A035490 -id:A035490 - cp's OEIS frontend

A035485 Card on top of deck at n-th stage of R. K. Guy's shuffling problem.

1, 2, 3, 1, 6, 5, 9, 1, 4, 2, 16, 10, 12, 14, 23, 16, 18, 20, 17, 27, 30, 33, 38, 10, 14, 37, 32, 6, 11, 19, 53, 37, 25, 21, 12, 34, 38, 8, 50, 48, 46, 14, 18, 23, 47, 53, 84, 52, 31, 49, 1, 51, 91, 61, 42, 79, 4, 29, 6, 49, 26, 23, 115, 4, 70, 93, 109, 11, 16, 19, 49, 18, 124, 97, 70, 10, 134, 111, 7, 38, 14, 79, 11, 129

Offset: 0

N. J. A. Sloane, Clark Kimberling

At n-th step, pick up top n cards and interlace them with the next n.
Here is the deck after steps 0,1,2,3,4,5:
1,2,3,4,5,6,7,...
2,1,3,4,5,6,7,...
3,2,4,1,5,6,7,...
1,3,5,2,6,4,7,8,9,...
6,1,4,3,7,5,8,2,9,10,...
It is conjectured that eventually every number appears on top of the deck.
See A035491 for (the relevant part of) the deck after the n-th step. - M. F. Hasler, Aug 13 2022

D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.

Lars Blomberg, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Perfect Shuffle.

See A035491 for the array, also A035490, A035492.

A035485(n)=A035491_row(n+!n)[1]-!n \\ M. F. Hasler, Aug 13 2022

def aupton(terms):
  alst, deck = [1], list(range(1, 2*terms+1))
  for n in range(1, terms+1):
    first, next = deck[:n], deck[n:2*n]
    deck[0:2*n:2] = next
    deck[1:2*n:2] = first
    alst.append(deck[0])
  return alst
print(aupton(83)) # Michael S. Branicky, Feb 01 2021

a(n) = A035491(n,1), i.e., the first element of the n-th row of that table, for all n > 0. - M. F. Hasler, Aug 13 2022

More terms from Jud McCranie

A035494 Order in which record high new cards appear for first time on top of deck in Guy's shuffling problem A035485.

Original entry on oeis.org

1, 2, 3, 6, 9, 16, 23, 27, 30, 33, 38, 53, 84, 91, 115, 124, 134, 157, 178, 222, 241, 267, 277, 298, 323, 368, 378, 407, 438, 450, 495, 496, 542, 546, 555, 561, 576, 581, 598, 619, 646, 665, 703, 750, 774, 782, 806, 860, 862, 864, 905, 909, 937, 976, 1005, 1052, 1056, 1121, 1152, 1197, 1241, 1269, 1316

Offset: 1

N. J. A. Sloane, Wouter Meeussen

nonn

D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.

Lars Blomberg, Table of n, a(n) for n = 1..3633
Eric Weisstein's World of Mathematics, Perfect shuffle.

Cf. A035485, A035490, A035491, A035492, A035493, A035500, A035501.

from itertools import count, islice
def agen(): # generator of terms
    deck = []; record = 1; yield 1
    for n in count(1):
        deck += [2*n-1, 2*n]
        first, next = deck[:n], deck[n:2*n]
        deck[0:2*n:2], deck[1:2*n:2] = next, first
        if deck[0] > record: record = deck[0]; yield record
print(list(islice(agen(), 63))) # Michael S. Branicky, Aug 11 2022

Monotonic subsequence of A035493.

More terms from Jud McCranie

A035493 Order in which new cards appear on top of deck in Guy's shuffling problem A035485.

Original entry on oeis.org

1, 2, 3, 6, 5, 9, 4, 16, 10, 12, 14, 23, 18, 20, 17, 27, 30, 33, 38, 37, 32, 11, 19, 53, 25, 21, 34, 8, 50, 48, 46, 47, 84, 52, 31, 49, 51, 91, 61, 42, 79, 29, 26, 115, 70, 93, 109, 124, 97, 134, 111, 7, 129, 131, 157, 107, 123, 117, 96, 94, 72, 178, 86, 35, 121

Offset: 1

N. J. A. Sloane, Wouter Meeussen

nonn

D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.

Lars Blomberg, Table of n, a(n) for n = 1..6679
Eric Weisstein's World of Mathematics, Perfect Shuffle.

Cf. A035485, A035490, A035491, A035492, A035494, A035500.

from itertools import count, islice
def agen(): # generator of terms
    deck = []; tops = {1}; yield 1
    for n in count(1):
        deck += [2*n-1, 2*n]
        first, next = deck[:n], deck[n:2*n]
        deck[0:2*n:2], deck[1:2*n:2] = next, first
        if deck[0] not in tops: tops.add(deck[0]); yield deck[0]
print(list(islice(agen(), 65))) # Michael S. Branicky, Aug 11 2022

Extended (and corrected) by Jud McCranie

A035492 Position of card 1 after n-th shuffle in Guy's shuffling problem (A035485).

Original entry on oeis.org

1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, 7, 14, 28, 25, 17, 34, 31, 23, 5, 10, 20, 40, 31, 11, 22, 44, 31, 3, 6, 12, 24, 48, 27, 54, 35, 70, 63, 47, 13, 26, 52, 17, 34, 68, 43, 86, 75, 51, 1, 2, 4, 8, 16, 32, 64, 13, 26, 52, 104, 85, 45, 90, 51

Offset: 0

Wouter Meeussen

nonn

D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.

Lars Blomberg, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Perfect Shuffle.

Cf. A035485, A035490, A035491, A035493, A035494.

Transpose[Position[NestList[riguy, {}, 64], 1]][[2]] (* See A035490. *)

from itertools import count, islice
def agen(): # generator of terms
    deck = []; yield 1
    for n in count(1):
        deck += [2*n-1, 2*n]
        first, next = deck[:n], deck[n:2*n]
        deck[0:2*n:2], deck[1:2*n:2] = next, first
        yield deck.index(1) + 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Aug 11 2022

A060750 Step at which card n appears on top of deck for first time in Guy's shuffling problem A035485.

Original entry on oeis.org

3, 1, 2, 8, 5, 4, 78, 37, 6, 11, 28, 12, 349, 13, 383, 10, 18, 16, 29, 17, 33, 210, 14, 133, 32, 60, 19, 106, 57, 20, 48, 26, 21, 35, 97, 217, 25, 22, 13932, 863, 205, 54, 30452, 306, 2591, 40, 44, 39, 49, 38, 51, 47, 30, 252992198, 2253, 101, 112, 246, 402, 119, 53, 139

Offset: 1

David W. Wilson, Apr 22 2001

Card #1 is initially at the top of the deck and next appears at the top of the deck after 3 shuffles. Here we do not accept 0 as a valid number of shuffles and so we say that card #1 first shows up on top after 3 shuffles. A060751 and A060752 also adopt this convention. Alternatively, we can say that card #1 first shows up on top after 0 shuffles; this leads to sequences A035490, A057983, A057984, etc.

See A035490 for references, links and programs.

Cf. A035485, A035490, A035491, A035492, A035493, A035494, A060751, A060752.

def a(n):
  deck = list(range(1, 2*maxcards+1))
  for step in range(1, maxcards+1):
    first, next = deck[:step], deck[step:2*step]
    deck[0:2*step:2] = next
    deck[1:2*step:2] = first
    if deck[0] == n: return step
  return '>' + str(step)
maxcards = 31000
print([a(n) for n in range(1, 54)]) # Michael S. Branicky, Mar 01 2021

A035491 Relevant part of deck in Guy's shuffling problem (A035485): row n of the table lists the first 2n "cards" (numbers) after the n-th shuffle.

Original entry on oeis.org

2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 6, 1, 4, 3, 7, 5, 8, 2, 5, 6, 8, 1, 2, 4, 9, 3, 10, 7, 9, 5, 3, 6, 10, 8, 7, 1, 11, 2, 12, 4, 1, 9, 11, 5, 2, 3, 12, 6, 4, 10, 13, 8, 14, 7, 4, 1, 10, 9, 13, 11, 8, 5, 14, 2, 7, 3, 15, 12, 16, 6, 2, 4, 7, 1, 3, 10, 15, 9, 12, 13, 16, 11, 6, 8, 17, 5, 18, 14

Offset: 1

Wouter Meeussen

			{}, {2, 1}, {3, 2, 4, 1}, {1, 3, 5, 2, 6, 4}, {6, 1, 4, 3, 7, 5, 8, 2}, ...
From _M. F. Hasler_, Aug 11 2022: (Start)
The first rows of the table are: (sequence = right part of the following table)
  row | first 2n cards (followed in the deck by 2n+1, 2n+2, ...)
------+---------------------------------------------------------
   0  |  -     (followed by 1, 2, 3, ...)
   1  |  2 1       (followed by 3, 4, 5, ...)
   2  |  3 2 4 1     (followed by 5, 6, 7, ...)
   3  |  1 3 5 2 6 4   (followed by 7, 8, 9, ...)
   4  |  6 1 4 3 7 5 8 2 (followed by 9, 10, 11, ...)
   5  |  5 6 8 1 2 4 9 3 10 7 (followed by 11, 12, 13, ...)
   6  |  9 5 3 6 10 8 7 1 11 2 12 4 (followed by 13, 14, 15, ...)
   7  |  1 9 11 5 2 3 12 6 4 10 13 8 14 7 (followed by 15, 16, 17, ...)
   8  |  4 1 10 9 13 11 8 5 14 2 7 3 15 12 16 6 (followed by 17, 18, 19, ...)
   (...)
The largest numbers in row n are 2n - k, located at column 2n + 1 - d(k) with d(k) = 2*A027383(k) = A347789(k+2) = 2, 4, 8, 12, 20, 28, ..., for k >= 0, d(k) <= 2n. (End)

D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.

Cf. A035485, A035490 - A035494.

Flatten[NestList[riguy, {}, 12]] (* See A035490. *)

A35491=Map(); d=[]; A035491_row(n)={while(#dM. F. Hasler, Aug 11 2022

from itertools import count, islice
def agen(): # generator of terms
    deck = []
    for n in count(1):
        deck += [2*n-1, 2*n]
        first, next = deck[:n], deck[n:2*n]
        deck[0:2*n:2], deck[1:2*n:2] = next, first
        yield from deck
print(list(islice(agen(), 90))) # Michael S. Branicky, Aug 11 2022

T[n, 2*n + 1 - 2*A027383(k)] = 2n - k for all n and k >= 0, A027383(k) <= n. - M. F. Hasler, Aug 13 2022

A057983 These numbers take a record number of steps to reach the top of the deck in Guy's shuffle (see A035485).

Original entry on oeis.org

1, 2, 3, 4, 7, 13, 15, 39, 43, 54, 1227, 1796, 2674, 3464, 6057, 7650, 17083

Offset: 0

Jud McCranie, Oct 24 2000

See A057984 for the number of steps for these to reach the top of the deck.

			In Guy's shuffle, 4 takes 8 shuffles to reach the top, no smaller number takes more shuffles, so 4 is in the sequence.

D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, Springer, 1998, chapter 5.
D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.

Cf. A035485, A035490, A035493, A035494, A057984, A060750-A060752.

A057984 = A035490(A057983).

A057984 These numbers are the record number of steps for the numbers in A057983 to reach the top of the deck in Guy's shuffle (see A035485).

Original entry on oeis.org

0, 1, 2, 8, 78, 349, 383, 13932, 30452, 252992198, 364873753, 850627701, 914514429, 17493745212, 471008431072, 1027221839335, 14568535208344

Offset: 0

Jud McCranie, Oct 24 2000

See A057983 for the numbers that take these number of steps to reach the top of the deck. Gale gives a(12) as 21879255397 (incorrect).

			In Guy's shuffle, 4 takes 8 shuffles to reach the top, no smaller number takes more shuffles, so 8 is in the sequence.

D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, Springer, 1998, chapter 5.
D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.

Cf. A035485, A035490, A035493, A035494, A057983, A060750-A060752.

A057984 = A035490(A057983).

A060751 These numbers take a record number of steps to reach the top of the deck in Guy's shuffle (see A060750).

Original entry on oeis.org

1, 4, 7, 13, 15, 39, 43, 54, 1227, 1796, 2674, 3464, 6057, 7650, 17083

Offset: 0

Jud McCranie, Oct 24 2000

See A060752 for the number of steps for these to reach the top of the deck.

See A057983 for references, links and programs.

Cf. A035485, A035490, A035493, A035494, A057984, A060750, A060752.

A060752 These numbers are the record number of steps for the numbers in A060751 to reach the top of the deck in Guy's shuffle (see A060750).

Original entry on oeis.org

3, 8, 78, 349, 383, 13932, 30452, 252992198, 364873753, 850627701, 914514429, 17493745212, 471008431072, 1027221839335, 14568535208344

Offset: 0

Jud McCranie, Oct 24 2000

See A057984 for references, links and programs.

Cf. A035485, A035490, A035493, A035494, A057983, A060750, A060751.

cp's OEIS Frontend

A035485 Card on top of deck at n-th stage of R. K. Guy's shuffling problem.

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A035494 Order in which record high new cards appear for first time on top of deck in Guy's shuffling problem A035485.

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A035493 Order in which new cards appear on top of deck in Guy's shuffling problem A035485.

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A035492 Position of card 1 after n-th shuffle in Guy's shuffling problem (A035485).

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A060750 Step at which card n appears on top of deck for first time in Guy's shuffling problem A035485.

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A035491 Relevant part of deck in Guy's shuffling problem (A035485): row n of the table lists the first 2n "cards" (numbers) after the n-th shuffle.

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A057983 These numbers take a record number of steps to reach the top of the deck in Guy's shuffle (see A035485).

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A057984 These numbers are the record number of steps for the numbers in A057983 to reach the top of the deck in Guy's shuffle (see A035485).

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