OFFSET
0,8
COMMENTS
a(n) takes the value k for the first time at n = 2^(k+1)-1. Cf. A000225. - Robert G. Wilson v, Apr 02 2009
a(n) = A213629(n,3) for n > 2. - Reinhard Zumkeller, Jun 17 2012
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J.-P. Allouche, On an Inequality in a 1970 Paper of R. L. Graham, INTEGERS 21A (2021), #A2.
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
John Brillhart and L. Carlitz, Note on the Shapiro Polynomials, Proceedings of the American Mathematical Society, volume 25, number 1, May 1970, pages 114-118 (see A001782 for a scanned copy), with a(n) = exponent in theorem 4.
Helmut Prodinger, Generalizing the sum of digits function, SIAM J. Algebraic Discrete Methods, Vol. 3, No. 1 (1982), pp. 35-42. MR0644955 (83f:10009). [See B_2(11,n) on p. 35. - N. J. A. Sloane, Apr 06 2014]
Michel Rigo and Manon Stipulanti, Revisiting regular sequences in light of rational base numeration systems, arXiv:2103.16966 [cs.FL], 2021. Mentions this sequence.
Bartosz Sobolewski and Lukas Spiegelhofer, Block occurrences in the binary expansion, arXiv:2309.00142 [math.NT], 2023.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Eric Weisstein's World of Mathematics, Digit Block.
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence.
FORMULA
a(4n) = a(4n+1) = a(n), a(4n+2) = a(2n+1), a(4n+3) = a(2n+1) + 1. - Ralf Stephan, Aug 21 2003
G.f.: (1/(1-x)) * Sum_{k>=0} t^3/((1+t)*(1+t^2)), where t = x^(2^k). - Ralf Stephan, Sep 10 2003
EXAMPLE
The binary expansion of 15 is 1111, which contains three occurrences of 11, so a(15)=3.
MAPLE
# To count occurrences of 11..1 (k times) in binary expansion of v:
cn := proc(v, k) local n, s, nn, i, j, som, kk;
som := 0;
kk := convert(cat(seq(1, j = 1 .. k)), string);
n := convert(v, binary);
s := convert(n, string);
nn := length(s);
for i to nn - k + 1 do
if substring(s, i .. i + k - 1) = kk then som := som + 1 fi od;
som; end; # This program no longer worked. Corrected by N. J. A. Sloane, Apr 06 2014.
[seq(cn(n, 2), n=0..300)];
# Alternative:
A014081 := proc(n) option remember;
if n mod 4 <= 1 then procname(floor(n/4))
elif n mod 4 = 2 then procname(n/2)
else 1 + procname((n-1)/2)
fi
end proc:
A014081(0):= 0:
map(A014081, [$0..1000]); # Robert Israel, Sep 04 2015
MATHEMATICA
f[n_] := Count[ Partition[ IntegerDigits[n, 2], 2, 1], {1, 1}]; Table[ f@n, {n, 0, 104}] (* Robert G. Wilson v, Apr 02 2009 *)
Table[SequenceCount[IntegerDigits[n, 2], {1, 1}, Overlaps->True], {n, 0, 120}] (* Harvey P. Dale, Jun 06 2022 *)
PROG
(Haskell)
import Data.Bits ((.&.))
a014081 n = a000120 (n .&. div n 2) -- Reinhard Zumkeller, Jan 23 2012
(PARI) A014081(n)=sum(i=0, #binary(n)-2, bitand(n>>i, 3)==3) \\ M. F. Hasler, Jun 06 2012
(PARI) a(n) = hammingweight(bitand(n, n>>1)) ;
vector(105, i, a(i-1)) \\ Gheorghe Coserea, Aug 30 2015
(Python)
def a(n): return sum([((n>>i)&3==3) for i in range(len(bin(n)[2:]) - 1)]) # Indranil Ghosh, Jun 03 2017
(Python)
from re import split
def A014081(n): return sum(len(d)-1 for d in split('0+', bin(n)[2:]) if d != '') # Chai Wah Wu, Feb 04 2022
CROSSREFS
KEYWORD
nonn,base,easy,changed
AUTHOR
STATUS
approved