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.

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A225232 The number of FO3C2 moves required to restore a packet of n playing cards to its original state (order and orientation).

Original entry on oeis.org

2, 4, 4, 12, 6, 24, 8, 40, 10, 60, 12, 84, 14, 112, 16, 144, 18, 180, 20, 220, 22, 264, 24, 312, 26, 364, 28, 420, 30, 480, 32, 544, 34, 612, 36, 684, 38, 760, 40, 840, 42, 924, 44, 1012, 46, 1104, 48, 1200, 50, 1300, 52, 1404, 54, 1512, 56, 1624, 58, 1740, 60, 1860, 62, 1984
Offset: 3

Views

Author

Colm Mulcahy, May 03 2013

Keywords

Comments

Each FO3C2 move Flips Over the top 3 cards as a unit and then Cuts 2 cards from the top to bottom. - Mulcahy

References

  • Colm Mulcahy, Mathematical Card Magic: Fifty-Two New Effects, A K Peters, 2013, chapter 9.

Links

Crossrefs

The even numbered terms are A046092.
Cf. A106232.

Programs

  • PARI
    a(n)={
    if(n<6,return(if(n>3,4,2)));
    n--;
    my(deck=vector(n,i,i),original=deck,steps);
    while(1,
        steps+=2;
        deck=concat(deck[5..n],-[deck[2],deck[1],deck[4],deck[3]]);
        if(deck==original,return(steps))
    )
}; \\ Charles R Greathouse IV, May 03 2013
  • PARI
    a(n)=if(n%2,n-1,n*(n-2)/2) \\ Charles R Greathouse IV, May 06 2013
  • PARI
    Vec(2*x^3*(x^2-2*x-1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Jun 04 2014

Formula

Assume n >=3. For odd n we have a(n) = n-1 and for even n we have a(n) = (n-2)n/2. Equivalently, a(2k+1) = 2k and a(2k) = 2k(k-1).
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). - Colin Barker, Jun 04 2014
G.f.: 2*x^3*(x^2-2*x-1) / ((x-1)^3*(x+1)^3). - Colin Barker, Jun 04 2014

Extensions

a(10), a(12)-a(64) from Charles R Greathouse IV, May 03 2013

A328180 a(n) is the maximum number of 5-cycles possible in an n-vertex planar graph.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 24, 41, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684, 760, 840, 924, 1012, 1104, 1200, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3280, 3444, 3612, 3784, 3960, 4140, 4324, 4512
Offset: 0

Views

Author

Stefano Spezia, Oct 06 2019

Keywords

Comments

All the terms are even numbers except for a(7) = 41 which is also the only prime.
For n >= 5, also the number of 5-cycles in the (n-2)-dipyramidal graph. - Eric W. Weisstein, Dec 07 2023

Links

Crossrefs

Cf. A005843, A046092, A106232, A185721.

Programs

  • Magma
    I:=[0, 0, 0, 0, 0, 6, 24, 41, 60, 84, 112]; [n le 11 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..51]];
  • Magma
    R:=PowerSeriesRing(Integers(),51); [0,0,0,0,0] cat Coefficients(R!(x^5*(-6-6*x+13*x^2-3*x^3-3*x^4+x^5)/(-1+x)^3)); // Marius A. Burtea, Oct 16 2019
  • Maple
    gf := (1/5040)*x^7-(1/20)*x^5-(1/6)*x^4+2*exp(x)*x^2-8*exp(x)*x-4*x+12*exp(x)-12; ser := series(gf, x, 51); seq(factorial(n)*coeff(ser, x, n), n = 0..50)
  • Mathematica
    Join[{0,0,0,0,0,6,24,41},Table[2n^2-10n+12,{n,8,50}]]
LinearRecurrence[{3,-3,1},{0,0,0,0,0,6,24,41,60,84,112},60] (* Harvey P. Dale, Jan 10 2022 *)
  • PARI
    concat([0, 0, 0, 0, 0], Vec(x^5*(-6-6*x+13*x^2-3*x^3-3*x^4+x^5)/(-1+x)^3+O(x^51)))

Formula

O.g.f.: x^5*(-6 - 6*x + 13*x^2 - 3*x^3 - 3*x^4 + x^5)/(-1 + x)^3.
E.g.f.: x^7/5040 - x^5/20 - x^4/6 + 2*exp(x)*x^2 - 8*exp(x)*x - 4*x + 12*exp(x) - 12.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 10.
a(n) = 0 for n < 5, a(5) = 6, a(6) = 24, a(7) = 41, a(n) = 2*n^2 - 10*n + 12 for n > 7 (see Theorem 1 in Győri et al.).
a(n) = A046092(n-3) for n > 7.
a(n) = A106232(n-2) for n > 7.

A106231 Least j > 1 such that j^2 = (4*n^2 + 2)*(k^2) + (4*n^2 + 2)*k + 1.

Original entry on oeis.org

11, 19, 77, 199, 409, 731, 1189, 1807, 2609, 3619, 4861, 6359, 8137, 10219, 12629, 15391, 18529, 22067, 26029, 30439, 35321, 40699, 46597, 53039, 60049, 67651, 75869, 84727, 94249, 104459, 115381, 127039, 139457, 152659, 166669, 181511, 197209, 213787
Offset: 1

Views

Author

Pierre CAMI, Apr 26 2005

Keywords

Comments

For j there is always a recurrence.
For n=1, j(1,1) = 1, j(2,1) = 10*j(1,1) + 1, then j(i,1) = 10*j(i-1,1) - j(i-3).
For n>1, j(1,n) = 1, j(2,n) = 4*n^3 - 4*n^2 + 2*n - 1, j(3,n) = 4*n^3 + 4*n^2 + 2*n+1, j(4,n) = (8*n^2+2)*j(2,n) + 1 then j(i,n) = (8*n^2+2)*j(i-2) - j(i-4,n).

Links

Crossrefs

Cf. A106232.

Programs

  • PARI
    a(n) = if(n==1, 11, 4*n^3-4*n^2+2*n-1); \\ Jinyuan Wang, Apr 07 2020

Formula

a(1) = 11, a(n) = 4*n^3 - 4*n^2 + 2*n - 1 for n > 1, k sequence = A106232.
G.f.: x*(10*x^4-39*x^3+67*x^2-25*x+11) / (x-1)^4. - Colin Barker, Mar 06 2013

Extensions

More terms from Colin Barker, Mar 06 2013
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