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 A033485 a(n) = a(n-1) + a(floor(n/2)), a(1) = 1. (Formerly N0236) 35
 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 (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 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 a(n) = A001401(n), for 1..14. A001401(15) = 84. - Wolfdieter Lang, Jan 09 2023 REFERENCES 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 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. FORMULA a(n) = A000123(n)/2, for n >= 1. 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=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 MAPLE 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 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 (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 CROSSREFS 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 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 December 8 15:20 EST 2023. Contains 367680 sequences. (Running on oeis4.)