login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A241773 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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

LINKS

Jonathan Deane, Table of n, a(n) for n = 1..10000

Jonathan Deane, An integer sequence whose members obey a given p.d.f.

MAPLE

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);

CROSSREFS

Sequence in context: A286477 A277230 A218534 * A205790 A279820 A235791

Adjacent sequences:  A241770 A241771 A241772 * A241774 A241775 A241776

KEYWORD

cofr,easy,nonn

AUTHOR

Jonathan Deane, Apr 28 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 12:47 EDT 2019. Contains 328257 sequences. (Running on oeis4.)