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A006336
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a(n) = a(n-1) + a(n - 1 - number of even terms so far).
(Formerly M0684)
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10
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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)
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
Max Alekseyev, Proof of Paul Hanna's formula
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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
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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} ]
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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 (Maximilian.Hasler(AT)gmail.com), 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
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CROSSREFS
| Cf. A007604.
Sequence in context: A071424 A008762 A101018 * A175831 A070228 A173599
Adjacent sequences: A006333 A006334 A006335 * A006337 A006338 A006339
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KEYWORD
| nonn,easy,nice
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AUTHOR
| D. R. Hofstadter
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 07 2001
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