

A078822


Number of distinct binary numbers contained as substrings in the binary representation of n.


28



1, 1, 3, 2, 4, 4, 5, 3, 5, 5, 5, 6, 7, 7, 7, 4, 6, 6, 6, 7, 7, 6, 8, 8, 9, 9, 9, 9, 10, 10, 9, 5, 7, 7, 7, 8, 7, 8, 9, 9, 9, 9, 7, 9, 11, 10, 11, 10, 11, 11, 11, 11, 12, 11, 11, 12, 13, 13, 13, 13, 13, 13, 11, 6, 8, 8, 8, 9, 8, 9, 10, 10, 9, 8, 10, 11, 11, 12, 12, 11, 11, 11, 11, 12, 10, 8
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OFFSET

0,3


COMMENTS

For k>0: a(2^k2)=2*(k1)+1, a(2^k1)=k, a(2^k)=k+2;
for k>1: a(2^k+1)=k+2;
for k>0: a(2^k1)=A078824(2^k1), a(2^k)=A078824(2^k).
For n>0: 0<a(2*n)a(n)<=A070939(n)+1, 0<a(2*n+1)a(n) < A070939(n).  Reinhard Zumkeller, Mar 07 2008
Row lengths in triangle A119709.  Reinhard Zumkeller, Aug 14 2013


LINKS

R. Zumkeller, Table of n, a(n) for n = 0..1000


EXAMPLE

n=10 > '1010' contains 5 different binary numbers: '0' (b0bb or bbb0), '1' (1bbb or bb1b), '10' (10bb or bb10), '101' (101b) and '1010' itself, therefore a(10)=5.


MATHEMATICA

a[n_] := (id = IntegerDigits[n, 2]; nd = Length[id]; Length[ Union[ Flatten[ Table[ id[[j ;; k]], {j, 1, nd}, {k, j, nd}], 1] //. {0, b__} :> {b}]]); Table[ a[n], {n, 0, 85}] (* JeanFrançois Alcover, Dec 01 2011 *)


PROG

(Haskell)
a078822 = length . a119709_row
import Numeric (showIntAtBase)
 Reinhard Zumkeller, Aug 13 2013, Sep 14 2011
(PARI) a(n) = {vb = binary(n); vf = []; for (i=1, #vb, for (j=1, #vb  i + 1, pvb = vector(j, k, vb[i+k1]); f = subst(Pol(pvb), x, 2); vf = Set(concat(vf, f)); ); ); #vf; } \\ Michel Marcus, May 08 2016


CROSSREFS

Cf. A078823, A078826, A078824, A007088, A144623, A144624.
Sequence in context: A322349 A322348 A321232 * A224980 A154392 A069745
Adjacent sequences: A078819 A078820 A078821 * A078823 A078824 A078825


KEYWORD

nonn,base,nice,look


AUTHOR

Reinhard Zumkeller, Dec 08 2002


STATUS

approved



