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A241773
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A sequence constructed so that the probability of occurrence of integer i > 0 is given by p(i) = log_2[(i+1)^2/(i^2+2*i)], which is the Gauss-Kuzmin distribution.
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1
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1, 2, 3, 1, 4, 1, 5, 2, 1, 6, 1, 7, 1, 2, 3, 1, 8, 1, 9, 2, 1, 4, 1, 10, 1, 3, 2, 1, 11, 1, 2, 5, 1, 12, 1, 3, 1, 2, 4, 1, 13, 1, 2, 6, 1, 14, 1, 3, 1, 2, 15, 1, 16, 1, 2, 4, 1, 3, 1, 5, 7, 1, 2, 1, 17, 1, 2, 3, 1, 18, 1, 8, 2, 1, 4, 1, 6, 1, 2, 3, 1, 5, 1, 19
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OFFSET
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1,2
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COMMENTS
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Let the sequence be A = {a(i)}, i = 1, 2, 3,... and define p(i) =
log_2[(i+1)^2/(i^2+2*i)]. Additionally, define u(j, k) = k*p(j) - N(j, k), where N(j, k) is the number of occurrences of j in {a(i)}, i = 1,..., k-1. Refer to the first argument of u as the "index" of u. Then A is defined by a(1) = 1 and, for i > 1, a(i) = m, where m is the index of the maximal element of the set {u(j, i)}, j = 1, 2, 3,... That there is a single maximal element for all i is guaranteed by the fact that p(i) - p(j) is irrational for all i not equal to j.
Interpreting sequence A as the partial coefficients of the continued fraction expansion of a real number C, say, then C = 1.44224780173510148... which is, by construction, normal (in the continued fraction sense).
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LINKS
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MAPLE
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pdf := i -> -log[2](1 - 1/(i+1)^2);
gen_seq := proc(n)
local i, j, N, A, u, mm, ndig;
ndig := 40; N := 'N';
for i from 1 to n do N[i] := 0; end do;
A := 'A'; A[1] := 1; N[1] := 1;
for i from 2 to n do
u := 'u';
for j from 1 to n do
u[j] := i*pdf(j) - N[j];
end do;
mm := max_maxind(evalf(convert(u, list), ndig));
if mm[3] then
A[i] := mm[1];
N[mm[1]] := N[mm[1]] + 1;
else
return();
end if;
end do;
return(convert(A, list));
end:
max_maxind := proc(inl)
local uniq, mxind, mx, i;
uniq := `true`;
if nops(inl) = 1 then return([1, inl[1], uniq]); end if;
mxind := 1; mx := inl[1];
for i from 2 to nops(inl) do
if inl[i] > mx then
mxind := i;
mx := inl[i];
uniq := `true`;
elif inl[i] = mx then
uniq := `false`;
end if;
end do;
return([mxind, mx, uniq]);
end:
gen_seq(100);
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CROSSREFS
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KEYWORD
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cofr,easy,nonn
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AUTHOR
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STATUS
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approved
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