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 1-8 of 8 results.

A159721 Number of permutations of 3 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

6, 36, 192, 960, 4608, 21504, 98304, 442368, 1966080, 8650752, 37748736, 163577856, 704643072, 3019898880, 12884901888, 54760833024, 231928233984, 979252543488, 4123168604160, 17317308137472, 72567767433216, 303465209266176, 1266637395197952, 5277655813324800
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A159715, A159727, A159733, A159736, A159738, A159739, A159740.

Programs

  • Magma
    [3*n*4^(n-2): n in [2..30]]; // G. C. Greubel, Jun 01 2018
  • Mathematica
    LinearRecurrence[{8,-16}, {6,36}, 30] (* or *) Table[3*n*4^(n-2), {n, 2, 30}] (* G. C. Greubel, Jun 01 2018 *)
  • PARI
    for(n=2,30, print1(3*n*4^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

Formula

a(n) = (copies*n)*(copies+1)^(n-2), here: copies = 3.
From Colin Barker, Mar 23 2018: (Start)
G.f.: 6*x^2*(1 - 2*x) / (1 - 4*x)^2.
a(n) = 8*a(n-1) - 16*a(n-2) for n>3. (End)
E.g.f.: 3*x*exp(4*x)/4. - G. C. Greubel, Jun 01 2018
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (16/3)*log(4/3) - 3/2.
Sum_{n>=2} (-1)^n/a(n) = (16/3)*log(5/4) - 7/6. (End)

A159727 Number of permutations of 4 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

8, 60, 400, 2500, 15000, 87500, 500000, 2812500, 15625000, 85937500, 468750000, 2539062500, 13671875000, 73242187500, 390625000000, 2075195312500, 10986328125000, 57983398437500, 305175781250000, 1602172851562500
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A159715, A159721, A159733, A159736, A159738, A159739, A159740.

Programs

  • Magma
    [ 4*n*5^(n-2) : n in [2..30]]; // G. C. Greubel, Jun 01 2018
  • Mathematica
    LinearRecurrence[{10,-25}, {8,60}, 30] (* or *) Table[4*n*5^(n-2), {n,2,30}] (* G. C. Greubel, Jun 01 2018 *)
  • PARI
    for(n=2, 30, print1(4*n*5^(n-2) , ", ")) \\ G. C. Greubel, Jun 01 2018

Formula

a(n) = (4*n)*(4+1)^(n-2).
From Colin Barker, Mar 23 2018: (Start)
G.f.: 4*x^2*(2 - 5*x) / (1 - 5*x)^2.
a(n) = 10*a(n-1) - 25*a(n-2) for n>3. (End)
E.g.f.: 4*x*exp(5*x)/5. - G. C. Greubel, Jun 01 2018
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (25/4)*log(5/4) - 5/4.
Sum_{n>=2} (-1)^n/a(n) = 5/4 - (25/4)*log(6/5). (End)

A159739 Number of permutations of 8 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

16, 216, 2592, 29160, 314928, 3306744, 34012224, 344373768, 3443737680, 34093003032, 334731302496, 3263630199336, 31632108085872, 305023899399480, 2928229434235008, 28001193964872264, 266834907194665104, 2534931618349318488, 24015141647519859360
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A159715, A159721, A159727, A159733, A159736, A159738, A159740.

Programs

  • Magma
    I:=[16, 216]; [n le 2 select I[n] else 18*Self(n-1) - 81*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 01 2018
  • Mathematica
    LinearRecurrence[{18,-81}, {16,216}, 30] (* G. C. Greubel, Jun 01 2018 *)
  • PARI
    Vec(8*x^2*(2 - 9*x) / (1 - 9*x)^2 + O(x^25)) \\ Colin Barker, Feb 26 2018

Formula

a(n) = (copies*n)*(copies+1)^(n-2) with copies = 8.
a(n) = 8*n*9^(n-2).
From Colin Barker, Feb 26 2018: (Start)
G.f.: 8*x^2*(2 - 9*x) / (1 - 9*x)^2.
a(n) = 18*a(n-1) - 81*a(n-2) for n>3.
(End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (81/8)*log(9/8) - 9/8.
Sum_{n>=2} (-1)^n/a(n) = 9/8 - (81/8)*log(10/9). (End)

A159715 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

4, 18, 72, 270, 972, 3402, 11664, 39366, 131220, 433026, 1417176, 4605822, 14880348, 47829690, 153055008, 487862838, 1549681956, 4907326194, 15496819560, 48814981614, 153418513644, 481176247338, 1506290861232, 4707158941350
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A027261, A159721, A159727, A159733, A159736, A159738, A159739, A159740.

Programs

  • Magma
    [2*n*3^(n-2): n in [2..30]]; // G. C. Greubel, Jun 01 2018
  • Mathematica
    LinearRecurrence[{6,-9}, {}, 30] (* or *) Table[2*n*3^(n-2), {n, 2, 30}] (* G. C. Greubel, Jun 01 2018 *)
  • PARI
    for(n=2, 30, print1(2*n*3^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

Formula

a(n) = (copies*n)*(copies+1)^(n-2), here: copies = 2.
Apparently a(n) = A027261(n-1), n > 2. - R. J. Mathar, Apr 21 2009
Conjectures from Colin Barker, Mar 23 2018: (Start)
G.f.: 2*x^2*(2 - 3*x) / (1 - 3*x)^2.
a(n) = 2*3^(n-2)*n for n>1.
a(n) = 6*a(n-1) - 9*a(n-2) for n>3. (End)
E.g.f.: 2*x*exp(3*x)/3. - G. C. Greubel, Jun 01 2018
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (9/2)*log(3/2) - 3/2.
Sum_{n>=2} (-1)^n/a(n) = 3/2 - (9/2)*log(4/3). (End)

A159736 Number of permutations of 6 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

12, 126, 1176, 10290, 86436, 705894, 5647152, 44471322, 345888060, 2663338062, 20338217928, 154231485954, 1162668124884, 8720010936630, 65109414993504, 484251274014186, 3589156501516908, 26519878594541598, 195409631749253880
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A159715, A159721, A159727, A159733, A159738, A159739, A159740.

Programs

  • Magma
    I:=[12, 126]; [n le 2 select I[n] else 14*Self(n-1) - 49*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 01 2018
  • Mathematica
    LinearRecurrence[{14,-49}, {12, 126}, 30] (* or *) Table[6*n*7^(n-2), {n, 2, 30}] (* G. C. Greubel, Jun 01 2018 *)
  • PARI
    for(n=2, 30, print1(6*n*7^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

Formula

a(n) = (copies*n)*(copies+1)^(n-2).
From G. C. Greubel, Jun 01 2018: (Start)
a(n) = 6*n*7^(n-2).
a(n) = 14*a(n-1) - 49*a(n-2).
G.f.: x^2*(12-42*x)/(1-14*x+49*x^2).
E.g.f.: 6*x*exp(7*x)/7. (End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (49/6)*log(7/6) - 7/6.
Sum_{n>=2} (-1)^n/a(n) = 7/6 - (49/6)*log(8/7). (End)

A159738 Number of permutations of 7 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

14, 168, 1792, 17920, 172032, 1605632, 14680064, 132120576, 1174405120, 10334765056, 90194313216, 781684047872, 6734508720128, 57724360458240, 492581209243648, 4186940278571008, 35465847065542656, 299489375220137984
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A159715, A159721, A159727, A159733, A159736, A159739, A159740.

Programs

  • Magma
    I:=[14, 168]; [n le 2 select I[n] else 16*Self(n-1) - 64*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 01 2018
  • Mathematica
    LinearRecurrence[{16,-64}, {14,168}, 30] (* or *) Table[7*n*8^(n-2), {n,2,30}] (* G. C. Greubel, Jun 01 2018 *)
  • PARI
    for(n=2,30, print1(7*n*8^(n-2), ", ")) \\ G. C. Greubel, Jun 01 2018

Formula

a(n) = (copies*n)*(copies+1)^(n-2).
From G. C. Greubel, Jun 01 2018: (Start)
a(n) = 7*n*8^(n-2).
a(n) = 16*a(n-1) - 64*a(n-2).
G.f.: x^2*(14-56*x)/(1-16*x+64*x^2).
E.g.f.: 7*x*exp(8*x)/8. (End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (64/7)*log(8/7) - 8/7.
Sum_{n>=2} (-1)^n/a(n) = 8/7 - (64/7)*log(9/8). (End)

A159740 Number of permutations of 9 indistinguishable copies of 1..n arranged in a circle with exactly 1 local maximum.

Original entry on oeis.org

18, 270, 3600, 45000, 540000, 6300000, 72000000, 810000000, 9000000000, 99000000000, 1080000000000, 11700000000000, 126000000000000, 1350000000000000, 14400000000000000, 153000000000000000
Offset: 2

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A159715, A159721, A159727, A159733, A159736, A159738, A159739.

Programs

  • Magma
    I:=[18, 270]; [n le 2 select I[n] else 20*Self(n-1) - 100*Self(n-2): n in [1..10]]; // G. C. Greubel, Jun 01 2018
  • Mathematica
    LinearRecurrence[{20,-100}, {18,270}, 30] (* or *) Table[9*n*10^(n-2), {n,2,30}] (* G. C. Greubel, Jun 01 2018 *)
  • PARI
    m=30; v=concat([18, 270], vector(m-2)); for(n=3, m, v[n]=20*v[n-1] -100*v[n-2]); v \\ G. C. Greubel, Jun 01 2018

Formula

a(n) = (copies*n)*(copies+1)^(n-2).
From G. C. Greubel, Jun 01 2018: (Start)
a(n) = 9*n*10^(n-2).
G.f.: x^2*(18-90*x)/(1-20*x+100*x^2).
E.g.f.: 9*x*exp(10*x)/10. (End)
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = (100/9)*log(10/9) - 10/9.
Sum_{n>=2} (-1)^n/a(n) = 10/9 - (100/9)*log(11/10). (End)

A159735 Number of permutations of 5 indistinguishable copies of 1..n arranged in a circle with exactly 3 local maxima.

Original entry on oeis.org

0, 120, 104160, 43965760, 14780546400, 4552085003160, 1344988823772440, 387866444580875520, 109997436597621865920, 30801926079814805641600, 8538476664330640027144320, 2347317237607372501196513280, 640818430658412951758052362240, 173908317523621059703082725693440
Offset: 1

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A159733, A159734, A334778.

Programs

  • PARI
    \\ CircPeaksBySig defined in A334778.
a(n) = {CircPeaksBySig(vector(n, i, 5), [3])[1]} \\ Andrew Howroyd, May 14 2020

Extensions

Terms a(7) and beyond from Andrew Howroyd, May 14 2020
Showing 1-8 of 8 results.

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