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A033485
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a(n) = a(n-1) + a([n/2]), a(1) = 1.
(Formerly N0236)
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25
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1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 83, 101, 119, 142, 165, 195, 225, 262, 299, 346, 393, 450, 507, 577, 647, 730, 813, 914, 1015, 1134, 1253, 1395, 1537, 1702, 1867, 2062, 2257, 2482, 2707, 2969
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OFFSET
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1,2
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COMMENTS
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Sequence gives the number of partitions of 2n into "strongly decreasing" parts (see the function s*(n) in the paper by Bessenrodt, Olsson, and Sellers); see the example in A040039.
a(A036554(n)) is even, a(A003159(n)) is odd. - Benoit Cloitre, Oct 23 2002
Partial sums of the sequence a(1)=1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), ... example : a(1) = 1, a(2) = 1+1= 2, a(3) = 1+1+1= 3, a(4) = 1+1+1+2= 5, a(5) = 1+1+1+2+2= 7, ... - Philippe Deléham, Jan 02 2004
The number of odd numbers before the n-th even number in this sequence is A003156(n). - Philippe Deléham, Mar 27 2004
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REFERENCES
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Fibonacci Quarterly, Vol. 9 (1971), page 135 (author and title needed!).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Christine Bessenrodt, Jorn B. Olsson, and James A. Sellers, Unique path partitions: Characterization and Congruences, arXiv preprint
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FORMULA
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Conjecture: lim n ->infinity a(2n)/a(n)*log(n)/n = c = 1.64.... and a(n)/A(n) is bounded where A(n)=1 if n is a power of 2, otherwise A(n)=sqrt(n)*product(k<log2(n), n/2^k/log(n/2^k)). - Benoit Cloitre, Jan 26 2003
G.f.: A(x) satisfies x + (1+x)*A(x^2) = (1-x)*A(x). a(n) modulo 2 = A035263(n) . - Philippe Deléham, Feb 25 2004
G.f.:(1/2)*(((1-x)*Product_{n>=0}(1-x^(2^n)))^(-1)-1). a(n) modulo 4 = A007413(n). - Philippe Deléham, Feb 28 2004
Sum_{k=1..n} a(k) = (a(2n+1)-1)/2. - Philippe Deléham, Mar 18 2004
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MATHEMATICA
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b[1]=1; b[n_] := b[n]=Sum[b[k], {k, 1, n/2}]; Table[b[n], {n, 3, 105, 2}] (Robert G. Wilson v, Apr 22 2001)
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PROG
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(PARI) a(n)=if(n<2, 1, a(floor(n/2))+a(n-1))
(Haskell)
import Data.List (transpose)
a033485 n = a033485_list !! (n-1)
a033485_list = 1 : zipWith (+)
a033485_list (concat $ transpose [a033485_list, a033485_list])
-- Reinhard Zumkeller, Nov 15 2012
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CROSSREFS
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Cf. A040039. Also half of A000123 (with first term omitted).
Cf. A022907.
Sequence in context: A008766 A103232 A062684 * A026811 A001401 A008628
Adjacent sequences: A033482 A033483 A033484 * A033486 A033487 A033488
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Philippe Deléham [BP 29, Coconi, 97670 Ouangani, Mayotte] (kolotoko(AT)wanadoo.fr). Entry revised by N. J. A. Sloane, Jun 10 2012
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STATUS
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approved
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