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A240388 A sequence related to the Stern sequence s(n) (A002487), defined by w(n) = s(3n)/2.

%I A240388 #49 Jan 25 2024 05:22:45
%S A240388 0,1,1,2,1,2,2,4,1,4,2,3,2,5,4,6,1,6,4,5,2,3,3,7,2,9,5,7,4,9,6,8,1,8,
%T A240388 6,9,4,7,5,9,2,7,3,4,3,8,7,11,2,13,9,12,5,8,7,15,4,17,9,11,6,13,8,10,
%U A240388 1,10,8,13,6,11,9,17,4,15,7,8,5,12,9,13,2,11,7,8,3,4,4,10,3,14,8,12,7,16,11,15,2,17,13,20,9,16,12,22,5,18,8,10,7,18,15,23,4,25,17,22,9,14,11,23,6,25,13,15,8,17,10,12,1
%N A240388 A sequence related to the Stern sequence s(n) (A002487), defined by w(n) = s(3n)/2.
%C A240388 The even terms in the Stern sequence, divided by 2.
%H A240388 Alois P. Heinz, <a href="/A240388/b240388.txt">Table of n, a(n) for n = 0..16384</a>
%H A240388 Jennifer Lansing, <a href="http://jointmathematicsmeetings.org/amsmtgs/2160_abstracts/1096-11-1220.pdf">On the Stern sequence and a related sequence</a>, Joint Mathematics Meetings, Baltimore, 2014.
%H A240388 Jennifer Lansing, <a href="/A240388/a240388.pdf">Dissertation: On the Stern sequence and a related sequence</a>, PhD dissertation, University of Illinois, 2014.
%H A240388 Jennifer Lansing, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Lansing/lansing2.html">Largest Values for the Stern Sequence</a>, J. Integer Seqs., 17 (2014), #14.7.5.
%F A240388 w(0)=0, w(1)=1, and w(3)=2.  For n >= 1, w(n) satisfies the recurrences w(2n)=w(n), w(8n +/- 1)=w(4n +/- 1) + 2w(n), w(8n +/- 3)=w(4n +/- 1) + w(2n +/- 1) -w(n).
%F A240388 a(n) = A002487(3*n) / 2. - _Joerg Arndt_, Jun 20 2022
%e A240388 w(7) = w(8-1) = w(3)+2w(1) = 2+2 = 4.
%e A240388 w(11) = w(8+3) = w(4+1)+w(2+1)-w(1)=w(5)+w(3)-w(1) = 2+2-1 = 3.
%e A240388 Comment from _N. J. A. Sloane_, Jul 01 2014: (Start)
%e A240388 May be arranged as a triangle:
%e A240388   0
%e A240388   1
%e A240388   1
%e A240388   2 1 2
%e A240388   2 4 1 4 2
%e A240388   3 2 5 4 6 1 6 4 5 2 3
%e A240388   3 7 2 9 5 7 4 9 6 8 1 8 6 9 4 7 5 9 2 7 3
%e A240388   ... (End)
%p A240388 A240388 := proc(n)
%p A240388     option remember;
%p A240388     local nloc;
%p A240388     if n <=1  then
%p A240388         n;
%p A240388     elif n = 3 then
%p A240388         2;
%p A240388     elif type(n,'even') then
%p A240388         procname(n/2) ;
%p A240388     elif modp(n,8) = 1 then
%p A240388         nloc := (n-1)/8 ;
%p A240388         procname(4*nloc+1)+2*procname(nloc) ;
%p A240388     elif modp(n,8) = 7 then
%p A240388         nloc := (n+1)/8 ;
%p A240388         procname(4*nloc-1)+2*procname(nloc) ;
%p A240388     elif modp(n,8) = 3 then
%p A240388         nloc := (n-3)/8 ;
%p A240388         procname(4*nloc+1)+procname(2*nloc+1)-procname(nloc) ;
%p A240388     else
%p A240388         nloc := (n+3)/8 ;
%p A240388         procname(4*nloc-1)+procname(2*nloc-1)-procname(nloc) ;
%p A240388     end if;
%p A240388 end proc: # _R. J. Mathar_, Jul 05 2014
%p A240388 # second Maple program:
%p A240388 b:= proc(n) option remember; `if`(n<2, n,
%p A240388       (q-> b(q)+(n-2*q)*b(n-q))(iquo(n, 2)))
%p A240388     end:
%p A240388 a:= n-> b(3*n)/2:
%p A240388 seq(a(n), n=0..128);  # _Alois P. Heinz_, Jun 20 2022
%t A240388 Clear[s]; s[0] = 0; s[1] = 1; s[n_?EvenQ] := s[n] = s[n/2];
%t A240388 s[n_?OddQ] :=
%t A240388 s[n] = s[(n + 1)/2] + s[(n - 1)/2] (* For the Stern sequence *)
%t A240388 Clear[w]; w[n_] = 1/2 s[3 n]
%o A240388 (PARI) a(n)=my(a=1, b=0); n*=3; while(n>0, if(n%2, b+=a, a+=b); n>>=1); b/2 \\ _Charles R Greathouse IV_, May 27 2014
%o A240388 (Python)
%o A240388 from functools import reduce
%o A240388 def A240388(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(3*n)[-1:2:-1],(1,0)))//2 # _Chai Wah Wu_, Jun 20 2022
%Y A240388 Cf. A002487.
%K A240388 nonn,look,easy
%O A240388 0,4
%A A240388 _Jennifer Lansing_, Apr 04 2014