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A078822
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Number of distinct binary numbers contained as substrings in the binary representation of n.
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30
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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
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0,3
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COMMENTS
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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
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..16384 (first 1001 terms from Reinhard Zumkeller)
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FORMULA
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For k>0: a(2^k-2) = 2*(k-1)+1, a(2^k-1) = k, a(2^k) = k+2;
for k>1: a(2^k+1) = k+2;
for k>0: a(2^k-1) = A078824(2^k-1), a(2^k) = A078824(2^k).
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EXAMPLE
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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.
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MAPLE
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a:= n-> (s-> nops({seq(seq(parse(s[i..j]), i=1..j),
j=1..length(s))}))(""||(convert(n, binary))):
seq(a(n), n=0..85); # Alois P. Heinz, Jan 20 2021
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MATHEMATICA
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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}] (* Jean-François Alcover, Dec 01 2011 *)
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PROG
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(Haskell)
a078822 = length . a119709_row
import Numeric (showIntAtBase)
-- Reinhard Zumkeller, Aug 13 2013, Sep 14 2011
(PARI) a(n) = {if (n==0, 1, vb = binary(n); vf = []; for (i=1, #vb, for (j=1, #vb - i + 1, pvb = vector(j, k, vb[i+k-1]); f = subst(Pol(pvb), x, 2); vf = Set(concat(vf, f)); ); ); #vf); } \\ Michel Marcus, May 08 2016; corrected Jun 13 2022
(Python)
def a(n): return 1 if n == 0 else len(set(((((2<<l)-1)<<i)&n)>>i for i in range(n.bit_length()) for l in range(n.bit_length()-i)))
print([a(n) for n in range(64)]) # Michael S. Branicky, Jul 28 2022
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CROSSREFS
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Cf. A078823, A078826, A078824, A007088, A141297, A144623, A144624.
Sequence in context: A322348 A321232 A326015 * A224980 A154392 A069745
Adjacent sequences: A078819 A078820 A078821 * A078823 A078824 A078825
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KEYWORD
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nonn,base,nice,look
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AUTHOR
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Reinhard Zumkeller, Dec 08 2002
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STATUS
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approved
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