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
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Hans Havermann, May 26 2002
STATUS
approved