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A033485 a(n) = a(n-1) + a([n/2]), a(1) = 1.
(Formerly N0236)
25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

REFERENCES

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).

LINKS

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

FORMULA

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

MATHEMATICA

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)

PROG

(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

CROSSREFS

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

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Philippe Deléham. Entry revised by N. J. A. Sloane, Jun 10 2012

STATUS

approved

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Last modified October 31 14:00 EDT 2014. Contains 248867 sequences.