derivative of n

[N. J. A. Sloane]

Dear Sequence Fans,   Here is a new class of sequences.

The derivative of a binary sequence is obtained by adding the bits
in pairs:   the derivative of 1 1 1 0 1 0 0 0 1 1 is
                               0 0 1 1 1 0 0 1 0
The periodic derivative includes the sum of the last bits:
1 1 1 0 1 0 0 0 1 0 has periodic derivative
0 0 1 1 1 0 0 1 1 1

One can do the same thing in other bases.

This suggested the following sequences (and many more
which I leave it you people to produce!).
NJAS

%I A038554
%S A038554 0,0,1,0,2,3,1,0,4,5,7,6,2,3,1,0,8,9,11,10,14,15,13,12,4
%N A038554 Derivative of n.
%R A038554
%O A038554 0,5
%K A038554 nonn,nice,easy,more
%F A038554 Write n in binary, replace each bit by sum of its 2 neighbors (a(0)=a(1)=0 by convention).
%e A038554 18=10010, derivative is 1011, so a(18)=11.
%A A038554 njas

%I A038558
%S A038558 2,4,5,8,9,11,10,16,17,19,18,23,22,20,21
%N A038558 Smallest number with derivative n.
%R A038558
%O A038558 1,1
%K A038558 nonn,nice,easy,more
%Y A038558 Cf. A038554-A038556.
%A A038558 njas

%I A038570
%S A038570 0,0,0,0,1,0,0,0,2,3,0,1,1,0,0,0,4,5,6,7,1,0,3,2,2
%N A038570 Second derivative of n.
%R A038570
%O A038570 0,9
%K A038570 nonn,easy,nice,more
%Y A038570 Cf. A038554.
%A A038570 njas

%I A038571
%S A038571 0,1,2,1,3,2,2,1,4,3,2,3,3,2,2,1,5,4,4,3,3,2,3,4,4
%N A038571 Number of times n must be differentiated to reach 0.
%R A038571
%O A038571 0,3
%K A038571 nonn,easy,nice,more
%Y A038571 Cf. A038554.
%A A038571 njas

%I A038555
%S A038555 0,0,0,1,2,0,2,0,1,3,4,5,7,8,6,2,0,1,6,7,8,1,2,0,5
%N A038555 Derivative of n in base 3.
%R A038555
%O A038555 0,5
%K A038555 nonn,nice,easy,more
%F A038555 Write n in ternary, replace each digit by sum of its 2 neighbors.
%Y A038555 Cf. A038554.
%e A038555 15 = 120 in ternary, derivative is 02 = 2, so a(15)=2.
%A A038555 njas

%I A038556
%S A038556 0,0,3,0,5,6,3,0,9,10,15,12,5,6,3,0
%N A038556 Periodic derivative of n.
%R A038556
%O A038556 0,3
%K A038556 nonn,nice,easy,more
%D A038556 Simmons, G. J. The structure of the differentiation digraphs of binary sequences. Ars Combin. 35 (1993), A, 71--88. Math. Rev 95f:05052.
%F A038556 If n=b_k b_{k-1} ... b_0 in base 2, a(n) is number with binary expansion (b_k+b_{k-1}) (b_{k-1}+b_{k-2}) ... (b_1+b_0) (b_0+b_{k}).
%Y A038556 Cf. A038554.
%e A038556 11=1011->1100 so a(11)=12.
%A A038556 njas

%I A038557
%S A038557 0,0,1,4,8,0,8,0,4,10,14,15,22,26,18,7,2
%N A038557 Periodic derivative of n in base 3.
%R A038557
%O A038557 0,4
%K A038557 nonn,nice,easy,more
%D A038557 Simmons, G. J. The structure of the differentiation digraphs of binary sequences. Ars Combin. 35 (1993), A, 71--88. Math. Rev 95f:05052.
%F A038557 If n=b_k b_{k-1} ... b_0 in base 3, a(n) is number with ternary expansion (b_k+b_{k-1}) (b_{k-1}+b_{k-2}) ... (b_1+b_0) (b_0+b_{k}).
%Y A038557 Cf. A038554-A038556.
%e A038557 14=112->200 so a(14)=18.
%A A038557 njas