Andrew Niedermaier - cp's OEIS frontend

A153741 Number of elements in wreath product C_2 wr S_n that alternate up/not-up with respect to a weak product ordering.

2, 3, 14, 49, 376, 1987, 21328, 150337, 2074624, 18279971, 308317184, 3259985969, 64981320704, 801591982115, 18436312819712, 259914703640065, 6774998673915904, 107452993132016323, 3130412454801965056

Offset: 1

Andrew Niedermaier, Dec 31 2008

nonn

			Viewing elements in one-line notation as a list of ordered pairs with first entries in [2] and second entries forming a permutation in S_n, two of the 6 up/not-up elements for n=3 are (1,2) (2,3) (1,1) and (1,1) (1,3) (2,2). Note that the first element goes up/down and the second goes up/not-up with respect to the weak product ordering on ordered pairs.

G. C. Greubel, Table of n, a(n) for n = 1..400
A. Niedermaier and J. Remmel, Analogues of Up-down Permutations for Colored Permutations, J. Int. Seq. 13 (2010), 10.5.6.

Cf. A069855.

Rest[CoefficientList[Series[(1+Sin[x]+x*Cos[x])/(Cos[x]-x*Sin[x]), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 25 2013 *)

E.g.f.: (1 + sin(x) + x*cos(x))/(cos(x) - x*sin(x)).

a(n) ~ c * n! / r^(n+1), where r = 0.860333589... (=A069855) is the root of the equation sin(r)*r = cos(r), and c = 2/((2+r^2)*sin(r)) = 0.9628268573779... if n is even and c = 2-2/(r^2+2*r*tan(r)) = 1.2701193119933... if n is odd. - Vaclav Kotesovec, Sep 25 2013

A153743 Number of elements in wreath product C_4 wr S_n that alternate up/not-up with respect to a weak product ordering.

Original entry on oeis.org

4, 10, 100, 565, 9356, 79584, 1844492, 20922625, 623457040, 8840131486, 321957866768, 5478133336309, 235789017471008, 4680625831294820, 232457094647793632, 5273696164520751265, 296832635265929103616

Offset: 1

Andrew Niedermaier, Dec 31 2008

nonn

			Viewing elements in one-line notation as a list of ordered pairs with first entries in [4] and second entries forming a permutation in S_n, two of the 100 up/not-up elements for n=3 are (1,2) (4,3) (3,1) and (1,1) (1,3) (4,2). Note that the first element goes up/down and the second goes up/not-up with respect to the weak product ordering on ordered pairs.

G. C. Greubel, Table of n, a(n) for n = 1..400
A. Niedermaier and J. Remmel, Analogues of Up-down Permutations for Colored Permutations, J. Int. Seq. 13 (2010), 10.5.6.

Rest[CoefficientList[Series[(6 + 6*Sin[x] + 18*x*Cos[x] - 9 x^2*Sin[x] - x^3*Cos[x])/(6*Cos[x] - 18*x*Sin[x] - 9 x^2*Cos[x] + x^3*Sin[x]), {x, 0, 40}], x]*Range[0, 40]!] (* G. C. Greubel, Aug 27 2016 *)

E.g.f.: (6 + 6*sin(x) + 18*x*cos(x) - 9*x^2*sin(x) - x^3*cos(x)) / (6*cos(x) - 18*x*sin(x) - 9*x^2*cos(x) + x^3*sin(x)).

A153742 Number of elements in wreath product C_3 wr S_n that alternate up/not-up with respect to a weak product ordering.

Original entry on oeis.org

3, 6, 44, 201, 2436, 16768, 284388, 2610633, 56926096, 653221506, 17409078576, 239721136817, 7550440414752, 121296879540684, 4408222329882272, 80934331054201905, 3333529520918540544, 68853515512316939422

Offset: 1

Andrew Niedermaier, Dec 31 2008

nonn

			Viewing elements in one-line notation as a list of ordered pairs with first entries in [3] and second entries forming a permutation in S_n, two of the 44 up/not-up elements for n=3 are (1,2) (3,3) (1,1) and (1,1) (1,3) (2,2). Note that the first element goes up/down and the second goes up/not-up with respect to the weak product ordering on ordered pairs.

G. C. Greubel, Table of n, a(n) for n = 1..400
A. Niedermaier and J. Remmel, Analogues of Up-down Permutations for Colored Permutations, J. Int. Seq. 13 (2010), 10.5.6.

Rest[CoefficientList[Series[(2 + 2*Sin[x] + 4 x*Cos[x] - x^2*Sin[x])/(2*Cos[x] - 4*x*Sin[x] - x^2*Cos[x]), {x, 0, 50}], x]*Range[0, 50]!] (* G. C. Greubel, Aug 27 2016 *)

E.g.f.: (2 + 2*sin(x) + 4*x*cos(x) - x^2*sin(x))/(2*cos(x) - 4*x*sin(x) -x^2*cos(x)).

a(n)/n! ~ c / r^(n+1) where r = 0.59974142102782394317972557684 is the root of the equation 4*r*tan(r) = (2-r^2), c = 4*sqrt(4 + 12*r^2 + r^4)/(12 + 16*r^2 + r^4) = 1.0837719267197115958973167583838141520381872675225558954477173... if n is even and c = (8 + 24*r^2 + 2*r^4)/(12 + 16*r^2 + r^4) = 1.5747968742391725511892660696837072745667493434277868133205599... if n is odd. - Vaclav Kotesovec, Aug 27 2016

A120436 Complete list of numbers that can be represented both as a product of 2 consecutive integers and as a product of 3 consecutive integers.

Original entry on oeis.org

0, 6, 210

Offset: 1

Andrew Niedermaier, Jul 16 2006

Mordell shows that the only integer points on the elliptic curve y*(y+1) = x*(x+1)*(x+2) have x = -2, -1, 0, 1, 5, corresponding to the products 0, 6, 210. However, there are infinitely many rational points generated from (x,y) = (0,0) by the chord-and-tangent process. - Jonathan Sondow, Oct 12 2013

			210 = 14*15 = 5*6*7.

Louis J. Mordell, Diophantine Equations, Academic Press 1969, p. 257.

Intersection of A002378 and A007531.

Module[{nn=20,p2,p3},p2=Times@@@Partition[Range[0,nn],2,1];p3= Times@@@ Partition[ Range[0,nn],3,1];Intersection[p2,p3]] (* Harvey P. Dale, Oct 05 2019 *)

Terms can be derived from the integral solutions to the elliptic curve y^2 = x^3 - 16*x + 16.

A092249 Positions of the integers in the standard diagonal enumeration of the rationals (with the integers in the first column and diagonals moving up to the right).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, 650, 696, 712, 754, 774, 806, 830, 882, 900, 940, 964

Offset: 1

Andrew Niedermaier, Feb 20 2004

nonn

A002088 without the leading zero. [R. J. Mathar, Jul 20 2009]

			The first few terms of the full enumeration are 1, 2, 1/2, 3, 1/3, 4, 3/2, 2/3, 1/4, 5, giving a(n) = 1, 2, 4, 6, 10,...
Contribution from _R. J. Mathar_, Jul 20 2009: (Start)
The positions in the first column of the table
....1..1/2..1/3..1/4..1/5..1/6..1/7..1/8..1/9.1/10.1/11.1/12
....2.......2/3.......2/5.......2/7.......2/9......2/11.....
....3..3/2.......3/4..3/5.......3/7..3/8......3/10.3/11.....
....4.......4/3.......4/5.......4/7.......4/9......4/11.....
....5..5/2..5/3..5/4.......5/6..5/7..5/8..5/9......5/11.5/12
....6.................6/5.......6/7................6/11.....
....7..7/2..7/3..7/4..7/5..7/6.......7/8..7/9.7/10.7/11.7/12
....8.......8/3.......8/5.......8/7.......8/9......8/11.....
....9..9/2.......9/4..9/5.......9/7..9/8......9/10.9/11.....
...10......10/3................10/7......10/9.....10/11.....
...11.11/2.11/3.11/4.11/5.11/6.11/7.11/8.11/911/10.....11/12
...12................12/5......12/7...............12/11.....
if scanned along rising antidiagonals, as defined by the ratios A038566(i)/A020653(i). (End)

Paolo Xausa, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A000010, A002088, A020653, A038566.

Accumulate[EulerPhi[Range[100]]] (* Paolo Xausa, Oct 19 2023 *)

a(11) and a(12) corrected by R. J. Mathar, Jul 20 2009
Incorrect recurrence formula removed by R. J. Mathar, Jul 29 2009
More terms (using A002088) from Michel Marcus, Sep 10 2018

A085735 Lyrics of "Aquarius", by Boards of Canada.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 44, 68, 27, 35, 42, 58, 47, 63, 85, 74, 67, 66, 51, 79, 42, 24, 45, 60, 10, 67, 56, 65, 44, 53, 44, 17, 15, 23

Offset: 1

Andrew Niedermaier, Jul 20 2003

From the 1998 album "Music Has the Right to Children".

Boards of Canada, Album information
Index entries for sequences related to songs

A085736 Numbers n such that all groups of order n are solvable.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Andrew Niedermaier, Jul 20 2003

nonn

			The symmetric and alternating groups on 5 elements are not solvable and have orders 60 and 120 respectively.

Eric Weisstein's World of Mathematics, Solvable groups

See A056866, the complementary set of numbers, which is the main entry for this question.

A076121 Complete list of possible cribbage hands.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 28, 29

Offset: 0

Andrew Niedermaier, Jul 21 2003

A 19-point cribbage hand is impossible, whereas 23 is possible: 5 5 J J (5) with knobs, for example. The single best cribbage hand is 5 5 5 J (5), where the cut card - the (5) - is the same suit as the Jack held. It is worth 29 points.

An interested game-theoretic approach at computing "average" cribbage hand values can be found here.
American Cribbage Congress Website.

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User: Andrew Niedermaier

A153741 Number of elements in wreath product C_2 wr S_n that alternate up/not-up with respect to a weak product ordering.

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A153743 Number of elements in wreath product C_4 wr S_n that alternate up/not-up with respect to a weak product ordering.

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A153742 Number of elements in wreath product C_3 wr S_n that alternate up/not-up with respect to a weak product ordering.

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A120436 Complete list of numbers that can be represented both as a product of 2 consecutive integers and as a product of 3 consecutive integers.

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A092249 Positions of the integers in the standard diagonal enumeration of the rationals (with the integers in the first column and diagonals moving up to the right).

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A085735 Lyrics of "Aquarius", by Boards of Canada.

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A085736 Numbers n such that all groups of order n are solvable.

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A076121 Complete list of possible cribbage hands.

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