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A300857 Base-7 based twisted permutation of the nonnegative integers. 5
0, 1, 2, 3, 4, 5, 6, 13, 20, 27, 34, 41, 40, 39, 38, 37, 36, 29, 22, 15, 16, 17, 18, 25, 24, 23, 30, 31, 32, 33, 26, 19, 12, 11, 10, 9, 8, 7, 14, 21, 28, 35, 42, 43, 44, 45, 46, 47, 48, 97, 146, 195, 244, 293, 292, 291, 290, 289, 288, 239, 190, 141, 142, 143, 144, 193, 192 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Base-7 variant of Knuth's A220952, i.e., two numbers a, b are adjacent iff for all i > j, the pairs (a_i,a_j) and (b_i,b_j) (where indices denote base-7 digits: a = Sum_{k>=0} a_k*7^k), are equal or neighbors in the following graph:

  .

  (0,6)--(1,6)--(2,6)--(3,6)--(4,6)--(5,6)  (6,6)

    |                                  |      |

    |                                  |      |

  (0,5)  (1,5)--(2,5)--(3,5)--(4,5)  (5,5)  (6,5)

    |      |                    |      |      |

    |      |                    |      |      |

  (0,4)  (1,4)  (2,4)--(3,4)  (4,4)  (5,4)  (6,4)

    |      |      |      |      |      |      |

    |      |      |      |      |      |      |

  (0,3)  (1,3)  (2,3)  (3,3)  (4,3)  (5,3)  (6,3)

    |      |      |      |      |      |      |

    |      |      |      |      |      |      |

  (0,2)  (1,2)  (2,2)  (3,2)--(4,2)  (5,2)  (6,2)

    |      |      |                    |      |

    |      |      |                    |      |

  (0,1)  (1,1)  (2,1)--(3,1)--(4,1)--(5,1)  (6,1)

    |      |                                  |

    |      |                                  |

  (0,0)  (1,0)--(2,0)--(3,0)--(4,0)--(5,0)--(6,0)

  .

It turns out that for any positive integer there are exactly two other adjacent nonnegative integers, and this sequence in which (a(n),a(n+1)) are pairs of adjacent integers, defines a permutation of the nonnegative integers.

The analog graph for base-3 would yield more than two other adjacent numbers for some n, e.g., n = 5 would be adjacent to 3, 4, 6, 7, and 8. For even bases there is not an exact analog of this graph.

LINKS

M. F. Hasler, Table of n, a(n) for n = 0..1000

Index entries related to permutations.

PROG

(PARI) isAdj(a, b)={a=Vec(digits(min(a, b), 7), -#b=concat(0, digits(max(a, b), 7))); normlp(a-b, 1)<2 && !for(j=2, #b, for(i=1, j-1, if(a[i]==b[i], !a[i] || a[i]==6 || (a[i]==5 && min(a[j], b[j])) || (a[i]==1 && max(a[j], b[j])<6) || (a[i]==2 && !#setminus(Set([a[j], b[j]]), [1, 2, 3, 4])) || (a[i]==4 && !#setminus(Set([a[j], b[j]]), [2, 3, 4, 5])) || (a[i]==3 && !#setminus(Set([a[j], b[j]]), [2, 3, 4])) || a[j]==b[j], (!a[j] && min(a[i], b[i])) || (a[j]==6 && max(a[i], b[i])<6) || (a[j]==1 && !#setminus(Set([a[i], b[i]]), [2, 3, 4, 5])) || (a[j]==5 && !#setminus(Set([a[i], b[i]]), [1, 2, 3, 4])) || (a[j]==4 && Set([a[i], b[i]])==[2, 3]) || (a[j]==2 && Set([a[i], b[i]])==[3, 4]) || a[i]==b[i]) || return))}

u=[]; for(n=a=0, 100, print1(a", "); u=setunion(u, [a]); while(#u>1&&u[2]==u[1]+1, u=u[^1]); for(k=u[1]+1, oo, !setsearch(u, k)&&isAdj(a, k)&&(a=k)&&next(2)))

CROSSREFS

Cf. A220952 (Knuth's original base-5 variant), A300856 (inverse permutation), A300855 (inverse of A220952).

Sequence in context: A222194 A057224 A153695 * A255261 A181303 A276529

Adjacent sequences:  A300854 A300855 A300856 * A300858 A300859 A300860

KEYWORD

nonn,base,nice

AUTHOR

M. F. Hasler, Mar 13 2018

STATUS

approved

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Last modified July 26 15:49 EDT 2021. Contains 346294 sequences. (Running on oeis4.)