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A300857
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Base-7 based twisted permutation of the nonnegative integers.
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5
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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nonn,base,nice
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AUTHOR
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STATUS
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approved
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