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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A057976 Number of shuffles where card 1 reaches a record depth (A057975) under the shuffle in A035485.

Original entry on oeis.org

1, 2, 6, 11, 14, 17, 23, 27, 33, 35, 37, 47, 60, 71, 89, 108, 110, 119, 124, 141, 149, 156, 174, 189, 209, 246, 275, 279, 301, 320, 347, 350, 364, 388, 452, 459, 509, 566, 571, 618, 638, 667, 675, 711, 737, 771, 797, 870, 906, 931, 947, 952, 1017, 1100, 1143
Offset: 0

Views

Author

Jud McCranie, Oct 22 2000

Keywords

Comments

The depths of card 1 after this number of shuffles are given in A057975.

Examples

			Card 1 reaches a depth of 8 after 6 shuffles, so 6 is in the sequence.

Crossrefs

Cf. A035485, A035492, A057975.

A080346 First card number to reach the top of the deck n times in Guy's shuffle (see A035485).

Original entry on oeis.org

2, 1, 14, 14, 38, 14, 38, 157, 1, 16929, 170, 170, 170, 170, 170, 170, 170896, 170896, 170
Offset: 1

Views

Author

Jud McCranie, Feb 16 2003, revised Mar 08 2003

Keywords

Comments

Card #1 being at the top before the first shuffle is not counted. The number of shuffles is in A080347. Card 170 reaches the top for the 15th time after 5886828 shuffles and for the 16th time after 9140809 shuffles, so a(15) and a(16) may be 170. [This was confirmed by performing 10^7 shuffles. - Frank A. Stevenson, Feb 04 2024]
Possible next term is 170896 which appears for the 20th time after 158538523553 shuffles. - Frank A. Stevenson, Feb 09 2024

Examples

			Card #14 is the first card to reach the top position 3 times (after 41 shuffles) and is the first card to do so, so a(3)=14.

Crossrefs

Cf. A035485, A057976, A080347.

Programs

  • Python
    # outputs: n, A080346(n), A080347(n)
from collections import Counter
def find_terms(max_shuffles):
  deck, times_on_top = [i for i in range(1, 2*max_shuffles+1)], Counter()
  alst, shuffles, n = [1], 0, 1
  while shuffles < max_shuffles:
    shuffles += 1
    first, next = deck[:shuffles], deck[shuffles:2*shuffles]
    deck[0:2*shuffles:2], deck[1:2*shuffles:2] = next, first
    times_on_top[deck[0]] += 1
    if times_on_top[deck[0]] == n: print(n, deck[0], shuffles); n += 1
find_terms(35000) # Michael S. Branicky, Feb 01 2021

Extensions

Definition corrected by Lambert Klasen (vim(AT)gmx.li), Feb 08 2008
a(15)-a(16) confirmed by Frank A. Stevenson, Feb 04 2024
a(17)-a(19) from Frank A. Stevenson, Feb 09 2024

A081058 Number of shuffles in Guy's shuffle (see A035485) for the card that is at the top of the deck after n shuffles to come to the top again.

Original entry on oeis.org

8, 263, 4, 23, 370, 790, 43, 48, 218, 5, 12, 22, 11, 29, 53, 26, 231, 583, 90, 271, 1988, 14, 52, 17, 6, 286, 31, 39, 40, 15, 320158, 354, 812, 487, 118, 43, 758, 470, 78, 689, 39, 29, 18, 12552564, 70, 249, 53, 2054, 10, 43, 409, 309, 13519, 224, 26, 7, 35, 800
Offset: 1

Views

Author

Jud McCranie, Mar 04 2003

Keywords

Comments

Corresponding shuffle number is in A081059. A081059[n]=A081058[n]+n.

Examples

			After 2 shuffles, card #3 is on top. It reaches the top again on shuffle # 265 (263 shuffles later), so a(2)=263.

Crossrefs

Cf. A035485, A060750, A060751, A081059.

A081059 Shuffle number in Guy's shuffle (see A035485) for the card that is at the top of the deck after n shuffles to come to the top again.

Original entry on oeis.org

9, 265, 7, 27, 375, 796, 50, 56, 227, 15, 23, 34, 24, 43, 68, 42, 248, 601, 109, 291, 2009, 36, 75, 41, 31, 312, 58, 67, 69, 45, 320189, 386, 845, 521, 153, 79, 795, 508, 117, 729, 80, 71, 61, 12552608, 115, 295, 100, 2102, 59, 93, 460, 361, 13572, 278, 81
Offset: 1

Views

Author

Jud McCranie, Mar 04 2003

Keywords

Comments

Number of shuffles later is in A081058. A081059[n]=A081058[n]+n.

Examples

			After 2 shuffles, card #3 is on top. It reaches the top again on shuffle # 265, so a(2)=265.

Crossrefs

Cf. A035485, A060750, A060751, A081058.

A035499 Turns at which card 1 surfaces in Guy's shuffling problem (A035485).

Original entry on oeis.org

0, 3, 7, 50, 93, 307, 832, 3345, 7438, 32327, 12165645, 861940477, 2151642067, 29933164804, 39288453372, 713937018048, 4426276078818, 19966143995933
Offset: 0

Views

Author

N. J. A. Sloane, Wouter Meeussen

Keywords

References

  • 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.

Crossrefs

Cf. A035485, A035490-A035494. Gives terms where 1's appear in A035492.

Extensions

More terms from David W. Wilson; and later from Jud McCranie, Oct 22 2000
a(15)-a(17) from Mark R. Diamond, Apr 20 2014

A057975 Record depths reached by card 1 in the shuffle in A035485.

Original entry on oeis.org

2, 4, 8, 16, 28, 34, 40, 44, 48, 54, 70, 86, 104, 112, 148, 160, 202, 204, 234, 272, 286, 312, 324, 366, 418, 474, 518, 530, 540, 564, 608, 678, 706, 748, 816, 902, 996, 1100, 1106, 1220, 1266, 1302, 1320, 1418, 1422, 1456, 1574, 1736, 1764, 1816, 1838, 1862
Offset: 0

Views

Author

Jud McCranie, Oct 22 2000

Keywords

Comments

The shuffle number reaching these terms are in A057976.

Examples

			Card 1 reaches a depth of 8 after 6 shuffles, so 8 is in the sequence.

Crossrefs

Cf. A035485, A035492, A057976.

A080347 Number of shuffles in Guy's shuffle (A035485) corresponding to the terms in A080346.

Original entry on oeis.org

1, 7, 41, 80, 219, 313, 1579, 4250, 32327, 96057, 203774, 321975, 394664, 1430433, 5886828, 9140809, 35697590, 101647281, 453696237
Offset: 1

Views

Author

Jud McCranie, Feb 16 2003, revised Mar 08 2003

Keywords

Comments

Card #1 being at the top before the first shuffle is not counted. The card at the top is in A080346. Card 170 reaches the top for the 15th time after 5886828 shuffles and for the 16th time after 9140809 shuffles, so a(15) <= 5886828 and a(16) <= 9140809.

Examples

			Card #14 is the first card to reach the top position 3 times after 41 shuffles and it is the first card to do so, so a(3)=41.

Crossrefs

Cf. A035485, A057976, A080346.

Programs

Extensions

a(2) and a(9) corrected by Michael S. Branicky, Feb 01 2021
a(15)-a(16) from Frank A. Stevenson, Feb 04 2024
a(17)-a(19) from Frank A. Stevenson, Feb 09 2024

A057974 Deepest position in the deck reached by card 1 before returning to the top in the shuffle in A035485 and A035499.

Original entry on oeis.org

4, 8, 86, 148, 540, 1574, 6344, 14406, 63836, 24317046, 1723718810, 4303212632, 59865566356, 78575930302
Offset: 0

Views

Author

Jud McCranie, Oct 22 2000

Keywords

Comments

The number of shuffles to return to the top is A035499. This sequence gives the deepest card 1 gets between returns to the top.

Examples

			Card 1 gets to the 4th position before returning to the top after 3 shuffles, so 4 is in the sequence. Then it gets to the 8th position before returning to the top on the 7th shuffle, so 8 is in the sequence, etc.

Crossrefs

Cf. A035485, A035499.

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

Views

Author

Jud McCranie, Oct 24 2000

Keywords

Comments

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

References

  • See A057983 for references, links and programs.

Crossrefs

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

Views

Author

Jud McCranie, Oct 24 2000

Keywords

References

  • See A057984 for references, links and programs.

Crossrefs

Cf. A035485, A035490, A035493, A035494, A057983, A060750, A060751.
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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