detective work needed

[FRANCISCO SALINAS]

N. J. A. Sloane wrote:

>A correspondent, "Thomas Sundquist" <thomas.sundquist at normandale.edu>
>write to point out that A062122 does not match its description,
>which is:
>
>%I A062122
>%S A062122 
>1,2,5,19,65,259,1161,5927,33946,216090,1513051,11558614,95658445,
>%T A062122 852571616,8141450460,82928132445,897492637757,10284508144797,
>%U A062122 124399102620413,1583898570128380
>%N A062122 Boustrophedon transform of Fibonacci numbers (A000045).
>%H A062122 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on 
>sequences: the
>  Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 (<a 
>href="http://www.rese
>arch.att.com/~njas/doc/bous.txt">Abstract</a>, <a 
>href="http://www.research.att
>%H A062122 <a 
>href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#boustrophe
>don">Index entries for sequences related to boustrophedon transform</a>
>%F A062122 a_n = Sum_{k=0..n} C(n,k) F_k E_{n-k}, where F_{k+1} = F_k + 
>F_{k-1}, F_0 = F
>_1 = 1 (A000045) and the E_k are as in A000111.
>%e A062122 a_0 = F_0 = 1, a_1 = F_0 + F_1 = 2, a_2 = F_2 + a_1 + F_1 = 5, 
>...
>%Y A062122 Sequence in context: A072468 A062666 A047116 this_sequence 
>A047022 A014273 A0
>58131
>%Y A062122 Adjacent sequences: A062119 A062120 A062121 this_sequence 
>A062123 A062124 A06
>2125
>%K A062122 nonn
>%O A062122 0,2
>%A A062122 Orleo Marinaro (KDNM21(AT)bipop.it), Jun 01 2001
>
>
>The %F line does indeed define a Boustrophedon transform
>of the Fibonacci numbers 1,1,2,3,5,8,13,...
>
>The trouble is it does not give 1,2,5,19,65,259,1161,...
>
>Rather than just say this is wrong, I would like to find
>out how it really is defined.  I tried a few things without success,
>and suggested to Thomas Sundquist that he write to the author.
>
>But in the mean time, it makes a nice puzzle.  Here we have 20 terms
>and some hints.  What is the sequence?  I would like to know
>the answer!
>
>Neil


Appears to be the Boustrophedon transform of:  0, 1, 0, 2, 3, 5, 8, 13, 21, 
34, 55, 89, 144,... the Fibonacci numbers(F_0=0,F_1=1) with an erroneous 
term (F_2=0 instead of 1).

>with(combinat, fibonacci):a:=[0,1,0,seq(fibonacci(i),i=3..22)]:BOUS2(a);

  [0, 1, 2, 5, 19, 65, 259, 1161, 5927, 33946, 216090, 1513051,

        11558614, 95658445, 852571616, 8141450460, 82928132445,

        897492637757, 10284508144797, 124399102620413,

        1583898570128385, 21175164077080102, 296571619014584968]

By the way, the last term in A062122 (1583898570128380) is wrong too.

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