

A071055


Number of 0's in nth row of triangle in A071038.


1



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
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OFFSET

0,3


COMMENTS

a(n) is also the number of pairs of consecutive entries in the nth 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..n1, 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

Eric Weisstein's World of Mathematics, Rule 182


FORMULA

a(n)=b(n+1), with b(0)=0, b(2n)=b(n), b(2n+1)=2b(n)+22[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

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],


PROG

(PARI) A011371(n)=my(s); while(n>>=1, s+=n); s


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



