A359655 (edges), A359656 (k-gons), A005728, A358887, A358883, A355799, A358949, A006842, A006843.

a(n) = A359655(n) - A359653(n) + 1 by Euler's formula.

A359968 Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions equal the Farey series of order n.

Original entry on oeis.org

3, 6, 37, 195, 1467, 3408, 17113, 40435, 109638, 191718, 572939, 842487, 2139708, 3231583, 5261013

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

The number of vertices along the shorter edges is A005728(n). No formula for a(n) is known. The sequence is inspired by the Farey fan; see A360042.

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.

Cf. A359969 (regions), A359970 (edges), A359971 (k-gons), A005728, A360042, A359974, A359690, A358949, A358887.

a(n) = A359970(n) - A359969(n) + 1 by Euler's formula.

A359974 Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions on one edge equal the Farey series of order n while on the other they divide its length into n equal segments.

Original entry on oeis.org

3, 6, 26, 93, 424, 876, 2785, 5542, 11575, 18761, 40249, 57399, 109376, 155965, 227884, 322377, 532454, 676282, 1056010, 1334975, 1767798, 2240664, 3252047, 3882192, 5226897

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

The number of vertices on the edge with point positions equaling the Farey series of order n is A005728(n). No formula for a(n) is known.

This graph is related to the 'Farey fan' given in the reference.

McIlroy, M. D. "A Note on Discrete Representation of Lines". AT&T Technical Journal, 64 (1985), 481-490.

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.

Cf. A359975 (regions), A359976 (edges), A359977 (k-gons), A005728, A359968, A359690, A358949, A358887.

a(n) = A359976(n) - A359975(n) + 1 by Euler's formula.

A048134 Number of colors that can be mixed with up to n units of yellow, blue, red.

Original entry on oeis.org

0, 3, 6, 13, 22, 40, 55, 88, 118, 163, 205, 280, 334, 436, 517, 625, 733, 901, 1018, 1225, 1381, 1591, 1786, 2083, 2287, 2617, 2887, 3238, 3544, 4006, 4306, 4831, 5239, 5749, 6205, 6817, 7267, 8005, 8572, 9280, 9880, 10780, 11374, 12361

Offset: 0

Jurjen N.E. Bos

			a(2)=6: primary and secondary colors (Y^1, B^1, R^1, Y^1*B^1, Y^1*R^1, B^1*R^1).

T. D. Noe, Table of n, a(n) for n = 0..1000

Two colors gives A005728.

Cf. A048240, A048241.

Accumulate[ Table[ Sum[ MoebiusMu[n/d]*(d + 1)*(d + 2)/2, {d, Divisors[n]}], {n, 0, 43}]] (* Jean-François Alcover, Jul 16 2012, after Vladeta Jovovic *)

a(n) = number of triples (i, j, k) with 1 <= i+j+k <= n and gcd(i, j, k) = 1.

Cumulative sums of A048240(k) for k>0.

More terms from Robin Trew (trew(AT)hcs.harvard.edu).

A119983 Number of ways to partition 1 into reduced fractions i/j with j <= n.

Original entry on oeis.org

1, 2, 4, 7, 13, 22, 36, 59, 107, 189, 244, 494, 594, 1063, 3276, 5508, 5804, 12427, 12916, 42411, 131773, 167588, 168842, 428013, 839368, 1015502, 1968162, 5787287, 5791851, 15163759, 15170600, 28838713, 75983560, 82753548, 486356263, 1158442727, 1158464363

Offset: 1

Franklin T. Adams-Watters, Aug 01 2006

nonn

The reduced fractions are the Farey fractions of order n (A005728). - Robert G. Wilson v, Aug 30 2010

			a(3) = 4; 1 = 1/1 = 1/2 + 1/2 = 2/3 + 1/3 = 1/3 + 1/3 + 1/3.

Cf. A000041, A020473, A115855 (one less), A115856.

Cf. A154886, A154888. - Reinhard Zumkeller, Jan 17 2009

Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, b}]; f[n_] := Length@ IntegerPartitions[1, All, Farey@ n]; Array[f, 27] (* Robert G. Wilson v, Aug 30 2010 *)

For p prime, a(p) = a(p-1) + P(p) - 1, where P is the partition function (A000041).

Definition corrected by Reinhard Zumkeller, Jan 17 2009
a(21)-a(27) from Robert G. Wilson v, Aug 30 2010
More terms from Jinyuan Wang, Dec 12 2024

A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4sin^2(Pij*x) dx.

Original entry on oeis.org

2, 4, 6, 10, 12, 20, 24, 34, 44, 64, 78, 116, 148, 208, 286, 410, 556, 808, 1120, 1620, 2308, 3352, 4784, 6980, 10064, 14680, 21296, 31128, 45276, 66288, 96712, 141654, 207156, 303716, 444748, 652612, 956884, 1404920, 2062080, 3029564, 4450120

Offset: 1

Thomas Ward, Jan 07 2008

nonn

This quantity arises in some examples associated to the dynamical Mertens's theorem for quasihyperbolic toral automorphisms.
The function being integrated to compute a_n vanishes on the set of points in the Farey sequence of level n. I am particularly interested in knowing how large the sequence is asymptotically.
a(n) = coefficient of x^(n*(n+1)/2) in the polynomial (-1)^n*Product_{k=1..n} (1-x^k)^2, and is the maximal such coefficient as well. - Steven Finch, Feb 03 2009

			a(2) = 4 since Integral_{0..1} sin^2(Pi*x) sin^2(2*Pi*x) dx = 1/4.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000 (terms 1..174 from Robert Israel)
Miklos Bóna, R. Gómez, M. D. Ward, Workshop in Analytic and Probabilistic Combinatorics BIRS-16w5048 2016.
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
Jeffrey Gaither, Guy Louchard, Stephan Wagner, and Mark Daniel Ward, Resolution of T. Ward's Question and the Israel-Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics, Combinatorics, Probability and Computing, 24 (2015), 195-215. Special Issue Honouring the Memory of Philippe Flajolet.
S. Jaidee, S. Stevens and T. Ward, Mertens' theorem for toral automorphisms, arXiv:0801.2082 [math.DS], 2008-2010.
S. Jaidee, S. Stevens and T. Ward, Mertens' theorem for toral automorphisms, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1819-1824.
Yasuhiko Kamiyama, The Euler characteristic of the fiber product of Morse functions, Bull. Korean Math. Soc. (2025) Vol. 62, No. 1, pp. 71-80. See pp. 73, 75.
Mark Daniel Ward, Resolution of T. Ward's Question and the Israel-Finch Conjecture. Precise Asymptotic Analysis of an Integer Sequence Motivated by the Dynamical Mertens' Theorem for Quasihyperbolic Toral Automorphisms, Slides, 2013.
T. Ward, D. W. Cantrell and R. Israel, sci.math.research discussion, 2008.

Cf. A005728, A047653.

a:= n->int(product(4*(sin(Pi*j*x))^2, j=1..n), x=0..1); seq(a(n), n=1..10);
# second Maple program:
A133871:= k -> (-1)^k*coeff(mul((t^j-1)^2,j=1..k),t,k*(k+1)/2);
# Robert Israel, Mar 15 2013

p = 1; Table[p = Expand[p*(1 - x^n)^2]; Max[(-1)^n*CoefficientList[p, x]], {n, 1, 100}] (* Vaclav Kotesovec, May 03 2018 *)
(* The constant "d" *) Chop[-E^(-I*(Pi^2*(1 + 6*x^2) - 6*PolyLog[2, E^(2*I*Pi*x)]) / (6*Pi*x)) /. x -> (x /. FindRoot[Pi*(Pi*(-1 + 6*x^2) + 12*I*x*Log[1 - E^(2*I*Pi*x)]) + 6*PolyLog[2, E^(2*I*Pi*x)], {x, 4/5}, WorkingPrecision -> 100])] (* Vaclav Kotesovec, May 04 2018 *)

a(n)=sum(k=0, n*(n+1)/2, polcoeff(prod(m=1, n, 1-x^m+x*O(x^k)), k)^2) \\ Paul D. Hanna

a(n) = sum of squares of coefficients in Product_{k=1..n} (1-x^k). - Paul D. Hanna, Nov 30 2010

a(n) ~ c * d^n / sqrt(n), where d = 1.48770584269062356180051131... and c = 2.40574583936181024... [Ward, 2013]. - Vaclav Kotesovec, May 03 2018

More terms from Steven Finch, Feb 03 2009

A226999 Inverse Euler transform of A005169 (fountains of coins).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 35, 55, 93, 149, 248, 403, 671, 1098, 1827, 3013, 5013, 8313, 13859, 23063, 38534, 64341, 107715, 180355, 302565, 507784, 853507, 1435415, 2416941, 4072272, 6868062, 11590807, 19577555, 33088481, 55964327, 94712212

Offset: 1

R. J. Mathar, Jun 26 2013

nonn

If G005169(x) = Sum_{i>=0} A005169(n)*x^n is the generating function of A005169, the a(n) are defined through G005169(x) = Product_{n>=1} 1/(1-x^n)^a(n), the inverse Euler transform of A005169.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Alois P. Heinz, Table of n, a(n) for n = 1..4178
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

Cf. A005169, A005170.

max = 100;
A005169 = Series[1 - Fold[Function[1 - x^#2/#1], 1, Range[max, 0, -1]], {x, 0, max}] // CoefficientList[#, x]&;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
EULERi[A005169 // Rest] (* Jean-François Alcover, Jan 06 2020 *)

a(n) ~ 1 / (n * r^n), where r = A347901 = 0.57614876914275660229786857371993878235472466311897446868515653431946822937499... - Vaclav Kotesovec, Oct 09 2019

A359692 Number of regions in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

Original entry on oeis.org

2, 12, 94, 382, 2486, 4946, 24100, 53152, 138158, 233254, 700720, 999364, 2559344, 3785044, 6027148, 9210820

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 11 2023

nonn

The number of vertices along each edge is A005728(n). No formula for a(n) is known.

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Wikipedia, Farey sequence.

Cf. A359690 (vertices), A359691 (crossings), A359693 (edges), A359694 (k-gons), A005728, A290131, A359653, A358886, A358882, A006842, A006843.

a(n) = A359693(n) - A359690(n) + 1 by Euler's formula.

A359969 Number of regions formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions equal the Farey series of order n.

Original entry on oeis.org

1, 5, 48, 239, 1798, 3950, 19953, 46007, 123338, 213793, 637960, 930635, 2361080, 3542822, 5736344

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

The number of vertices along the shorter edges is A005728(n). No formula for a(n) is known. The sequence is inspired by the Farey fan; see A360042.

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.

Cf. A359968 (vertices), A359970 (edges), A359971 (k-gons), A005728, A360042, A359975, A359690, A358948, A358886.

a(n) = A359970(n) - A359968(n) + 1 by Euler's formula.

A359970 Number of edges formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions equal the Farey series of order n.

Original entry on oeis.org

3, 10, 84, 433, 3264, 7357, 37065, 86441, 232975, 405510, 1210898, 1773121, 4500787, 6774404, 10997356

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

The number of vertices along the shorter edges is A005728(n). No formula for a(n) is known. The sequence is inspired by the Farey fan; see A360042.

See A359968 and A359969 for images of the triangle.

Cf. A359968 (vertices), A359969 (regions), A359971 (k-gons), A005728, A360042, A359976, A359693, A358950, A358888.

a(n) = A359968(n) + A359969(n) - 1 by Euler's formula.

cp's OEIS Frontend

A359654 Number of vertices formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

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A359968 Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions equal the Farey series of order n.

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A359974 Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions on one edge equal the Farey series of order n while on the other they divide its length into n equal segments.

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A048134 Number of colors that can be mixed with up to n units of yellow, blue, red.

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Mathematica

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A119983 Number of ways to partition 1 into reduced fractions i/j with j <= n.

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A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4*sin^2(Pi*j*x) dx.

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A226999 Inverse Euler transform of A005169 (fountains of coins).

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A359692 Number of regions in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

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A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4sin^2(Pij*x) dx.