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Number of i such that d(i)>d(i-1), where Sum{d(i)*4^i: i=0,1,....,m} is base 4 representation of n.
2

%I #22 May 15 2016 11:28:30

%S 0,0,0,1,0,0,0,1,1,0,0,1,1,1,0,1,1,1,1,1,0,0,0,1,1,0,0,1,1,1,0,1,1,1,

%T 1,2,1,1,1,1,1,0,0,1,1,1,0,1,1,1,1,2,1,1,1,2,2,1,1,1,1,1,0,1,1,1,1,2,

%U 1,1,1,2,2,1,1,2,2,2,1,1,1,1,1,1,0,0,0,1,1,0

%N Number of i such that d(i)>d(i-1), where Sum{d(i)*4^i: i=0,1,....,m} is base 4 representation of n.

%C From _Jeffrey Shallit_, May 15 2016: (Start)

%C A "2-regular" sequence, satisfying the recurrence relations:

%C a(4n+3) = a(n)

%C a(16n) = a(16n+1) = a(16n+2) = a(4n)

%C a(16n+5) = a(16n+6) = a(4n+1)

%C a(16n+8) = a(16n+9) = a(4n+2) + 1

%C a(16n+10) = a(4n+2)

%C a(16n+12) = a(16n+13) = a(16n+14) = a(4n+1) + 1

%C a(64n+4) = a(4n) + 1

%C a(64n+20) = a(16n+4)

%C a(64n+36) = a(4n+2) + 2

%C a(64n+52) = a(n) + 2

%C (End)

%H Michael De Vlieger, <a href="/A037819/b037819.txt">Table of n, a(n) for n = 1..10000</a>

%p A037819 := proc(n)

%p a := 0 ;

%p dgs := convert(n,base,4);

%p for i from 2 to nops(dgs) do

%p if op(i,dgs)>op(i-1,dgs) then

%p a := a+1 ;

%p end if;

%p end do:

%p a ;

%p end proc: # _R. J. Mathar_, Oct 15 2015

%t Table[Count[Differences@ IntegerDigits[n, 4], k_ /; k < 0], {n, 120}] (* _Michael De Vlieger_, May 15 2016 *)

%Y Cf. A037802.

%K nonn,base

%O 1,36

%A _Clark Kimberling_

%E Sign in Name corrected by _R. J. Mathar_, Oct 15 2015