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A033485 a(n) = a(n-1) + a(floor(n/2)), a(1) = 1.

%I A033485 N0236 #90 Jan 05 2025 19:51:35
%S A033485 1,2,3,5,7,10,13,18,23,30,37,47,57,70,83,101,119,142,165,195,225,262,
%T A033485 299,346,393,450,507,577,647,730,813,914,1015,1134,1253,1395,1537,
%U A033485 1702,1867,2062,2257,2482,2707,2969,3231,3530,3829,4175,4521,4914,5307,5757
%N A033485 a(n) = a(n-1) + a(floor(n/2)), a(1) = 1.
%C A033485 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.
%C A033485 a(A036554(n)) is even, a(A003159(n)) is odd. - _Benoit Cloitre_, Oct 23 2002
%C A033485 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
%C A033485 The number of odd numbers before the n-th even number in this sequence is A003156(n). - _Philippe Deléham_, Mar 27 2004
%C A033485 There are no terms divisible by 4 and there are infinitely many terms divisible by {2,3,5,6,7,9,10,11,13,14,15} (see van Doorn link). - _Ivan N. Ianakiev_, Aug 06 2022 and _Wouter van Doorn_, Sep 17 2024
%C A033485 a(n) = A001401(n), for 1..14. A001401(15) = 84. - _Wolfdieter Lang_, Jan 09 2023
%D A033485 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H A033485 T. D. Noe, <a href="/A033485/b033485.txt">Table of n, a(n) for n=1..1000</a>
%H A033485 J. Arkin, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/9-2/advanced9-2-a.pdf">Problem H-102: Gone but not forgotten</a>, Fibonacci Quarterly, Vol. 9 (1971), page 135.
%H A033485 Christine Bessenrodt, Jorn B. Olsson, and James A. Sellers, <a href="http://arxiv.org/abs/1107.1156">Unique path partitions: Characterization and Congruences</a>, arXiv:1107.1156 [math.CO], 2011-2012.
%H A033485 Philippe Deléham, <a href="/A033485/a033485.pdf">Letter to N. J. A. Sloane, Apr 20 1998</a>
%H A033485 William Gasarch, <a href="https://blog.computationalcomplexity.org/2022/07/what-is-known-about-that-sequence.html">What is known about that sequence</a>, Computational Complexity blog, Jul 20 2022.
%H A033485 Wouter van Doorn, <a href="https://arxiv.org/abs/2406.09532">On the congruence properties and growth rate of a recursively defined sequence</a>, arXiv:2406.09532 [math.NT], 2024.
%F A033485 a(n) = A000123(n)/2, for n >= 1.
%F A033485 Conjecture: lim_{n->oo} 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
%F A033485 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
%F A033485 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
%F A033485 Sum_{k=1..n} a(k) = (a(2n+1)-1)/2 = A178855(n). - _Philippe Deléham_, Mar 18 2004
%F A033485 a(2n-1) = A131205(n). - _Jean-Paul Allouche_, Aug 11 2021
%F A033485 There exists a function f(n) such that n^f(n) < a(n) < n^(f(n) + epsilon) for all epsilon > 0 and all large enough n. - _Wouter van Doorn_, Sep 17 2024
%p A033485 a:= proc(n) option remember;
%p A033485       `if`(n<2, n, a(n-1)+a(iquo(n, 2)))
%p A033485     end:
%p A033485 seq(a(n), n=1..60);  # _Alois P. Heinz_, Dec 16 2019
%t A033485 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 *)
%o A033485 (PARI) a(n)=if(n<2,1,a(floor(n/2))+a(n-1))
%o A033485 (Haskell)
%o A033485 import Data.List (transpose)
%o A033485 a033485 n = a033485_list !! (n-1)
%o A033485 a033485_list = 1 : zipWith (+)
%o A033485    a033485_list (concat $ transpose [a033485_list, a033485_list])
%o A033485 -- _Reinhard Zumkeller_, Nov 15 2012
%o A033485 (Magma) [n le 1 select 1 else Self(n-1) + Self(Floor(n/2)) : n in [1..60]]; // _Vincenzo Librandi_, Nov 20 2015
%o A033485 (Python)
%o A033485 from itertools import islice
%o A033485 from collections import deque
%o A033485 def A033485_gen(): # generator of terms
%o A033485     aqueue, f, b, a = deque([2]), True, 1, 2
%o A033485     yield from (1, 2)
%o A033485     while True:
%o A033485         a += b
%o A033485         yield a
%o A033485         aqueue.append(a)
%o A033485         if f: b = aqueue.popleft()
%o A033485         f = not f
%o A033485 A033485_list = list(islice(A033485_gen(),40)) # _Chai Wah Wu_, Jun 07 2022
%Y A033485 Cf. A040039 (first differences), A178855 (partial sums).
%Y A033485 Also half of A000123 (with first term omitted).
%Y A033485 Cf. A022907.
%Y A033485 Cf. A036554, A003159, A003156.
%Y A033485 Cf. A007413, A178855, A131205.
%K A033485 nonn,nice,easy
%O A033485 1,2
%A A033485 _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