Colm Mulcahy - cp's OEIS frontend

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

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

Colm Mulcahy, May 03 2013

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

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

Charles R Greathouse IV, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

The even numbered terms are A046092.

Cf. A106232.

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

a(n)=if(n%2,n-1,n*(n-2)/2) \\ Charles R Greathouse IV, May 06 2013

Vec(2*x^3*(x^2-2*x-1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Jun 04 2014

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

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

A221564 The number of FO4C3 moves required to restore a packet of n playing cards to its original state (order and orientation), where each move Flips Over the top four (4) as a unit and then Cuts three (3) cards from the top to bottom.

Original entry on oeis.org

2, 4, 4, 4, 12, 12, 6, 24, 24, 8, 40, 40, 10, 60, 60, 12, 84, 84, 14, 112, 112, 16, 144, 144, 18, 180, 180, 20, 220, 220, 22, 264, 264, 24, 312, 312, 26, 364, 364, 28, 420, 420, 30, 480, 480, 32, 544, 544, 34, 612, 612, 36, 684, 684, 38, 760, 760, 40, 840, 840

Offset: 4

Colm Mulcahy, May 04 2013

Conjecture: a(3k+1) = 2k.

The top card remains on top but is flipped over with each move. The remaining cards split into three cycles either of length 2*floor((n-1)/3) or 2*ceiling((n-1)/3). - Andrew Howroyd, Apr 27 2020

Andrew Howroyd, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A225232.

a(n)={2*((n-1)\3)*if(n%3==1, 1, (n-1)\3+1)} \\ Andrew Howroyd, Apr 27 2020

Vec(2*x^4*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Apr 29 2020

a(3*n+1) = 2*n; a(3*n) = a(3*n-1) = 2*n*(n-1). - Andrew Howroyd, Apr 27 2020
From Colin Barker, Apr 29 2020: (Start)
G.f.: 2*x^4*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x + x^2)^3).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>12.
(End)

a(16) corrected and terms a(17) and beyond from Andrew Howroyd, Apr 27 2020

A202475 Decimal expansion of the real number x between 3 and 4 where 2^x = x!.

Original entry on oeis.org

3, 4, 5, 9, 8, 6, 5, 6, 4, 4, 0, 4, 4, 9, 9, 9, 1, 3, 4, 1, 8, 7, 8, 6, 1, 0, 8, 1, 0, 6, 8, 9, 8, 1, 2, 0, 2, 7, 7, 5, 1, 8, 4, 5, 9, 9, 0, 6, 4, 2, 8, 3, 1, 4, 5, 2, 9, 8, 0, 6, 8, 8, 7, 2, 8, 5, 8, 2, 5, 2, 2, 1, 2, 1, 1, 1, 4, 5, 1, 3, 1, 3, 8, 9, 7, 9, 2

Offset: 1

Colm Mulcahy, Dec 19 2011

2^3 > 3! but 2^4 < 4!. Since the exponential and generalized factorial (Gamma) functions are continuous, it follows that 2^x = x! ( = gamma(x+1) ) for some x between 3 and 4. It's about 3.45986564404500.

			3.459865644044999134187861081068981202775184599...

WolframAlpha, Find value

RealDigits[x/.FindRoot[2^x==x!,{x,3,4},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jan 22 2016 *)

solve(x=3, 4, 2^x-gamma(x+1)) \\ Michel Marcus, Aug 03 2013

More terms from D. S. McNeil, Dec 19 2011

A161172 a(n) is the order (or period) of the "Yummie" permutation applied to a set of n objects.

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 6, 7, 15, 20, 11, 24, 24, 14, 6, 28, 17, 120, 55, 180, 21, 18, 60, 42, 90, 153, 140, 429, 56, 152, 60, 70, 483, 3640, 180, 272, 72, 1260, 180, 252, 174, 1260, 36, 442, 1404, 660, 47, 496, 240, 481, 48, 98, 570, 572

Offset: 1

Colm Mulcahy, Jun 04 2009

nonn

The Yummie permutation is done as follows. Start with a packet of n cards (numbered 1 to n from top to bottom), and deal them into two piles, first to a spectator (pile A), and then to yourself (pile B), saying "You, me," silently to yourself over and over. Then, pick up pile B and deal again, first to the spectator, thereby adding to the existing pile A, and then to yourself, forming a new pile B. Repeat, picking up the diminished pile B, and dealing "You, me" as before. Eventually, just one card remains in pile B; place it on top of pile A. The sequence of cards in pile A determines the Yummie permutation ("You, me" said fast sounds like "Yummie").

			a(9) = 15, because when the Yummie permutation is applied to {1,2,3,4,5,6,7,8,9} we get {6,2,4,8,9,7,5,3,1}, which corresponds to the product of a disjoint five cycle and a three cycle, and hence has order 15.

Andrew Howroyd, Table of n, a(n) for n = 1..2048
Colm Mulcahy, The Yummie Deal and Variations , Card Colm, MAA Online, April 2009

Cf. A051732, A161173, A289386.

P(n,i)={if(i%2, n-(i\2), P(n\2, (n-i)\2+1))}
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
Cycles(n)={my(L=List()); for(i=1, n, my(k=Follow(i, j->P(n, j))); if(k, listput(L,k))); vecsort(Vec(L))}
a(n)={lcm(Cycles(n))} \\ Andrew Howroyd, Apr 28 2020

A161173 a(n) is the order (or period) of the "Cat's" permutation applied to a list of n objects.

Original entry on oeis.org

1, 1, 3, 4, 2, 4, 6, 10, 6, 10, 14, 12, 30, 36, 24, 14, 12, 56, 18, 66, 10, 60, 14, 110, 198, 126, 48, 133, 210, 78, 105, 18, 18, 110, 660, 396, 93, 552, 120, 616, 276, 345, 43, 108, 1122, 204, 702, 1904, 138, 598, 2310, 1080, 132, 330

Offset: 1

Colm Mulcahy, Jun 04 2009, Jun 07 2009

nonn

The Cat's permutation is done as follows. Start with a packet of n cards (numbered 1 to n from top to bottom), and deal them into two piles, first to yourself (pile B), and then to a spectator (pile A), saying "Me, you," silently to yourself over and over. Pick up pile B and deal again, first to yourself, forming a new pile B, and then to the spectator, thereby adding to the existing pile A. Repeat, picking up the diminished pile B, and dealing "Me, you" as before. Eventually, just one card remains in pile B; place it on top of pile A. The sequence of the cards in pile A determines the Cat's permutation ("Me, you" said fast sounds like something a cat says).

Values for n = 4, 19 and 27 given in the 'period' column of the table for the Cat's deal in the Colm Mulachy link are incorrect. However, the corresponding cycle decompositions are correct. - Andrew Howroyd, Apr 28 2020

			a(9) = 6, because when the Cat's permutation is applied to {1,2,3,4,5,6,7,8,9} we get {9,1,5,3,7,8,6,4,2}, which corresponds to the product of a disjoint six cycle and a three cycle, and hence has order lcm(6,3)=6.

Andrew Howroyd, Table of n, a(n) for n = 1..2048
Colm Mulcahy, The Yummie Deal and Variations, Card Colm, MAA Online, April 2009.

Cf. A161172.

P(n,i)={if(n==1, 1, if(i%2==0, n+1-i\2, P((n+1)\2, (n+1)\2-i\2)))}
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
Cycles(n)={my(L=List()); for(i=1, n, my(k=Follow(i, j->P(n, j))); if(k, listput(L,k))); vecsort(Vec(L))}
a(n)={lcm(Cycles(n))} \\ Andrew Howroyd, Apr 28 2020

Some terms corrected by Andrew Howroyd, Apr 28 2020

A138181 Largest Fibonacci number not exceeding the n-th prime.

Original entry on oeis.org

2, 3, 5, 5, 8, 13, 13, 13, 21, 21, 21, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 233, 233

Offset: 1

Colm Mulcahy, Mar 04 2008

a(n) = largest summand in the Zeckendorf representation of the n-th prime

			a(4)=5 because 5 is the largest Fibonacci number not exceeding 7 (the 4th prime)

Cf. A138182, A138183, A000040, A000045.

With[{rf=Reverse[Fibonacci[Range[20]]]},Flatten[Table[Select[rf,#<=Prime[ n]&,1],{n,60}]]] (* Harvey P. Dale, May 25 2013 *)

Corrected by Harvey P. Dale, May 25 2013

A138185 Smallest prime >= n-th Fibonacci number.

Original entry on oeis.org

2, 2, 2, 2, 3, 5, 11, 13, 23, 37, 59, 89, 149, 233, 379, 613, 991, 1597, 2591, 4201, 6779, 10949, 17713, 28657, 46381, 75029, 121403, 196429, 317827, 514229, 832063, 1346273, 2178313, 3524603, 5702897, 9227479, 14930387, 24157823, 39088193

Offset: 0

Colm Mulcahy, Mar 04 2008

			a(6) = 11 because 11 is the smallest prime not less than 8 (the 6th Fibonacci number).

Harry J. Smith, Table of n, a(n) for n = 0..657

Cf. A138184.

with(combinat): a:=proc(n) if isprime(fibonacci(n))=true then fibonacci(n) else nextprime(fibonacci(n)) end if end proc: seq(a(n),n=0..35); # Emeric Deutsch, Mar 31 2008

fib[0] = 0; fib[1] = 1; fib[n_] := fib[n] = fib[n - 1] + fib[n - 2] nextprime[n_] := Module[{k = n},While[Not[PrimeQ[k]], k++ ]; k] Table[nextprime[fib[n]], {n, 0, 50}] (* Erich Friedman, Mar 26 2008 *)
NextPrime/@(Fibonacci[Range[0,50]]-1) (* Harvey P. Dale, Nov 23 2011 *)

More terms from Erich Friedman and Emeric Deutsch, Mar 26 2008

Changed the definition of Fibonacci number to F(0) = 0, F(1) = 1, which is the standard definition. - Harry J. Smith, Jan 06 2009

A138184 Largest prime not exceeding Fibonacci(n) = A000045(n).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 31, 53, 89, 139, 233, 373, 607, 983, 1597, 2579, 4177, 6763, 10939, 17707, 28657, 46351, 75017, 121379, 196387, 317797, 514229, 832003, 1346249, 2178283, 3524569, 5702867, 9227443, 14930341, 24157811, 39088157, 63245971

Offset: 3

Colm Mulcahy, Mar 04 2008

			a(8) = 19 because 19 is the largest prime not exceeding 21 = A000045(8).

Harry J. Smith, Table of n, a(n) for n = 3..657

Cf. A138181, A138182, A138183, A138185.

A138184 := proc(n) prevprime(combinat[fibonacci](n)+1) ; end: seq(A138184(n),n=3..45) ; # R. J. Mathar, Apr 22 2008

PrimePrev[n_]:=Module[{k=n},While[ !PrimeQ[k],k-- ];k];f[n_]:=Fibonacci[n];lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)

a(n) = A000040(A054782(n)). - R. J. Mathar, Apr 22 2008

Edited and extended by R. J. Mathar, Apr 22 2008

Offset changed from 4 to 3 by Harry J. Smith, Jan 02 2009

A138183 Smallest Fibonacci number not less than the n-th prime.

Original entry on oeis.org

2, 3, 5, 8, 13, 13, 21, 21, 34, 34, 34, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 377, 377

Offset: 1

Colm Mulcahy, Mar 04 2008

			a(4) = 8 because the 8 is the smallest Fibonacci number not less than 7 (the 4th prime).

Cf. A138182, A138185.

With[{fibs=Fibonacci[Range[20]]},Table[SelectFirst[fibs,#>=n&],{n,Prime[ Range[60]]}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 22 2018 *)

a(n) = {p = prime(n); i = 0; until ((f = fibonacci(i)) >= p, i++); f;} \\ Michel Marcus, Aug 31 2013

A138182 Smallest summand in the Zeckendorf representation of the n-th prime.

Original entry on oeis.org

2, 3, 5, 2, 3, 13, 1, 1, 2, 8, 2, 3, 2, 1, 13, 1, 1, 1, 1, 3, 5, 3, 2, 89, 8, 1, 1, 5, 2, 3, 1, 8, 1, 3, 5, 2, 13, 1, 2, 8, 1, 3, 13, 2, 1, 55, 1, 3, 2, 1, 233, 1, 8, 5, 3, 1, 2, 1, 2, 1, 3, 5, 1, 2, 1, 8, 1, 2, 1, 1, 2, 3, 3, 1, 2, 1, 1, 2, 3, 3, 8, 2, 2, 1, 2, 3, 1, 1, 8, 2, 1, 13, 21, 1, 1, 3, 1, 144, 2, 2

Offset: 1

Colm Mulcahy, Mar 04 2008

			a(5) = 3 because the Zeckendorf representation of the 5th prime is 11 = 3 + 8.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A138182, A138183.

from sympy import prime
def A138182(n):
    m, tlist = prime(n), [1,2]
    while tlist[-1]+tlist[-2] <= m:
        tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d == m:
            return d
        elif d < m:
            m -= d # Chai Wah Wu, Jun 14 2018

a(n) = A139764(A000040(n)). [From R. J. Mathar, Oct 23 2010]

a(8) replaced by 1. Sequence extended beyond a(18) - R. J. Mathar, Oct 23 2010

cp's OEIS Frontend

User: Colm Mulcahy

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

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A221564 The number of FO4C3 moves required to restore a packet of n playing cards to its original state (order and orientation), where each move Flips Over the top four (4) as a unit and then Cuts three (3) cards from the top to bottom.

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A202475 Decimal expansion of the real number x between 3 and 4 where 2^x = x!.

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Mathematica

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A161172 a(n) is the order (or period) of the "Yummie" permutation applied to a set of n objects.

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A161173 a(n) is the order (or period) of the "Cat's" permutation applied to a list of n objects.

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PARI

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A138181 Largest Fibonacci number not exceeding the n-th prime.

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A138185 Smallest prime >= n-th Fibonacci number.

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