%I A006336 M0684
%S A006336 1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,443,531,
%T A006336 640,749,887,1054,1221,1428,1635,1893,2202,2511,2887,3330,3773,4304,
%U A006336 4835,5475,6224,6973,7860,8747,9801,11022,12243,13671,15306,16941
%N A006336 a(n) = a(n-1) + a(n - 1 - number of even terms so far).
%C A006336 Comments from T. D. Noe, Jul 27 2007: (Start) This is similar to A00123 
               and A005704, which both have a recursion a(n)=a(n-1)+a([n/k]), where 
               k is 2 and 3, respectively. Those sequences count "partitions of 
               k*n into powers of k". For the present sequence, k=phi. Does A006336(n) 
               count the partitions of n*phi into powers of phi?
%C A006336 Answering my own question: If the recursion starts with a(0)=1, then 
               I think we obtain "number of partitions of n*phi into powers of phi" 
               (see A131882).
%C A006336 Here we need negative powers of phi also: letting p=phi and q=1/phi, 
               we have
%C A006336 n=0: 0*p = {} for 1 partition,
%C A006336 n=1: 1*p = p = 1+q for 2 partitions,
%C A006336 n=2: 2*p = p+p = 1+p+q = 1+1+q+q = p^2+q for 4 partitions, etc.
%C A006336 So the present sequence, which starts with a(1)=1, counts 1/2 of the 
               "number of partitions of n*phi into powers of phi". (End)
%D A006336 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006336 T. D. Noe, <a href="b006336.txt">Table of n, a(n) for n=1..1000</a>
%H A006336 Max Alekseyev, <a href="a006336.txt">Proof of Paul Hanna's formula</a>
%F A006336 Comment from Paul D. Hanna, Jul 22 2007: It seems that A006336 can be 
               generated by a rule using the golden ratio phi: a(n) = a(n-1) + a([n/
               Phi]) for n>1 with a(1)=1 where phi = (sqrt(5)+1)/2, I.e. the number 
               of even terms up to position n-1 equals n-1 - [n/Phi] for n>1 where 
               Phi = (sqrt(5)+1)/2. (This is true - see the Alekseyev link.)
%F A006336 a(n)=a(n-1)+a(A060143(n)) for n>1; subsequence of A134409; A134408 and 
               A134409 give first and second differences; A001950(n)=Min(m:A134409(m)=a(n)). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 24 2007
%t A006336 a[n_Integer] := a[n] = Block[{c, k}, c = 0; k = 1; While[k < n, If[ EvenQ[ 
               a[k] ], c++ ]; k++ ]; Return[a[n - 1] + a[n - 1 - c] ] ]; a[1] = 
               1; a[2] = 2; Table[ a[n], {n, 0, 60} ]
%o A006336 (PARI) A006336(N=99) = local(a=vector(N,i,1), e=0); for(n=2,#a,e+=0==(a[n]=a[n-1]+a[n-1-e])%2);
               a \\ M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jul 23 2007
%Y A006336 Cf. A007604.
%Y A006336 Sequence in context: A071424 A008762 A101018 this_sequence A070228 A006304 
               A039847
%Y A006336 Adjacent sequences: A006333 A006334 A006335 this_sequence A006337 A006338 
               A006339
%K A006336 nonn,easy,nice
%O A006336 1,2
%A A006336 D. R. Hofstadter
%E A006336 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 07 2001