A300857 Base-7 based twisted permutation of the nonnegative integers.

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

Offset: 0

Author

M. F. Hasler, Mar 13 2018

Keywords

• nonn
• base
• nice

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:

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

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

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

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

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

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

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

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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.

Crossrefs

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

Programs

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)))