A092942 -id:A092942 - cp's OEIS frontend

A016789 a(n) = 3*n + 2.

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179

Offset: 0

N. J. A. Sloane

Except for 1, n such that Sum_{k=1..n} (k mod 3)*binomial(n,k) is a power of 2. - Benoit Cloitre, Oct 17 2002
The sequence 0,0,2,0,0,5,0,0,8,... has a(n) = n*(1 + cos(2*Pi*n/3 + Pi/3) - sqrt(3)*sin(2*Pi*n + Pi/3))/3 and o.g.f. x^2(2+x^3)/(1-x^3)^2. - Paul Barry, Jan 28 2004 [Artur Jasinski, Dec 11 2007, remarks that this should read (3*n + 2)*(1 + cos(2*Pi*(3*n + 2)/3 + Pi/3) - sqrt(3)*sin(2*Pi*(3*n + 2)/3 + Pi/3))/3.]
Except for 2, exponents e such that x^e + x + 1 is reducible. - N. J. A. Sloane, Jul 19 2005
The trajectory of these numbers under iteration of sum of cubes of digits eventually turns out to be 371 or 407 (47 is the first of the second kind). - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009
Union of A165334 and A165335. - Reinhard Zumkeller, Sep 17 2009
a(n) is the set of numbers congruent to {2,5,8} mod 9. - Gary Detlefs, Mar 07 2010
It appears that a(n) is the set of all values of y such that y^3 = k*n + 2 for integer k. - Gary Detlefs, Mar 08 2010
These numbers do not occur in A000217 (triangular numbers). - Arkadiusz Wesolowski, Jan 08 2012
A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
Also indices of even Bell numbers (A000110). - Enrique Pérez Herrero, Sep 10 2013
Central terms of the triangle A108872. - Reinhard Zumkeller, Oct 01 2014
A092942(a(n)) = 1 for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n-1), n >= 1, is also the complex dimension of the manifold E(S), the set of all second-order irreducible Fuchsian differential equations defined on P^1 = C U {oo}, having singular points at most in S = {a_1, ..., a_n, a_{n+1} = oo}, a subset of P^1. See the Iwasaki et al. reference, Proposition 2.1.3., p. 149. - Wolfdieter Lang, Apr 22 2016
Except for 2, exponents for which 1 + x^(n-1) + x^n is reducible. - Ron Knott, Sep 16 2016
The reciprocal sum of 8 distinct items from this sequence can be made equal to 1, with these terms: 2, 5, 8, 14, 20, 35, 41, 1640. - Jinyuan Wang, Nov 16 2018
There are no positive integers x, y, z such that 1/a(x) = 1/a(y) + 1/a(z). - Jinyuan Wang, Dec 31 2018
As a set of positive integers, it is the set sum S + S where S is the set of numbers in A016777. - Michael Somos, May 27 2019
Interleaving of A016933 and A016969. - Leo Tavares, Nov 16 2021
Prepended with {1}, these are the denominators of the elements of the 3x+1 semigroup, the numerators being A005408 prepended with {2}. See Applegate and Lagarias link for more information. - Paolo Xausa, Nov 20 2021
This is also the maximum number of moves starting with n + 1 dots in the game of Sprouts. - Douglas Boffey, Aug 01 2022 [See the Wikipedia link. - Wolfdieter Lang, Sep 29 2022]
a(n-2) is the maximum sum of the span (or L(2,1)-labeling number) of a graph of order n and its complement. The extremal graphs are stars and their complements. For example, K_{1,2} has span 3, and K_2 has span 2. Thus a(3-1) = 5. - Allan Bickle, Apr 20 2023

			G.f. = 2 + 5*x + 8*x^2 + 11*x^3 + 14*x^4 + 17*x^5 + 20*x^6 + ... - _Michael Somos_, May 27 2019

K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 149.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
H. Balakrishnan and N. Deo, Parallel algorithm for radiocoloring a graph, Congr. Numer. 160 (2003), 193-204.
Allan Bickle, Extremal Decompositions for Nordhaus-Gaddum Theorems, Discrete Math, 346 7 (2023), 113392.
L. Euler, Observatio de summis divisorum p. 9.
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 937
L. B. W. Jolley, Summation of Series, Dover, 1961, p. 16
Tanya Khovanova, Recursive Sequences
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Capped Triangular Frames
Wikipedia, Sprouts (game)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

First differences of A005449.
Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388, A005351.
Cf. A087370.
Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.
Cf. A000096, A000217.
Cf. A016933, A016969.

List([0..70],n->3*n+2); # Muniru A Asiru, Nov 02 2018

a016789 = (+ 2) . (* 3)  -- Reinhard Zumkeller, Jul 05 2013

[3*n+2: n in [0..80]]; // Vincenzo Librandi, Apr 14 2015

seq(3*n+2, n = 0 .. 50); # Matt C. Anderson, May 18 2017

Range[2, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
LinearRecurrence[{2,-1},{2,5},70] (* Harvey P. Dale, Aug 11 2021 *)

vector(100,n,3*n-1) \\ Derek Orr, Apr 13 2015

for n in range(0,100): print(3*n+2, end=', ') # Stefano Spezia, Nov 21 2018

G.f.: (2+x)/(1-x)^2.
a(n) = 3 + a(n-1).
a(n) = 1 + A016777(n).
a(n) = A124388(n)/9.
a(n) = A125199(n+1,1). - Reinhard Zumkeller, Nov 24 2006
Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) - log(2)). - Benoit Cloitre, Apr 05 2002
1/2 - 1/5 + 1/8 - 1/11 + ... = (1/3)*(Pi/sqrt(3) - log 2). [Jolley] - Gary W. Adamson, Dec 16 2006
Sum_{n>=0} 1/(a(2*n)*a(2*n+1)) = (Pi/sqrt(3) - log 2)/9 = 0.12451569... (see A196548). [Jolley p. 48 eq (263)]
a(n) = 2*a(n-1) - a(n-2); a(0)=2, a(1)=5. - Philippe Deléham, Nov 03 2008
a(n) = 6*n - a(n-1) + 1 with a(0)=2. - Vincenzo Librandi, Aug 25 2010
Conjecture: a(n) = n XOR A005351(n+1) XOR A005352(n+1). - Gilian Breysens, Jul 21 2017
E.g.f.: (2 + 3*x)*exp(x). - G. C. Greubel, Nov 02 2018
a(n) = A005449(n+1) - A005449(n). - Jinyuan Wang, Feb 03 2019
a(n) = -A016777(-1-n) for all n in Z. - Michael Somos, May 27 2019
a(n) = A007310(n+1) + (1 - n mod 2). - Walt Rorie-Baety, Sep 13 2021
a(n) = A000096(n+1) - A000217(n-1). See Capped Triangular Frames illustration. - Leo Tavares, Oct 05 2021

A032766 Numbers that are congruent to 0 or 1 (mod 3).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103

Offset: 0

Patrick De Geest, May 15 1998

Omitting the initial 0, a(n) is the number of 1's in the n-th row of the triangle in A118111. - Hans Havermann, May 26 2002
Binomial transform is A053220. - Michael Somos, Jul 10 2003
Smallest number of different people in a set of n-1 photographs that satisfies the following conditions: In each photograph there are 3 women, the woman in the middle is the mother of the person on her left and is a sister of the person on her right and the women in the middle of the photographs are all different. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Partial sums of A000034. - Richard Choulet, Jan 28 2010
Starting with 1 = row sums of triangle A171370. - Gary W. Adamson, Feb 15 2010
a(n) is the set of values for m in which 6k + m can be a perfect square (quadratic residues of 6 including trivial case of 0). - Gary Detlefs, Mar 19 2010
For n >= 2, a(n) is the smallest number with n as an anti-divisor. - Franklin T. Adams-Watters, Oct 28 2011
Sequence is also the maximum number of floors with 3 elevators and n stops in a "Convenient Building". See A196592 and Erich Friedman link below. - Robert Price, May 30 2013
a(n) is also the total number of coins left after packing 4-curves patterns (4c2) into a fountain of coins base n. The total number of 4c2 is A002620 and voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 26 2013
Number of partitions of 6n into two even parts. - Wesley Ivan Hurt, Nov 15 2014
Number of partitions of 3n into exactly 2 parts. - Colin Barker, Mar 23 2015
Nonnegative m such that floor(2*m/3) = 2*floor(m/3). - Bruno Berselli, Dec 09 2015
For n >= 3, also the independence number of the n-web graph. - Eric W. Weisstein, Dec 31 2015
Equivalently, nonnegative numbers m for which m*(m+2)/3 and m*(m+5)/6 are integers. - Bruno Berselli, Jul 18 2016
Also the clique covering number of the n-Andrásfai graph for n > 0. - Eric W. Weisstein, Mar 26 2018
Maximum sum of degeneracies over all decompositions of the complete graph of order n+1 into three factors. The extremal decompositions are characterized in the Bickle link below. - Allan Bickle, Dec 21 2021
Also the Hadwiger number of the n-cocktail party graph. - Eric W. Weisstein, Apr 30 2022
The number of integer rectangles with a side of length n+1 and the property: the bisectors of the angles form a square within its limits. - Alexander M. Domashenko, Oct 17 2024
The maximum possible number of 5-cycles in an outerplanar graph on n+4 vertices. - Stephen Bartell, Jul 10 2025

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Allan Bickle, Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Erich Friedman, Problem of the month November 2009
Z. Füredi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, Nordhaus-Gaddum-type theorems for decompositions into many parts, J. Graph Theory 50 (2005), 273-292.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 282. [Book's website]
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem #20. [Broken link; Cached copy]
Matroids Matheplanet, Number of d-generator groups of order 2^(d+1) and exponent-p class 2 (in German).
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), pp. 277-310, see p. 302.
Kival Ngaokrajang, Illustration of initial terms (U).
Eric Weisstein's World of Mathematics, Andrásfai Graph
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Hadwiger Number
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006578 (partial sums), A000034 (first differences), A016789 (complement).
Essentially the same: A049624.
Column 1 (the second leftmost) of triangular table A026374.
Column 1 (the leftmost) of square array A191450.
Row 1 of A254051.
Row sums of A171370.
Cf. A066272 for anti-divisors.
Cf. A253888 and A254049 (permutations of this sequence without the initial zero).
Cf. A254103 and A254104 (pair of permutations based on this sequence and its complement).
Cf. A000035, A000217, A000982, A001651, A002620, A002717, A003945.
Cf. A004396, A004526, A007310, A007494, A008619, A030308, A035360.
Cf. A047270, A049615, A052928, A053220, A070893, A084056, A092942.
Cf. A118111, A132463, A133083, A196592, A249745.

a032766 n = div n 2 + n  -- Reinhard Zumkeller, Dec 13 2014
(MIT/GNU Scheme) (define (A032766 n) (+ n (floor->exact (/ n 2)))) ;; Antti Karttunen, Jan 24 2015

&cat[ [n, n+1]: n in [0..100 by 3] ]; // Vincenzo Librandi, Nov 16 2014

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..69); # Zerinvary Lajos, Mar 16 2008
seq(floor(n/2)+n, n=0..69); # Gary Detlefs, Mar 19 2010
select(n->member(n mod 3,{0,1}), [$0..103]); # Peter Luschny, Apr 06 2014

a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 4; Array[a, 60, 0] (* Robert G. Wilson v, Mar 28 2011 *)
Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
Flatten[{#,#+1}&/@(3Range[0,40])] (* or *) LinearRecurrence[{1,1,-1}, {0,1,3}, 100] (* or *) With[{nn=110}, Complement[Range[0,nn], Range[2,nn,3]]] (* Harvey P. Dale, Mar 10 2013 *)
CoefficientList[Series[x (1 + 2 x) / ((1 - x) (1 - x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 16 2014 *)
Floor[3 Range[0, 69]/2] (* L. Edson Jeffery, Jan 14 2017 *)
Drop[Range[0,110],{3,-1,3}] (* Harvey P. Dale, Sep 02 2023 *)

PARI
```
{a(n) = n + n\2}
```

concat(0, Vec(x*(1+2*x)/((1-x)*(1-x^2)) + O(x^100))) \\ Altug Alkan, Dec 09 2015

[int(3*n//2) for n in range(101)] # G. C. Greubel, Jun 23 2024

G.f.: x*(1+2*x)/((1-x)*(1-x^2)).
a(-n) = -A007494(n).
a(n) = A049615(n, 2), for n > 2.
From Paul Barry, Sep 04 2003: (Start)
a(n) = (6n - 1 + (-1)^n)/4.
a(n) = floor((3n + 2)/2) - 1 = A001651(n) - 1.
a(n) = sqrt(2) * sqrt( (6n-1) (-1)^n + 18n^2 - 6n + 1 )/4.
a(n) = Sum_{k=0..n} 3/2 - 2*0^k + (-1)^k/2. (End)
a(n) = 3*floor(n/2) + (n mod 2) = A007494(n) - A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = 2 * A004526(n) + A004526(n+1). - Philippe Deléham, Aug 07 2006
a(n) = 1 + ceiling(3*(n-1)/2). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Row sums of triangle A133083. - Gary W. Adamson, Sep 08 2007
a(n) = (cos(Pi*n) - 1)/4 + 3*n/2. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008
A004396(a(n)) = n. - Reinhard Zumkeller, Oct 30 2009
a(n) = floor(n/2) + n. - Gary Detlefs, Mar 19 2010
a(n) = 3n - a(n-1) - 2, for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
a(n) = n + (n-1) - (n-2) + (n-3) - ... 1 = A052928(n) + A008619(n-1). - Jaroslav Krizek, Mar 22 2011
a(n) = a(n-1) + a(n-2) - a(n-3). - Robert G. Wilson v, Mar 28 2011
a(n) = Sum_{k>=0} A030308(n,k) * A003945(k). - Philippe Deléham, Oct 17 2011
a(n) = 2n - ceiling(n/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = A000217(n) - 2 * A002620(n-1). - Kival Ngaokrajang, Oct 26 2013
a(n) = Sum_{i=1..n} gcd(i, 2). - Wesley Ivan Hurt, Jan 23 2014
a(n) = 2n + floor((-n - (n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014
A092942(a(n)) = n for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = floor(3*n/2). - L. Edson Jeffery, Jan 18 2015
a(n) = A254049(A249745(n)) = (1+A007310(n)) / 2 for n >= 1. - Antti Karttunen, Jan 24 2015
E.g.f.: (3*x*exp(x) - sinh(x))/2. - Ilya Gutkovskiy, Jul 18 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Dec 04 2021

Better description from N. J. A. Sloane, Aug 01 1998

A251599 Centers of rows of the triangular array formed by the natural numbers.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 13, 18, 19, 25, 32, 33, 41, 50, 51, 61, 72, 73, 85, 98, 99, 113, 128, 129, 145, 162, 163, 181, 200, 201, 221, 242, 243, 265, 288, 289, 313, 338, 339, 365, 392, 393, 421, 450, 451, 481, 512, 513, 545, 578, 579, 613, 648, 649, 685, 722, 723

Offset: 1

Dave Durgin, Dec 05 2014

nonn

Forms a cascade of 3-number triangles down the center of the triangle array. Related to A000124 (left/west bank of same triangular array), A000217 (right/east bank) and A001844 (center column).
Sums of the mentioned cascading triangles: a(3*n-2) + a(3*n-1) + a(3*n) = A058331(n) + A001105(n) + A001844(n-1) = 2*A056106(n) = 2*(3*n^2-n+1). - Reinhard Zumkeller, Dec 13 2014
Union of A080827 and A000982. - David James Sycamore, Aug 09 2018

			First ten terms (1,2,3,5,8,9,13,18,19,25) may be read down the center of the triangular formation:
               1
             2   3
           4   5   6
         7   8   9  10
      11  12  13  14  15
    16  17  18  19  20  21
  22  23  24  25  26  27  28

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David James Sycamore, A080827 & A000982
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A000124, A000217, A001844.
Cf. A092942 (first differences).
Cf. A001105, A056106, A058331.
Cf. A080827, A000982.

a251599 n = a251599_list !! (n-1)
a251599_list = f 0 $ g 1 [1..] where
   f i (us:vs:wss) = [head $ drop i us] ++ (take 2 $ drop i vs) ++
                     f (i + 1) wss
   g k zs = ys : g (k + 1) xs where (ys,xs) = splitAt k zs
-- Reinhard Zumkeller, Dec 12 2014

a:= n-> (m-> 2*(m+1)^2-[2*m+1, 0, -1][1+r])(iquo(n-1, 3, 'r')):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 10 2014

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 5, 8, 9, 13}, 60] (* Jean-François Alcover, Jan 09 2016 *)

Vec(-x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3) + O(x^80)) \\ Michel Marcus, Jan 09 2016

Terms for n=1 (mod 3): 2m^2+2m+1, for n=2 (mod 3): 2m^2+4m+2, for n=0 (mod 3): 2m^2+4m+3, where m = floor((n-1)/3).

G.f.: -x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 10 2014

A336717 a(n) is the least k such that R(k) = n for some random Fibonacci sequence R (a random Fibonacci sequence R satisfies R(0) = 0, R(1) = 1, and for m > 0, R(m+1) = R(m) + (-1)^r(m) * R(m-1) where r is a sequence of integers).

Original entry on oeis.org

0, 1, 3, 4, 6, 5, 9, 7, 6, 8, 9, 8, 9, 7, 9, 10, 9, 10, 9, 10, 12, 8, 12, 10, 12, 10, 12, 11, 12, 10, 12, 11, 12, 13, 9, 13, 12, 11, 15, 11, 12, 11, 15, 13, 12, 13, 12, 11, 15, 13, 12, 13, 15, 13, 15, 10, 15, 13, 15, 13, 12, 14, 15, 14, 12, 13, 12, 14, 15, 14

Offset: 0

Rémy Sigrist, Aug 01 2020

nonn

This sequence is well defined as the random Fibonacci sequence A092942 contains every nonnegative integers.

			For n = 4:
- random Fibonacci sequences R may start as follows:
  R(0)  R(1)  R(2)  R(3)  R(4)  R(5)  R(6)
  ----  ----  ----  ----  ----  ----  ----
     0     1     1     2     3     5     8
     0     1     1     2     3     5     2
     0     1     1     2     3     1     4
     0     1     1     2     3     1    -2
     0     1     1     2     1     3     4
     0     1     1     2     1     3     2
     0     1     1     2     1    -1     0
     0     1     1     2     1    -1    -2
     0     1     1     0     1     1     2
     0     1     1     0     1     1     0
     0     1     1     0    -1    -1     0
     0     1     1     0    -1    -1    -2
- the value 4 first appears in column R(6),
- so a(4) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
James Grime, Random Fibonacci Numbers, Numberphile video (2020).
Rémy Sigrist, C program for A336717

Cf. A000045, A092942.

C
```
See Links section.
```

a(A000045(n)) = n for any n <> 2.

A376930 a(0)=0, a(1)=1; for n>1, a(n) = a(n-1)+a(n-2), except where a(n-1) is a prime greater than 2, in which case a(n) = a(n-1)-a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 4, 5, 1, 6, 7, 1, 8, 9, 17, 8, 25, 33, 58, 91, 149, 58, 207, 265, 472, 737, 1209, 1946, 3155, 5101, 1946, 7047, 8993, 16040, 25033, 8993, 34026, 43019, 8993, 52012, 61005, 113017, 52012, 165029, 217041, 382070, 599111, 981181, 1580292, 2561473

Offset: 0

Stuart Coe, Oct 11 2024

nonn

It is not clear whether this sequence continues to grow or whether it become stuck in a loop (which could happen if two primes occur in terms n and n-1 or terms n and n-2). Indeed, the sequence is stuck in a loop from around n=10 if we do not ignore the prime number 2.
Similarly, it is not known if the sequence contains any negative terms (which may happen if two primes are adjacent or separated by one other term).
If it continues to grow, it is not clear whether this sequence will contain an infinite number of prime numbers.
Beyond the trivial case of 1, it is not clear if any number will appear more than three times in the sequence. 8993 appears three times, due to several prime terms in close succession.
Also 4618239875200356592 appears three times, as a(111), a(114) and a(117). - Robert Israel, Nov 12 2024

			a(2) = a(1) + a(0) [as a(1) is not a prime > 2] = 1 + 0 = 1.
a(3) = a(2) + a(1) [as a(2) is not a prime > 2] = 1 + 1 = 2.
a(4) = a(3) + a(2) [as a(3) is not a prime > 2] = 2 + 1 = 3.
a(5) = a(4) - a(3) [as a(4) is a prime > 2]     = 3 - 2 = 1.

Robert Israel, Table of n, a(n) for n = 0..4810

Cf. A092942, A229137,

f:= proc(n) option remember;
      if procname(n-1) > 2 and isprime(procname(n-1)) then procname(n-1) - procname(n-2)
      else procname(n-1) + procname(n-2)
      fi
end proc:
f(0):= 0: f(1):= 1:
seq(f(i),i=0..100); # Robert Israel, Nov 12 2024

s={0,1};Do[If[PrimeQ[s[[-1]]]&&s[[-1]]>2,AppendTo[s,s[[-1]]-s[[-2]]],AppendTo[s,s[[-1]]+s[[-2]]]  ],{n,48}];s (* James C. McMahon, Nov 07 2024 *)

from sympy import isprime
from itertools import islice
def agen(): # generator of terms
    a = [0, 1]
    yield from a
    while True:
        an = a[-1]+a[-2] if a[-1] < 3 or not isprime(a[-1]) else a[-1]-a[-2]
        yield an
        a = [a[-1], an]
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 11 2024

cp's OEIS Frontend

A016789 a(n) = 3*n + 2.

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A032766 Numbers that are congruent to 0 or 1 (mod 3).

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A251599 Centers of rows of the triangular array formed by the natural numbers.

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A336717 a(n) is the least k such that R(k) = n for some random Fibonacci sequence R (a random Fibonacci sequence R satisfies R(0) = 0, R(1) = 1, and for m > 0, R(m+1) = R(m) + (-1)^r(m) * R(m-1) where r is a sequence of integers).

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A376930 a(0)=0, a(1)=1; for n>1, a(n) = a(n-1)+a(n-2), except where a(n-1) is a prime greater than 2, in which case a(n) = a(n-1)-a(n-2).

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