A056753 Only odd numbers occur and for all k there are k numbers between any two successive occurrences of k.

1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 9, 1, 3, 1, 7, 1, 3, 1, 11, 1, 3, 1, 7, 1, 3, 1, 13, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 17, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 19, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 21, 1, 3, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 3, 1, 23, 1, 3, 1, 7, 1, 3, 1

Offset: 0

Author

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 19 2001

Keywords

• nice
• nonn
• look

Comments

Only the numbers 2^m - 1 occur more than once.

a(A005843(n)) = 1; a(A016813(n)) = 3; a(A004771(n)) = 7; a(A008598(n) + 35) = 15; a(A008598(n) + 155) = 31. - Reinhard Zumkeller, Aug 23 2009

A181497(n) = smallest m such that A056753(m) = 2*n + 1. - Reinhard Zumkeller, Oct 24 2010

Links

• R. Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A004771, A005843, A008598, A016813, A164632, A181497.

Programs

Haskell

import Data.List (intercalate, group)
a056753 n = a056753_list !! n
a056753_list = [1] ++ odds [] where
   odds xs = xs ++ (intercalate xs' $ group [y+2,y+4..2*y+1]) ++ odds xs'
        where y = 2 * length xs + 1
              xs' = xs ++ [y] ++ xs
-- Reinhard Zumkeller, Feb 25 2012, Oct 24 2010

Magma

S:=[ 0: n in [1..100] ]; k:=1; p:=Position(S, 0, 1); while p gt 0 do for j in [p..#S by k+1] do if S[j] eq 0 then S[j]:=k; else break; end if; end for; f:=p; p:=Position(S, 0, f); k+:=2; end while; S; // Klaus Brockhaus, Oct 25 2010

Mathematica

a[n_] := a[n] = (ClearAll[f]; f[i_, x_, y_, z_] := f[i, x, y, z] = If[i == n, If[x == 1, a[n-z] + 2, a[n-z]], If[x == 1, If[y == 1, f[i+1, 2z, z, 2z], f[i+1, z, y-1, z]], f[i+1, x-1, y, z]]]; If[n == 0, 1, f[1, 1, 1, 1]]); Table[a[n], {n, 0, 98}] (* Jean-François Alcover, Dec 14 2011, after Reinhard Zumkeller *)

Formula

Let x = a(n - A164632(n)), a(n) = if (x occurred exactly once so far) then x+2 else x. - Reinhard Zumkeller, Aug 23 2009