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A360449
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The lexicographically earliest sequence a(n) = v(x[n]) where x[k], k >= 0, are distinct finite nonnegative integer sequences with |x[k] - x[k+1]| = 1, and v(x) = Sum_{k >= 0} x(k)*b^k + b^(b-1), b = max(deg(x), sup(x)) + 1.
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2
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1, 3, 5, 4, 15, 16, 17, 14, 11, 20, 19, 18, 21, 22, 23, 26, 25, 24, 33, 30, 27, 28, 29, 32, 31, 34, 35, 107, 91, 75, 71, 67, 83, 87, 103, 99, 115, 114, 113, 112, 116, 117, 118, 119, 123, 122, 121, 120, 124, 108, 92, 76, 77, 78, 79, 95, 94, 93, 109, 110, 111
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OFFSET
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0,2
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COMMENTS
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This is actually an enumeration and a Hamiltonian path in N[X] = N^(N) = L^p(N; N), the space of all polynomials (or finite sequences, or equivalently sequences with finite L^p norm for any 1 <= p < oo) with coefficients in N = {0, 1, 2, 3, ...}.
|x - y| = 1 means that x and y are nearest neighbors for any of these p-norms, i.e., they differ in exactly one component, and by exactly +-1 in that component.
The function v(x) assigns a unique positive integer to any polynomial or finite sequence x, and v(x) is strictly increasing with increasing b(x) = max(sup(x), deg(x)) + 1, where as usual deg(x) = sup { k >= 0 : x(k) != 0 }.
Thus, by construction, all finite nonnegative integer sequences are listed exactly once, usually in order of increasing b(x) - but it can happen that some not yet listed sequences with smallest b-value are not among the nearest neighbors.
Sequence A360450 is a variant where a(n) = Sum_{k>=0} x(k)*b^k with b = 10 instead, when b(x) <= 10. This makes the sequences more human readable and searchable: e.g., polynomials 1 + X, X, 2*X and 1+X*2 would then become 11, 10, 20, 21, ... which is definitely easier to "read".
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LINKS
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EXAMPLE
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The values a(n) correspond to the following sequences:
n | a(n) | deg(x) | sup(x) | b(x) | polynomial = sequence x = x[n].
---+------+--------+--------+------+---------------------
0 | 1 | -oo | 0 | 1 | 0 = (0, ...): Only seq. with b = 1.
1 | 3 | 0 | 1 | 2 | 1 = (1, 0, ...)
2 | 5 | 1 | 1 | 2 | 1 + X = (1, 1, 0, ...)
3 | 4 | 1 | 1 | 2 | X = (0, 1, 0, ...): Last seq. with b = 2.
4 | 15 | 1 | 2 | 3 | 2*X = (0, 2, 0, ...)
5 | 16 | 1 | 2 | 3 | 1 + 2*X = (1, 2, 0, ...)
6 | 17 | 1 | 2 | 3 | 2 + 2*X = (2, 2, 0, ...)
7 | 14 | 1 | 2 | 3 | 2 + X = (2, 1, 0, ...)
8 | 11 | 0 | 2 | 3 | 2 = (2, 0, ...)
9 | 20 | 2 | 2 | 3 | 2 + X^2 = (2, 0, 1, 0, ...)
10 | 19 | 2 | 1 | 3 | 1 + X^2 = (1, 0, 1, 0, ...)
11 | 18 | 2 | 1 | 3 | X^2 = (0, 0, 1, 0, ...)
12 | 21 | 2 | 1 | 3 | X + X^2 = (0, 1, 1, 0, ...)
13 | 22 | 2 | 1 | 3 | 1 + X + X^2 = (1, 1, 1, 0, ...)
14 | 23 | 2 | 2 | 3 | 2 + X + X^2 = (2, 1, 1, 0, ...)
15 | 26 | 2 | 2 | 3 | 2 + 2*X + X^2 = (2, 2, 1, 0, ...)
16 | 25 | 2 | 2 | 3 | 1 + 2*X + X^2 = (1, 2, 1, 0, ...)
17 | 24 | 2 | 2 | 3 | 2*X + X^2 = (0, 2, 1, 0, ...)
18 | 33 | 2 | 2 | 3 | 2*X + 2*X^2 = (0, 2, 2, 0, ...)
19 | 30 | 2 | 2 | 3 | X + 2*X^2 = (0, 1, 2, 0, ...)
20 | 27 | 2 | 2 | 3 | 2*X^2 = (0, 0, 2, 0, ...)
...
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PROG
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(PARI) L360449=List('x*0); A360449(n)={local(T, Q, P.m=if(P, max(vecmax(Vec(P))+1, #P), 1), P.v=subst(P, 'x, P=P.m)+P^(P-1), S, chk(P, V=P.v)=(!T||V<T)&& !setsearch(S, P)&& [T=V, Q=P]); for(n=#L360449, n, S=if(Q, setunion(S, [Q]), Set(L360449)); my(P=L360449[n], B=P.m); T=0; for(k=1, #P, polcoef(P, #P-k)&& chk(P-x^(#P-k))); T|| for(k=0, #P+1, chk(P+'x^k)&& polcoef(P, k)<B-1&& break); listput(L360449, Q)); L360449[n+1].v}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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