A173922 -id:A173922 - cp's OEIS frontend

A079944 A run of 2^n 0's followed by a run of 2^n 1's, for n=0, 1, 2, ...

0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Feb 21 2003

With offset 2, this is the second bit in the binary expansion of n. - Franklin T. Adams-Watters, Feb 13 2009

a(n) = A173920(n+2,2); in the sequence of nonnegative integers (cf. A001477) substitute all n by 2^floor(n/2) occurrences of (n mod 2). - Reinhard Zumkeller, Mar 04 2010

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. See Example 1.34.

R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions

Cf. A086694, A079882, A079945, A173922, A173923.

a079944 n = a079944_list !! n
a079944_list =  f [0,1] where f (x:xs) = x : f (xs ++ [x,x])
-- Reinhard Zumkeller, Oct 14 2010, Mar 28 2011

Table[IntegerDigits[n + 2, 2][[2]], {n, 0, 100}] (* Jean-François Alcover, Jul 26 2019 *)

a(n)=binary(n+2)[2] \\ Charles R Greathouse IV, Nov 07 2016

a(n) = floor(log[2](4*(n+2)/3)) - floor(log[2](n+2)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003
For n >= 2, a(n-2)=1+floor(log[2](n/3))-floor(log[2](n/2)) - Benoit Cloitre, Mar 03 2003
G.f.: 1/x^2/(1-x) * (1/x + sum(k>=0, x^(3*2^k)-x^2^(k+1))). - Ralf Stephan, Jun 04 2003
a(n) = A000035(A004526(A030101(n+2))). - Reinhard Zumkeller, Mar 04 2010

A173920 Triangle read by rows: T(n,k) = convolution of n with k in binary representation, 0<=k<=n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2

Offset: 0

Reinhard Zumkeller, Mar 04 2010

T(n,k) = SUM(bn(i)*bk(L-i-1): 0<=iA070939(n), n=SUM(bn(i)*2^i:0<=i

T(n,2*k+1) = T(n,2*k) + 1;

T(n,k) <= MIN{A000120(n),A000120(k)};

row sums give A173921; central terms give A159780;

T(n,0) = A000004(n);

T(n,1) = A000012(n) for n>0;

T(n,2) = A079944(n-2) for n>1;

T(n,3) = A079882(n-2) for n>2;

T(n,4) = A173922(n-4) for n>3;

T(n,8) = A173923(n-8) for n>7;

T(n,n) = A159780(n).

			T(13,10) = T('1101','1010') = 1*0 + 1*1 + 0*0 + 1*1 = 2;
T(13,11) = T('1101','1011') = 1*1 + 1*1 + 0*0 + 1*1 = 3;
T(13,12) = T('1101','1100') = 1*0 + 1*0 + 0*1 + 1*1 = 1;
T(13,13) = T('1101','1101') = 1*1 + 1*0 + 0*1 + 1*1 = 2.
Triangle begins:
  0;
  0, 1;
  0, 1, 0;
  0, 1, 1, 2;
  0, 1, 0, 1, 0;
  0, 1, 0, 1, 1, 2;
  ...

T[n_, k_] := Module[{bn, bk, lg},
     bn = IntegerDigits[n, 2];
     bk = IntegerDigits[k, 2];
     lg = Max[Length[bn], Length[bk]];
     ListConvolve[PadLeft[bn, lg], PadLeft[bk, lg]]][[1]];
Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

T(n,k) = c(A030101(n),k,0) with c(x,y,z) = if y=0 then z else c([x/2],[y/2],z+(x mod 2)*(y mod 2)).

A200675 Powers of 2 repeated 4 times.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 32, 32, 32, 32, 64, 64, 64, 64, 128, 128, 128, 128, 256, 256, 256, 256, 512, 512, 512, 512, 1024, 1024, 1024, 1024, 2048, 2048, 2048, 2048, 4096, 4096, 4096, 4096

Offset: 0

Jeremy Gardiner, Nov 20 2011

Run lengths in A173922.

			a(4) = 2^1 = 2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2)

Cf. A000079, A016116, A173862, A108105, A173922, A200678.

CoefficientList[Series[x*(1 + x)*(1 + x^2)/(1 - 2*x^4), {x,0,50}],x] (* or *) Table[2^(Floor[n/4]), {n,0,50}] (* G. C. Greubel, Apr 30 2017 *)

a(n)=2^(n\4) \\ Charles R Greathouse IV, Oct 03 2016

def A200675(n): return 1<<(n>>2) # Chai Wah Wu, Jan 30 2023

a(n) = 2^floor(n/4).

G.f.: x*(1+x)*(1+x^2)/(1-2*x^4 ). - R. J. Mathar, Nov 21 2011

Offset corrected by Charles R Greathouse IV, Oct 03 2016

A173923 In the sequence of nonnegative integers substitute all n by 2^floor(n/8) occurrences of (n mod 2).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 0

Reinhard Zumkeller, Mar 04 2010

nonn

a(n) = A173920(n+8,8).

a(n) = A000035(A132292(A030101(n+8)+1)).

Cf. A001477, A079944, A173922.

Table[PadRight[{},2^Floor[n/8],Mod[n,2]],{n,0,30}]//Flatten (* Harvey P. Dale, Aug 07 2021 *)

Sequence definition changed for clarity (see A173922).

Showing 1-4 of 4 results.

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A079944 A run of 2^n 0's followed by a run of 2^n 1's, for n=0, 1, 2, ...

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A173920 Triangle read by rows: T(n,k) = convolution of n with k in binary representation, 0<=k<=n.

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A200675 Powers of 2 repeated 4 times.

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A173923 In the sequence of nonnegative integers substitute all n by 2^floor(n/8) occurrences of (n mod 2).

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