A071055 Number of 0's in n-th row of triangle in A071038.

0, 0, 2, 0, 2, 2, 6, 0, 2, 2, 6, 2, 6, 6, 14, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 2, 6, 6, 14, 6, 14, 14, 30, 6, 14, 14, 30, 14, 30, 30, 62, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 2, 6, 6, 14, 6

Offset

0,3

Comments

a(n) is also the number of pairs of consecutive entries in the n-th row of Pascal's triangle with opposite parity.

All terms appear to be of the form 2^k - 2 (checked for n <= 10000). - Michael De Vlieger, Mar 02 2015

This appears to be equal to the number of previous values k, from 1..n-1, such that k AND n = k, where 'AND' is binary AND, and where the sequence starts at 1. For example, 1 AND 2 = 0, so a(2) = 0, while 1 AND 3 = 1 and 2 AND 3 = 2, so a(3) = 2. It follows from this that if n = 2^m - 1 then a(n) = n - 1 = 2^m - 2, giving the right border values noted below. - Scott R. Shannon, Apr 19 2023

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Rule 182

Index entries for sequences related to cellular automata

Formula

a(n)=b(n+1), with b(0)=0, b(2n)=b(n), b(2n+1)=2b(n)+2-2[n==0] (conjectured). - Ralf Stephan, Mar 05 2004

a(n) = pext(n, n + 1) (conjectured) where pext is the "parallel bits extract" instruction of the x86 CPU; pext(x, mask) extracts bits from x at the bit locations specified by mask to contiguous low bits. - Falk Hüffner, Jul 26 2019

Example

From Omar E. Pol, Mar 02 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the powers of 2, the sequence begins:
0;
0,2;
0,2,2,6;
0,2,2,6,2,6,6,14;
0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30;
0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30,2,6,6,14,6,14,14,30,6,14,14,30,14,30,30,62;
...
It appears that the right border gives the nonnegative terms of A000918.
It appears that the row sums give A056182.
(End)

Mathematica

Count[#, n_ /; n == 0] & /@
Flatten[CellularAutomaton[182, {{1}, 0}, {{#}}] & /@ Range[0, 100],
1] (* Michael De Vlieger, Mar 02 2015 *)

Prog

(PARI) A011371(n)=my(s); while(n>>=1, s+=n); s
a(n)=my(t=A011371(n)); sum(k=1, n, (A011371(k)+A011371(n-k)==t)!=(A011371(k-1)+A011371(n-k+1)==t)) \\ Charles R Greathouse IV, Mar 02 2015

Crossrefs

Cf. A071042.

Sequence in context: A301823 A301999 A171936 * A183034 A354101 A078052

Adjacent sequences: A071052 A071053 A071054 * A071056 A071057 A071058

Keyword

nonn

Author

Hans Havermann, May 26 2002

Status

approved