A006336 a(n) = a(n-1) + a(n - 1 - number of even terms so far). (Formerly M0684)

1, 2, 3, 5, 8, 11, 16, 21, 29, 40, 51, 67, 88, 109, 138, 167, 207, 258, 309, 376, 443, 531, 640, 749, 887, 1054, 1221, 1428, 1635, 1893, 2202, 2511, 2887, 3330, 3773, 4304, 4835, 5475, 6224, 6973, 7860, 8747, 9801, 11022, 12243, 13671, 15306, 16941

Offset

1,2

Comments

From T. D. Noe, Jul 27 2007: (Start)

This is similar to A000123 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)

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Max Alekseyev, Proof of Paul Hanna's formula.

D. R. Hofstadter, Eta-Lore. [Cached copy, with permission]

D. R. Hofstadter, Pi-Mu Sequences. [Cached copy, with permission]

D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991.

Formula

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.) - Paul D. Hanna, Jul 22 2007

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, Oct 24 2007

Maple

# Maple code for first M terms of a(n) and A060144, from N. J. A. Sloane, Oct 25 2014
M:=100;
v[1]:=1; v[2]:=2; w[1]:=0; w[2]:=1;
for n from 3 to M do
   v[n]:=v[n-1]+v[n-1-w[n-1]];
if v[n] mod 2 = 0 then w[n]:=w[n-1]+1 else w[n]:=w[n-1]; fi; od:
[seq(v[n], n=1..M)]; # A006336
[seq(w[n], n=1..M)]; # A060144 shifted

Mathematica

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} ]

Prog

(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, Jul 23 2007
(Haskell)
a006336 n = a006336_list !! (n-1)
a006336_list = 1 : h 2 1 0 where
  h n last evens = x : h (n + 1) x (evens + 1 - x `mod` 2) where
    x = last + a006336 (n - 1 - evens)
-- Reinhard Zumkeller, May 18 2011

Crossrefs

Cf. A007604, A000123, A005704, A131882, A134408, A134409, A001950.

"Number of even terms so far" is A060144(n+1).

Sequence in context: A363047 A101018 A320593 * A175831 A070228 A173599

Adjacent sequences: A006333 A006334 A006335 * A006337 A006338 A006339

Keyword

nonn,easy,nice

Author

D. R. Hofstadter, Jul 15 1977

Extensions

More terms from Robert G. Wilson v, Mar 07 2001

Entry revised by N. J. A. Sloane, Oct 25 2014

Status

approved