1,2

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?

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).

Here we need negative powers of phi also: letting p=phi and q=1/phi, we have

n=0: 0*p = {} for 1 partition,

n=1: 1*p = p = 1+q for 2 partitions,

n=2: 2*p = p+p = 1+p+q = 1+1+q+q = p^2+q for 4 partitions, etc.

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)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000

Max Alekseyev, Proof of Paul Hanna's formula

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.)

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

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} ]

(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

Cf. A007604.

Sequence in context: A071424 A008762 A101018 this_sequence A070228 A006304 A039847

Adjacent sequences: A006333 A006334 A006335 this_sequence A006337 A006338 A006339

nonn,easy,nice

D. R. Hofstadter

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 07 2001