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

%I A071055 #55 Feb 16 2025 08:32:46
%S A071055 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,
%T A071055 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,
%U A071055 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
%N A071055 Number of 0's in n-th row of triangle in A071038.
%C A071055 a(n) is also the number of pairs of consecutive entries in the n-th row of Pascal's triangle with opposite parity.
%C A071055 All terms appear to be of the form 2^k - 2 (checked for n <= 10000). - _Michael De Vlieger_, Mar 02 2015
%C A071055 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
%D A071055 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H A071055 Charles R Greathouse IV, <a href="/A071055/b071055.txt">Table of n, a(n) for n = 0..10000</a>
%H A071055 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule182.html">Rule 182</a>
%H A071055 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A071055 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
%F A071055 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
%e A071055 From _Omar E. Pol_, Mar 02 2015: (Start)
%e A071055 Also, written as an irregular triangle in which the row lengths are the powers of 2, the sequence begins:
%e A071055 0;
%e A071055 0,2;
%e A071055 0,2,2,6;
%e A071055 0,2,2,6,2,6,6,14;
%e A071055 0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30;
%e A071055 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;
%e A071055 ...
%e A071055 It appears that the right border gives the nonnegative terms of A000918.
%e A071055 It appears that the row sums give A056182.
%e A071055 (End)
%t A071055 Count[#, n_ /; n == 0] & /@
%t A071055 Flatten[CellularAutomaton[182, {{1}, 0}, {{#}}] & /@ Range[0, 100],
%t A071055 1] (* _Michael De Vlieger_, Mar 02 2015 *)
%o A071055 (PARI) A011371(n)=my(s); while(n>>=1, s+=n); s
%o A071055 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
%Y A071055 Cf. A071042.
%K A071055 nonn
%O A071055 0,3
%A A071055 _Hans Havermann_, May 26 2002