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A033485
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a(n) = a(n-1) + a(floor(n/2)), a(1) = 1.
(Formerly N0236)
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35
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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, 3231, 3530, 3829, 4175, 4521, 4914, 5307, 5757
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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
There are no terms divisible by 4 and there are infinitely many terms divisible by {2,3,5,7} (see Computational Complexity link). - Ivan N. Ianakiev, Aug 06 2022
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REFERENCES
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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
J. Arkin, Problem H-102: Gone but not forgotten, Fibonacci Quarterly, Vol. 9 (1971), page 135.
Christine Bessenrodt, Jorn B. Olsson, and James A. Sellers, Unique path partitions: Characterization and Congruences, arXiv:1107.1156 [math.CO], 2011-2012.
Philippe Deléham, Letter to N. J. A. Sloane, Apr 20 1998
William Gasarch, What is known about that sequence, Computational Complexity blog, Jul 20 2022.
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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<log_2(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 = A178855(n). - Philippe Deléham, Mar 18 2004
a(2n-1) = A131205(n). - Jean-Paul Allouche, Aug 11 2021
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MAPLE
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a:= proc(n) option remember;
`if`(n<2, n, a(n-1)+a(iquo(n, 2)))
end:
seq(a(n), n=1..60); # Alois P. Heinz, Dec 16 2019
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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
(Magma) [n le 1 select 1 else Self(n-1) + Self(Floor(n/2)) : n in [1..60]]; // Vincenzo Librandi, Nov 20 2015
(Python)
from itertools import islice
from collections import deque
def A033485_gen(): # generator of terms
aqueue, f, b, a = deque([2]), True, 1, 2
yield from (1, 2)
while True:
a += b
yield a
aqueue.append(a)
if f: b = aqueue.popleft()
f = not f
A033485_list = list(islice(A033485_gen(), 40)) # Chai Wah Wu, Jun 07 2022
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CROSSREFS
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Cf. A040039 (first differences), A178855 (partial sums).
Also half of A000123 (with first term omitted).
Cf. A022907.
Cf. A036554, A003159, A003156.
Cf. A007413, A178855, A131205.
Sequence in context: A103232 A062684 A341912 * A026811 A001401 A347549
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. Entry revised by N. J. A. Sloane, Jun 10 2012
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STATUS
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approved
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