decomposition in Odd & Even Fibs

[Wouter Meeussen]

is this too far fetched?
------------------------------------------------------------------------

%I A000001 
%S A000001 1,2,4,5,9,10,12,13,22,23,25,26,30,31,33,34,56,57,59,60,64,65,67,68,
%T A000001 77,78,80,81,85,86,88,89,145,146,148,149,153,154,156,157,166,167,169,
%U A000001 170,174,175,177,178,200,201,203,204,208,209,211,212,221,222,224,225
%N A000001 integers that can be decomposed into sums of different Fibonacci
numbers of even argument
%R A000001 
%Y A000001 cf. A000002
%A A000001 Wouter Meeussen, w.meeussen.vdmcc at vandemoortele.be
%O A000001 0,2
%t A000001
FoldList[#1+#2&,1,Fold[{#1,#2,#1}&,1,1+Table[Fibonacci[k],{k,2,15,2}]]//Flatten]
%K A000001 nonn


%I A000002 
%S A000002 1,2,3,4,6,7,8,9,14,15,16,17,19,20,21,22,35,36,37,38,40,41,42,43,48,
%T A000002 49,50,51,53,54,55,56,90,91,92,93,95,96,97,98,103,104,105,106,108,109,
%U A000002 110,111,124,125,126,127,129,130,131,132,137,138,139,140,142,143,144
%N A000002 integers that can be decomposed into sums of different Fibonacci
numbers of odd argument
%R A000002 
%Y A000002 cf. A000001
%A A000002 Wouter Meeussen, w.meeussen.vdmcc at vandemoortele.be
%O A000002 0,2
%t A000002
FoldList[#1+#2&,1,Fold[{#1,#2,#1}&,1,Table[Fibonacci[k],{k,1,13,2}]]//Flatten]
%K A000002 nonn

-----------------------------------------------------------------------
why? What's the relation between this Folding and "Fibo-partitions"?
-----------------------------------------------------------------------
it1=CoefficientList[Normal at Series[Product[1+z^Fibonacci[k],{k,2,17,2}],{z,0,
1596}],z];

counts the number of ways in which n can be decomposed into a sum Fibonacci
numbers of even argument {1,3,8,21,55,144,377,987,..}, whose running sum is
one less than a Fib of odd argument.

This is a list of many 0's and few 1's. The 1's occur at positions:

pit1=Position[it1,1]//Flatten

{1,2,4,5,9,10,12,13,22,23,25,26,30,31,33,34,56,57,59,60,64,65,
67,68,77,78,80,81,85,86,88,89,145,146,148,149,153,154,156,157,
166,167,169,170,174,175,177,178,200,201,203,204,208,209,211,
212,221,222,224,225,229,230,232,233,378,379,381,382,386,387,389,
390,399,400,402,403,407,408,410,411,433,434,436,

*** So these are the numbers that can be (uniquely) decomposed ***
*** into Fibs of even argument  *** 

This result can also be obtained by the running sum of the simple Folding
operation:

FoldList[#1+#2&,1,Fold[{#1,#2,#1}&,1,1+Table[Fibonacci[k],{k,2,15,2}]]//
Flatten] == pit1 
 gives 'True'
**********************************************************************
An analogous result holds for the Fibs of odd arument, 
{1,2,5,13,34,89,233,610,1597,..}, whose running sum is one less than a Fib
of even argument.

it2=CoefficientList[Normal at Series[Product[1+z^Fibonacci[k],{k,1,17,2}],{z,0,
1596}],z];
Now the 1's occur at positions:

pit2=Position[it2,1]//Flatten

{1,2,3,4,6,7,8,9,14,15,16,17,19,20,21,22,35,36,37,38,40,41,42,
43,48,49,50,51,53,54,55,56,90,91,92,93,95,96,97,98,103,104,105,
106,108,109,110,111,124,125,126,127,129,130,131,132,137,138,
139,140,142,143,144,145,234,235,236,237,239,240,241,242,247,
248,249,250,252,253,254,255,268,269,270,271,273,

*** So these are the numbers that can be (uniquely) decomposed ***
*** into Fibs of even argument *** 

and the running sum of the "equivalent" Folding operation is given by

FoldList[#1+#2&,1,Fold[{#1,#2,#1}&,1,Table[Fibonacci[k],{k,1,13,2}]]//Flatte
n] == pit2
 gives 'True'
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be