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A030067 The "Semi-Fibonacci sequence": a(1) = 1; a(n) = a(n/2) (n even); a(n) = a(n-1) + a(n-2) (n odd). 18
1, 1, 2, 1, 3, 2, 5, 1, 6, 3, 9, 2, 11, 5, 16, 1, 17, 6, 23, 3, 26, 9, 35, 2, 37, 11, 48, 5, 53, 16, 69, 1, 70, 17, 87, 6, 93, 23, 116, 3, 119, 26, 145, 9, 154, 35, 189, 2, 191, 37, 228, 11, 239, 48, 287, 5, 292, 53, 345, 16, 361, 69, 430, 1, 431, 70, 501, 17, 518, 87, 605, 6, 611, 93 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This is the "semi-Fibonacci sequence". The distinct numbers that appear are called "semi-Fibonacci numbers", and are given in A030068.

a(2n+1) >= a(2n-1) + 1 is monotonically increasing. a(2n)/n can be arbitrarily small, as a(2^n) = 1. There are probably an infinite number of primes in the sequence. - Jonathan Vos Post, Mar 28 2006

From Robert G. Wilson v, Jan 17 2014: (Start)

Positions where k occurs:

   k: sequence

   -:-----------------------------

   1: A000079;

   2: 3*A000079 = A007283;

   3: 5*A000079 = A020714;

   4: none in the first 10^6 terms;

   5: 7*A000079 = A005009;

   6: 9*A000079 = A005010;

   7: none in the first 10^6 terms;

   8: none in the first 10^6 terms;

   9: 11*A000079 = A005015;

  10: none in the first 10^6 terms;

  11: 13*A000079 = A005029;

  12: none in the first 10^6 terms;

(End)

Any integer N which occurs in this sequence first occurs as an odd-indexed term a(2k-1) = A030068(k-1), and thereafter at indices (2k-1)*2^j, j=1,2,3,... (Both of these statements follow immediately from the definition of even-indexed terms.) No N can occur a second time as an odd-indexed term: This follows from the definition of these terms, a(2n+1) = a(2n) + a(2n-1) = a(2n-1) + a(n), which shows that the subsequence of odd-indexed terms (A030068) is strictly increasing, and therefore equal to the range (or: set) of the semi-Fibonacci numbers. - M. F. Hasler, Mar 24 2017

The lines in the logarithmic scatterplot of the sequence corresponds to sets of indices with the same 2-adic valuation. - Rémy Sigrist, Nov 27 2017

Define the partition subsum polynomial of an integer partition m of n where m = (m_1, m_2, ...m_k) by ps(m,x) = Product_{i=1..k} (1+x^m_i). Expanding ps(m,x) gives 1+a_1 x+a_2 x^2+...+a_n x^n, where a_j is the number of ways to form the subsum j from the parts of m. Then the number of partitions m of n for which ps(m,x) has no repeated root is a(n). - George Beck, Nov 07 2018

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Abdulaziz M. Alanazi, Augustine O. Munagi and Darlison Nyirenda, Power Partitions and Semi-m-Fibonacci Partitions, arXiv:1910.09482 [math.CO], 2019.

George E. Andrews, Binary and Semi-Fibonacci Partitions, Journal of Ramanujan Society of Mathematics and Mathematics Sciences, honoring A.K. Agarwal's 70th birthday), 7:1(2019), 01-06.

George Beck, Semi-Fibonacci Partitions

Rémy Sigrist, Colored logarithmic scatterplot of the first 10000 terms (where the color is function of the 2-adic valuation of n)

FORMULA

Theorem: a(2n+1) - a(2n-1) = a(n). Proof: a(2n+1) - a(2n-1) = a(2n) + a(2n-1) - a(2n-2) - a(2n-3) = a(n) - a(n-1) + a(n-1) (induction) = a(n). - N. J. A. Sloane, May 02 2010

a(2^n - 1) = A129092(n) for n >= 1, where A129092 forms the row sums and column 0 of triangle A129100, which is defined by the nice property that column 0 of matrix power A129100^(2^k) = column k of A129100 for k > 0. - Paul D. Hanna, Dec 03 2008

G.f. g(x) satisfies (1-x^2) g(x) = (1+x-x^2) g(x^2) + x. - Robert Israel, Mar 23 2017

EXAMPLE

a(1) = 1 by definition.

a(2) = a(1) = 1.

a(3) = 1 + 1 = 2.

a(4) = a(2) = 1.

a(5) = 2 + 1 = 3.

a(6) = a(3) = 2.

a(7) = 3 + 2 = 5.

a(8) = a(4) = 1.

a(9) = 5 + 1 = 6.

a(10) = a(5) = 3.

MAPLE

f:=proc(n) option remember; if n=1 then RETURN(1) elif n mod 2 = 0 then RETURN(f(n/2)) else RETURN(f(n-1)+f(n-2)); fi; end;

MATHEMATICA

semiFibo[1] = 1; semiFibo[n_?EvenQ] := semiFibo[n] = semiFibo[n/2]; semiFibo[n_?OddQ] := semiFibo[n] = semiFibo[n - 1] + semiFibo[n - 2]; Table[semiFibo[n], {n, 80}] (* Jean-François Alcover, Aug 19 2013 *)

PROG

(Haskell)

import Data.List (transpose)

a030067 n = a030067_list !! (n-1)

a030067_list = concat $ transpose [scanl (+) 1 a030067_list, a030067_list]

-- Reinhard Zumkeller, Jul 21 2013, Jul 07 2013

(PARI) a(n) = if(n==1, 1, if(n%2 == 0, a(n/2), a(n-1) + a(n-2)));

vector(100, n, a(n)) \\ Altug Alkan, Oct 12 2015

(Python)

a=[1]; [a.append(a[-2]+a[-1] if n%2 else a[n//2-1]) for n in range(2, 75)]

print(a) # Michael S. Branicky, Jul 07 2022

CROSSREFS

Cf. A000045, A030068, A074364, A129092, A129100.

See A109671 for a variant.

Sequence in context: A332819 A321272 A321270 * A181363 A105800 A105602

Adjacent sequences:  A030064 A030065 A030066 * A030068 A030069 A030070

KEYWORD

nonn,nice,look

AUTHOR

David W. Wilson

STATUS

approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)