A064455 -id:A064455 - cp's OEIS frontend

A064433 Number of iterations of A064455 to reach 2 (or 1 in the case of 1).

1, 1, 2, 6, 3, 5, 7, 12, 4, 14, 6, 11, 8, 8, 13, 13, 5, 10, 15, 15, 7, 7, 12, 12, 9, 17, 9, 71, 14, 14, 14, 68, 6, 19, 11, 11, 16, 16, 16, 24, 8, 70, 8, 21, 13, 13, 13, 67, 10, 18, 18, 18, 10, 10, 72, 72, 15, 23, 15, 23, 15, 15, 69, 69, 7, 20, 20, 20, 12, 12, 12, 66, 17, 74, 17

Offset: 1

Jonathan Ayres (Jonathan.ayres(AT)btinternet.com), Oct 01 2001

Similar to 3x+1 series (A008908). Does this sequence converge to 2 for all values of n (true for all values of n up to 100000)? The inverse sequence using next n = n-int(n/2) for n even and n+int(n/2) for n odd leads to 3 (?) possible end sequences (1), (5, 7, 10) and (17, 25, 37, 55, 82, 41, 61, 91, 136, 68, 34)
Starting with a number n, the next value generated is n+int(n/2) if n is even, n-int(n/2) if n is odd; a(n) is the number of iteration for the initial value n to reach the limit of 1 to 2
Collatz's 3N+1 function as isometry over the dyadics is N->N/2 if even, but N->(3N+1)/2 if odd, including the (necessary) halving into each tripling step. Counting steps until reaching 1 in this way leads to this sequence instead of A008908. - Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009
The value at each step of a trajectory starting with n (n>1) is equal to the value plus one at the same step of the row starting with (n-1) of the irregular triangle of the abbreviated (Terras-modified) Collatz sequence (A070168). - K. Spage, Aug 07 2014

			a(4) = 6. Starting with 4, 4 is even so the next number is 4+int(4/2) = 6, 6 is even so next number is 6+int(6/2) = 9, 9 is odd so next number is 9-int(9/2) = 5, 5 is odd so next number is 5-int(5/2) = 3, 3 is odd so next number is 3-int(3/2)=2, so giving a sequence of 4,6,9,5,3,2: 6 numbers.
a(5) = 3. Starting with 5, A064455(5) = 3, A064455(3) = 2, so giving a trajectory of 5,3,2: 3 numbers. - _K. Spage_, Aug 07 2014

Antti Karttunen, Table of n, a(n) for n = 1..19683
M. del P. Canales Chacon and M. J. Vielhaber, Structural and Computational Complexity of Isometries and Their Shift Commutators, Electr. Colloq. on Computational Cpx., ECCC TR04-057, 2004. [From Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009]

Cf. A006666, A008908, A064455, A070168.

Table[Length@ NestWhileList[If[EvenQ@ #, 3 #/2, (# + 1)/2] &, n, # != 1 + Boole[n > 1] &], {n, 75}] (* Michael De Vlieger, Sep 24 2016 *)

A064455(n) = {if(n%2, (n + 1)/2, 3*n/2)}
A064433(n) = {my(c=1); if(n==1, 1, while(n!=2, n=A064455(n); c++); c)} \\ K. Spage, Aug 07 2014

a(n) = A006666(n-1) + 1. - K. Spage, Aug 04 2014

A014682 The Collatz or 3x+1 function: a(n) = n/2 if n is even, otherwise (3n+1)/2.

Original entry on oeis.org

0, 2, 1, 5, 2, 8, 3, 11, 4, 14, 5, 17, 6, 20, 7, 23, 8, 26, 9, 29, 10, 32, 11, 35, 12, 38, 13, 41, 14, 44, 15, 47, 16, 50, 17, 53, 18, 56, 19, 59, 20, 62, 21, 65, 22, 68, 23, 71, 24, 74, 25, 77, 26, 80, 27, 83, 28, 86, 29, 89, 30, 92, 31, 95, 32, 98, 33, 101, 34, 104

Offset: 0

Mohammad K. Azarian

This is the function usually denoted by T(n) in the literature on the 3x+1 problem. See A006370 for further references and links.
Intertwining of sequence A016789 '2,5,8,11,... ("add 3")' and the nonnegative integers.
a(n) = log_2(A076936(n)). - Amarnath Murthy, Oct 19 2002
The average value of a(0), ..., a(n-1) is A004526(n). - Amarnath Murthy, Oct 19 2002
Partial sums are A093353. - Paul Barry, Mar 31 2008
Absolute first differences are essentially in A014681 and A103889. - R. J. Mathar, Apr 05 2008
Only terms of A016789 occur twice, at positions given by sequences A005408 (odd numbers) and A016957 (6n+4): (1,4), (3,10), (5,16), (7,22), ... - Antti Karttunen, Jul 28 2017
a(n) represents the unique congruence class modulo 2n+1 that is represented an odd number of times in any 2n+1 consecutive oblong numbers (A002378). This property relates to Jim Singh's 2018 formula, as n^2 + n is a relevant oblong number. - Peter Munn, Jan 29 2022

			a(3) = -3*(-1) - 2*1 - 1*(-1) - 0*1 + 1*(-1) + 2*1 + 3*(-1) + 4*1 + 5*(-1) + 6*1 = 5. - _Bruno Berselli_, Dec 14 2015

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.

N. J. A. Sloane (terms 0..1000) & Antti Karttunen, Table of n, a(n) for n = 0..100000
C. J. Everett, Iteration of the number-theoretic function f(2n) = n, f(2n + 1) = 3n + 2, Advances in Mathematics, Volume 25, Issue 1, July 1977, Pages 42-45.
Jeffrey C. Lagarias, The 3x+1 Problem: An Overview, arXiv:2111.02635 [math.NT], 2021.
Keenan Monks, Kenneth G. Monks, Kenneth M. Monks, and Maria Monks, Strongly sufficient sets and the distribution of arithmetic sequences in the 3x+1 graph, arXiv:1204.3904v1 [math.DS], Apr 17 2012.
R. Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252.
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A002378, A004526, A076936, A139391, A016116, A126241, A060412, A060413, A006370, A070168 (iterations), A005408, A016957, A064455, A153285, A008619, A193356.
Bisections: A001477, A016789.
Cf. A093353.

a014682 n = if r > 0 then div (3 * n + 1) 2 else n'
            where (n', r) = divMod n 2
-- Reinhard Zumkeller, Oct 03 2014

[IsOdd(n) select (3*n+1)/2 else n/2: n in [0..52]]; // Vincenzo Librandi, Sep 28 2018

T:=proc(n) if n mod 2 = 0 then n/2 else (3*n+1)/2; fi; end; # N. J. A. Sloane, Jan 31 2011
A076936 := proc(n) option remember ; local apr,ifr,me,i,a ; if n <=2 then n^2 ; else apr := mul(A076936(i),i=1..n-1) ; ifr := ifactors(apr)[2] ; me := -1 ; for i from 1 to nops(ifr) do me := max(me, op(2,op(i,ifr))) ; od ; me := me+ n-(me mod n) ; a := 1 ; for i from 1 to nops(ifr) do a := a*op(1,op(i,ifr))^(me-op(2,op(i,ifr))) ; od ; if a = A076936(n-1) then me := me+n ; a := 1 ; for i from 1 to nops(ifr) do a := a*op(1,op(i,ifr))^(me-op(2,op(i,ifr))) ; od ; fi ; RETURN(a) ; fi ; end: A014682 := proc(n) log[2](A076936(n)) ; end: for n from 1 to 85 do printf("%d, ",A014682(n)) ; od ; # R. J. Mathar, Mar 20 2007

Collatz[n_?OddQ] := (3n + 1)/2; Collatz[n_?EvenQ] := n/2; Table[Collatz[n], {n, 0, 79}] (* Alonso del Arte, Apr 21 2011 *)
LinearRecurrence[{0, 2, 0, -1}, {0, 2, 1, 5}, 70] (* Jean-François Alcover, Sep 23 2017 *)
Table[If[OddQ[n], (3 n + 1) / 2, n / 2], {n, 0, 60}] (* Vincenzo Librandi, Sep 28 2018 *)

a(n)=if(n%2,3*n+1,n)/2 \\ Charles R Greathouse IV, Sep 02 2015

a(n)=if(n<2,2*n,(n^2-n-1)%(2*n+1)) \\ Jim Singh, Sep 28 2018

def a(n): return n//2 if n%2==0 else (3*n + 1)//2
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 29 2017

From Paul Barry, Mar 31 2008: (Start)
G.f.: x*(2 + x + x^2)/(1-x^2)^2.
a(n) = (4*n+1)/4 - (2*n+1)*(-1)^n/4. (End)
a(n) = -a(n-1) + a(n-2) + a(n-3) + 4. - John W. Layman
For n > 1 this is the image of n under the modified "3x+1" map (cf. A006370): n -> n/2 if n is even, n -> (3*n+1)/2 if n is odd. - Benoit Cloitre, May 12 2002
O.g.f.: x*(2+x+x^2)/((-1+x)^2*(1+x)^2). - R. J. Mathar, Apr 05 2008
a(n) = 5/4 + (1/2)*((-1)^n)*n + (3/4)*(-1)^n + n. - Alexander R. Povolotsky, Apr 05 2008
a(n) = Sum_{i=-n..2*n} i*(-1)^i. - Bruno Berselli, Dec 14 2015
a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(i,k) + (-1)^k. - Wesley Ivan Hurt, Sep 20 2017
a(n) = (n^2-n-1) mod (2*n+1) for n > 1. - Jim Singh, Sep 26 2018
The above formula can be rewritten to show a pattern: a(n) = (n*(n+1)) mod (n+(n+1)). - Peter Munn, Jan 29 2022
Binary: a(n) = (n shift left (n AND 1)) - (n shift right 1) = A109043(n) - A004526(n). - Rudi B. Stranden, Jun 15 2021
From Rudi B. Stranden, Mar 21 2022: (Start)
a(n) = A064455(n+1) - 1, relating the number ON cells in row n of cellular automaton rule 54.
a(n) = 2*n - A071045(n).
(End)
E.g.f.: (1 + x)*sinh(x)/2 + 3*x*cosh(x)/2 = ((4*x+1)*e^x + (2*x-1)*e^(-x))/4. - Rénald Simonetto, Oct 20 2022
a(n) = n*(n mod 2) + ceiling(n/2) = A193356(n) + A008619(n+1). - Jonathan Shadrach Gilbert, Mar 12 2023
a(n) = 2*a(n-2) - a(n-4) for n > 3. - Chai Wah Wu, Apr 17 2024

Edited by N. J. A. Sloane, Apr 26 2008, at the suggestion of Artur Jasinski

Edited by N. J. A. Sloane, Jan 31 2011

A052928 The even numbers repeated.

Original entry on oeis.org

0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 70, 72, 72

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) is also the binary rank of the complete graph K(n). - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 6, a(n) is the number of (0,1) n X n matrices A <= P^(-1)+I+P having exactly two 1's in every row and column with perA=2. - Vladimir Shevelev, Apr 12 2010
a(n+2) is the number of symmetry allowed, linearly independent terms at n-th order in the series expansion of the (E+A)xe vibronic perturbation matrix, H(Q) (cf. Eisfeld & Viel). - Bradley Klee, Jul 21 2015
The arithmetic function v_2(n,1) as defined in A289187. - Robert Price, Aug 22 2017
For n > 1, also the chromatic number of the n X n white bishop graph. - Eric W. Weisstein, Nov 17 2017
For n > 2, also the maximum vertex degree of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 23 2018
For n >= 2, a(n+2) gives the minimum weight of a Boolean function of algebraic degree at most n-2 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019

C. D. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 181. - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Beierle, A. Biryukov and A. Udovenko, On degree-d zero-sum sets of full rank, Cryptography and Communications, November 2019.
W. Eisfeld and A. Viel, Higher order (A+E)xe pseudo-Jahn-Teller coupling, J. Chem. Phys., 122, 204317 (2005).
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 914
J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature
Eric Weisstein's World of Mathematics, Maximum Vertex Degree
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Random Matrix
Eric Weisstein's World of Mathematics, White Bishop Graph
Index entries for linear recurrences with constant coefficients, signature (1,1,-1)
Index entries for Molien series

Cf. A000034, A000124, A004001, A004526, A005843, A007590, A008619, A008794, A032766, A064455, A099392, A109613, A118266, A123684, A124356, A192442, A289187, A342819.
First differences: A010673; partial sums: A007590; partial sums of partial sums: A212964(n+1).
Complement of A109613 with respect to universe A004526. - Guenther Schrack, Dec 07 2017
Is first differences of A099392. Fixed point sequence: A005843. - Guenther Schrack, May 30 2019
For n >= 3, A329822(n) gives the minimum weight of a Boolean function of algebraic degree at most n-3 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019

a052928 = (* 2) . flip div 2
a052928_list = 0 : 0 : map (+ 2) a052928_list
-- Reinhard Zumkeller, Jun 20 2015

[2*Floor(n/2) : n in [0..50]]; // Wesley Ivan Hurt, Sep 13 2014

spec := [S,{S=Union(Sequence(Prod(Z,Z)),Prod(Sequence(Z),Sequence(Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Flatten[Table[{2n, 2n}, {n, 0, 39}]] (* Alonso del Arte, Jun 24 2012 *)
With[{ev=2Range[0,40]},Riffle[ev,ev]] (* Harvey P. Dale, May 08 2021 *)
Table[Round[n + 1/2], {n, -1, 72}] (* Ed Pegg Jr, Jul 28 2025 *)

a(n)=n\2*2 \\ Charles R Greathouse IV, Nov 20 2011

a(n) = 2*floor(n/2).
G.f.: 2*x^2/((-1+x)^2*(1+x)).
a(n) + a(n+1) + 2 - 2*n = 0.
a(n) = n - 1/2 + (-1)^n/2.
a(n) = n + Sum_{k=1..n} (-1)^k. - William A. Tedeschi, Mar 20 2008
a(n) = a(n-1) + a(n-2) - a(n-3). - R. J. Mathar, Feb 19 2010
a(n) = |A123684(n) - A064455(n)| = A032766(n) - A008619(n-1). - Jaroslav Krizek, Mar 22 2011
For n > 0, a(n) = floor(sqrt(n^2+(-1)^n)). - Francesco Daddi, Aug 02 2011
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=0 and b(k)=2^k for k>0. - Philippe Deléham, Oct 19 2011
a(n) = A109613(n) - 1. - M. F. Hasler, Oct 22 2012
a(n) = n - (n mod 2). - Wesley Ivan Hurt, Jun 29 2013
a(n) = a(a(n-1)) + a(n-a(n-1)) for n>2. - Nathan Fox, Jul 24 2016
a(n) = 2*A004526(n). - Filip Zaludek, Oct 28 2016
E.g.f.: x*exp(x) - sinh(x). - Ilya Gutkovskiy, Oct 28 2016
a(-n) = -a(n+1); a(n) = A005843(A004526(n)). - Guenther Schrack, Sep 11 2018
From Guenther Schrack, May 29 2019: (Start)
a(b(n)) = b(n) + ((-1)^b(n) - 1)/2 for any sequence b(n) of offset 0.
a(a(n)) = a(n), idempotent.
a(A086970(n)) = A124356(n-1) for n > 1.
a(A000124(n)) = A192447(n+1).
a(n)*a(n+1)/2 = A007590(n), also equals partial sums of a(n).
A007590(a(n)) = 2*A008794(n). (End)

More terms from James Sellers, Jun 05 2000

Removed duplicate of recurrence; corrected original recurrence and g.f. against offset - R. J. Mathar, Feb 19 2010

A080512 a(n) = n if n is odd, a(n) = 3*n/2 if n is even.

Original entry on oeis.org

1, 3, 3, 6, 5, 9, 7, 12, 9, 15, 11, 18, 13, 21, 15, 24, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 29, 45, 31, 48, 33, 51, 35, 54, 37, 57, 39, 60, 41, 63, 43, 66, 45, 69, 47, 72, 49, 75, 51, 78, 53, 81, 55, 84, 57, 87, 59, 90, 61, 93, 63, 96, 65, 99, 67, 102

Offset: 1

Amarnath Murthy, Mar 20 2003

First differences of the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011
Last term in n-th row of A080511.
Also A005408 and positive terms of A008585 interleaved. - Omar E. Pol, May 28 2012
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized heptagonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A080511, A008619, A126988.

Cf. A005408, A008585, A064455, A085787, A126246

import Data.List (transpose)
a080512 n = if m == 0 then 3 * n' else n  where (n', m) = divMod n 2
a080512_list = concat $ transpose [[1, 3 ..], [3, 6 ..]]
-- Reinhard Zumkeller, Apr 06 2015

[n*(5+(-1)^n)/4: n in [1..60]]; // Vincenzo Librandi, Sep 11 2011

Table[If[EvenQ[n],3n/2,n],{n,68}] (* Jayanta Basu, May 20 2013 *)

a(n) = n if n is odd, a(n) = 3*n/2 if n is even.
a(n)*a(n+3) = -3 + a(n+1)*a(n+2).
From Paul Barry, Sep 04 2003: (Start)
G.f.: (1+3*x+x^2)/((1-x^2)^2);
a(n) = n*(5 + (-1)^n)/4. (End)
Multiplicative with a(2^e) = 3*2^(e-1), a(p^e) = p^e otherwise. - Christian G. Bower, May 17 2005
Equals A126988 * (1, 1, 0, 0, 0, ...) - Gary W. Adamson, Apr 17 2007
Dirichlet g.f.: zeta(s-1) * (1 + 1/2^s). - Amiram Eldar, Oct 25 2023
Sum_{d divides n} mu(n/d)*a(d) = A126246(n), where mu(n) = A008683(n) is the Möbius function. - Peter Bala, Dec 31 2023

A123684 Alternate A016777(n) with A000027(n).

Original entry on oeis.org

1, 1, 4, 2, 7, 3, 10, 4, 13, 5, 16, 6, 19, 7, 22, 8, 25, 9, 28, 10, 31, 11, 34, 12, 37, 13, 40, 14, 43, 15, 46, 16, 49, 17, 52, 18, 55, 19, 58, 20, 61, 21, 64, 22, 67, 23, 70, 24, 73, 25, 76, 26, 79, 27, 82, 28, 85, 29, 88, 30, 91, 31, 94, 32, 97, 33, 100, 34, 103, 35, 106, 36

Offset: 1

Alford Arnold, Oct 11 2006

a(n) is a diagonal of Table A123685.
The arithmetic average of the first n terms gives the positive integers repeated (A008619). - Philippe Deléham, Nov 20 2013
Images under the modified '3x-1' map: a(n) = n/2 if n is even, (3n-1)/2 if n is odd. (In this sequence, the numbers at even indices n are n/2 [A000027], and the numbers at odd indices n are 3((n-1)/2) + 1 [A016777] = (3n-1)/2.) The latter correspondence interestingly mirrors an insight in David Bařina's 2020 paper (see below), namely that 3(n+1)/2 - 1 = (3n+1)/2. - Kevin Ge, Oct 30 2024

			The natural numbers begin 1, 2, 3, ... (A000027), the sequence 3*n + 1 begins 1, 4, 7, 10, ... (A016777), therefore A123684 begins 1, 1, 4, 2, 7, 3, 10, ...
1/1 = 1, (1+1)/2 = 1, (1+1+4)/3 = 2, (1+1+4+2)/4 = 2, ... - _Philippe Deléham_, Nov 20 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
David Bařina, Convergence verification of the Collatz problem

Cf. A000027, A008619, A016777, A016825, A031878, A052928, A064455.
Cf. A071045, A092530, A093005, A105638, A123685, A137501, A225126.
Cf. A014682 (modified Collatz map)

import Data.List (transpose)
a123684 n = a123684_list !! (n-1)
a123684_list = concat $ transpose [a016777_list, a000027_list]
-- Reinhard Zumkeller, Apr 29 2013

&cat[ [ 3*n-2, n ]: n in [1..36] ]; // Klaus Brockhaus, May 12 2007

/* From the fourteenth formula: */ [&+[1+k*(-1)^k: k in [0..n]]: n in [0..80]]; // Bruno Berselli, Jul 16 2013

A123684:=n->n-1/4-(1/2*n-1/4)*(-1)^n: seq(A123684(n), n=1..70); # Wesley Ivan Hurt, Jul 26 2014

CoefficientList[Series[(1 +x +2*x^2)/((1-x)^2*(1+x)^2), {x,0,70}], x] (* Wesley Ivan Hurt, Jul 26 2014 *)
LinearRecurrence[{0,2,0,-1},{1,1,4,2},80] (* Harvey P. Dale, Apr 14 2025 *)

print(vector(72, n, if(n%2==0, n/2, (3*n-1)/2))) \\ Klaus Brockhaus, May 12 2007

print(vector(72, n, n-1/4-(1/2*n-1/4)*(-1)^n)); \\ Klaus Brockhaus, May 12 2007

[(n + (2*n-1)*(n%2))//2 for n in range(1,71)] # G. C. Greubel, Mar 15 2024

From Klaus Brockhaus, May 12 2007: (Start)
G.f.: x*(1+x+2*x^2)/((1-x)^2*(1+x)^2).
a(n) = (1/4)*(4*n - 1 - (2*n - 1)*(-1)^n).
a(2n-1) = A016777(n-1) = 3(n-1) + 1.
a(2n) = A000027(n) = n.
a(n) = A071045(n-1) + 1.
a(n) = A093005(n) - A093005(n-1) for n > 1.
a(n) = A105638(n+2) - A105638(n+1) for n > 1.
a(n) = A092530(n) - A092530(n-1) - 1.
a(n) = A031878(n+1) - A031878(n) - 1. (End)
a(2*n+1) + a(2*n+2) = A016825(n). - Paul Curtz, Mar 09 2011
a(n)= 2*a(n-2) - a(n-4). - Paul Curtz, Mar 09 2011
From Jaroslav Krizek, Mar 22 2011 (Start):
a(n) = n + a(n-1) for odd n; a(n) = n - A064455(n-1) for even n.
a(n) = A064455(n) - A137501(n).
Abs(a(n) - A064455(n)) = A052928(n). (End)
a(n) = A225126(n) for n > 1. - Reinhard Zumkeller, Apr 29 2013
a(n) = Sum_{k=1..n} (1 + (k-1)*(-1)^(k-1)). - Bruno Berselli, Jul 16 2013
a(n) = n + floor(n/2) for odd n; a(n) = n/2 for even n. - Reinhard Muehlfeld, Jul 25 2014

More terms from Klaus Brockhaus, May 12 2007

A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3).

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, 9, 18, 20, 11, 22, 24, 13, 26, 28, 15, 30, 32, 17, 34, 36, 19, 38, 40, 21, 42, 44, 23, 46, 48, 25, 50, 52, 27, 54, 56, 29, 58, 60, 31, 62, 64, 33, 66, 68, 35, 70, 72, 37, 74, 76, 39, 78, 80, 41, 82, 84, 43, 86, 88, 45

Offset: 0

Bruno Berselli, Dec 12 2015 - based on an idea by Paul Curtz

The inverse permutation is given by P(n) = A006368(n-1) + 1, for n >= 1, and P(0) = 0. - Wolfdieter Lang, Sep 21 2021

This permutation is given by A006369(n-1) + 1, with A006369(-1) = -1. Observed by Kevin Ryde. - Wolfdieter Lang, Sep 22 2021

			-------------------------------------------------------------------------
0, 1, 2, 3,  4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...
+  +  +  +   +  +  +   +   +   +   +   +   +   +   +   +   +   +   +
0, 0, 0, 1, -1, 1, 2, -2,  2,  3, -3,  3,  4, -4,  4,  5, -5,  5,  6, ...
-------------------------------------------------------------------------
0, 1, 2, 4,  3, 6, 8,  5, 10, 12,  7, 14, 16,  9, 18, 20, 11, 22, 24, ...
-------------------------------------------------------------------------

Bruno Berselli, Table of n, a(n) for n = 0..1000
Peter Lynch and Michael Mackey, Parity and Partition of the Rational Numbers, arXiv:2205.00565 [math.NT], 2022. See set F p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).
Index entries for permutations of the positive (or nonnegative) integers.

Cf. A001477, A006368, A006369, A008738, A032793.
Cf. A064455: n+floor(n/2)*(-1)^(n mod 2).
Cf. A265888: n+floor(n/4)*(-1)^(n mod 4).
Cf. A265734: n+floor(n/5)*(-1)^(n mod 5).

[n+Floor(n/3)*(-1)^(n mod 3): n in [0..70]];

Table[n + Floor[n/3] (-1)^Mod[n, 3], {n, 0, 70}]

[n+floor(n/3)*(-1)^mod(n,3) for n in (0..70)]

G.f.: x*(1 + 2*x + 4*x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-3) - a(n-6).
a(3*k)   = 4*k;
a(3*k+1) = 2*k+1, hence a(3*k+1) = a(3*k)/2 + 1;
a(3*k+2) = 4*k+2, hence a(3*k+2) = 2*a(3*k+1) = a(3*k) + 2.
Sum_{i=0..n} a(i) = A008738(A032793(n+1)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Mar 30 2023

A071030 Triangle read by rows giving successive states of cellular automaton generated by "Rule 54".

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1

Offset: 0

Hans Havermann, May 26 2002

Row n has length 2n+1.

Even rows r sum to r/2 + 1, odd rows r sum to 3r to produce the sequence {1, 3, 2, 6, 3, 9, 4, 12, ...} = A064455(n + 1). - Michael De Vlieger, Oct 07 2015

			From _Michael De Vlieger_, Oct 07 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells:
                        1
                      1 1 1
                    1 . . . 1
                  1 1 1 . 1 1 1
                1 . . . 1 . . . 1
              1 1 1 . 1 1 1 . 1 1 1
            1 . . . 1 . . . 1 . . . 1
          1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
        1 . . . 1 . . . 1 . . . 1 . . . 1
      1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
    1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1
  1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1
1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Visualization of rows 0 - 31 of Rule 54
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A064455. See A118108 and A118109 for two other versions of this sequence.

clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; clip /@ CellularAutomaton[54, {{1}, 0}, 8] // Flatten (* Michael De Vlieger, Oct 07 2015 *)

Corrected by Hans Havermann, Jan 07 2012

A137501 The even numbers repeated, with alternating signs.

Original entry on oeis.org

0, 0, 2, -2, 4, -4, 6, -6, 8, -8, 10, -10, 12, -12, 14, -14, 16, -16, 18, -18, 20, -20, 22, -22, 24, -24, 26, -26, 28, -28, 30, -30, 32, -32, 34, -34, 36, -36, 38, -38, 40, -40, 42, -42, 44, -44, 46, -46, 48, -48, 50, -50, 52, -52, 54, -54, 56, -56, 58, -58, 60, -60, 62, -62, 64, -64, 66, -66, 68, -68, 70, -70, 72, -72, 74

Offset: 0

Carlos Alberto da Costa Filho (cacau_dacosta(AT)hotmail.com), Apr 22 2008

The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=1 and k shifted one place, thus a(k) = (P(1,1)-(-1)^(k-1) P(1,2(k-1)+1))/2. - Peter Luschny, Jul 12 2009

With just one 0 at the beginning, this is a permutation of all the even integers. - Alonso del Arte, Jun 24 2012

Index entries for linear recurrences with constant coefficients, signature (-1,1,1).

Cf. A052928, A064455, A123684, A153641.

den:= n -> (n-1/2+1/2*(-1)^n)*(-1)^n: seq(den(n),n=-10..10);
a := n -> (1+(-1)^n*(2*n-1))/2; # Peter Luschny, Jul 12 2009

Flatten[Table[{2n, -2n}, {n, 0, 39}]] (* Alonso del Arte, Jun 24 2012 *)
With[{enos=2*Range[0,40]},Riffle[enos,-enos]] (* Harvey P. Dale, Oct 12 2014 *)

a(n) = ( n - (1/2) + (1/2)*(-1)^n )*(-1)^n.
From R. J. Mathar, Feb 14 2010: (Start)
a(n) = -a(n-1) + a(n-2) + a(n-3).
G.f.: 2*x^2/((1-x) * (1+x)^2). (End)
a(n) = A064455(n) - A123684(n). - Jaroslav Krizek, Mar 22 2011

A098557 Expansion of e.g.f. (1/2)(1+x)log((1+x)/(1-x)).

Original entry on oeis.org

0, 1, 2, 2, 8, 24, 144, 720, 5760, 40320, 403200, 3628800, 43545600, 479001600, 6706022400, 87178291200, 1394852659200, 20922789888000, 376610217984000, 6402373705728000, 128047474114560000, 2432902008176640000, 53523844179886080000, 1124000727777607680000

Offset: 0

Paul Barry, Sep 14 2004

G. C. Greubel, Table of n, a(n) for n = 0..450

From Johannes W. Meijer, Nov 12 2009: (Start)
Cf. A109613 (odd numbers repeated).
Equals the first left hand column of A167552.
Equals the first right hand column of A167556.
A098557(n)*A064455(n) equals the second right hand column of A167556(n).
(End)
Cf. A052558, A073742.

[0,1] cat [Factorial(n-1) + Factorial(n-2)*(1+(-1)^n)/2: n in [2..30]]; // G. C. Greubel, Jan 17 2018

Join[{0,1}, Table[(n-1)! + (n-2)!*(1+(-1)^n)/2, {n,2,30}]] (* or *) With[{nmax = 50}, CoefficientList[Series[(1/2)*(1 + x)*Log[(1 + x)/(1 - x)], {x,0,nmax}], x]*Range[0,nmax]!] (* G. C. Greubel, Jan 17 2018 *)

for(n=0, 30, print1(if(n==0,0, if(n==1, 1, (n-1)! + (n-2)!*(1 + (-1)^n)/2)), ", ")) \\ G. C. Greubel, Jan 17 2018

a(n+1) = n! + (n-1)! * (1-(-1)^n)/2.
a(n+2) = 2*A052558(n).
conjecture: -a(n) +a(n-1) +(n-1)*(n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
G.f.: 1-G(0), where G(k)= 1 + x*(2*k-1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013
Sum_{n>=1} 1/a(n) = sinh(1) + 1 = A073742 + 1. - Amiram Eldar, Jan 22 2023

A265734 Permutation of nonnegative integers: a(n) = n + floor(n/5)*(-1)^(n mod 5).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 8, 7, 10, 12, 9, 14, 11, 16, 18, 13, 20, 15, 22, 24, 17, 26, 19, 28, 30, 21, 32, 23, 34, 36, 25, 38, 27, 40, 42, 29, 44, 31, 46, 48, 33, 50, 35, 52, 54, 37, 56, 39, 58, 60, 41, 62, 43, 64, 66, 45, 68, 47, 70, 72, 49, 74, 51, 76, 78, 53, 80

Offset: 0

Bruno Berselli, Dec 15 2015 - based on an idea by Paul Curtz

			------------------------------------------------------------------------
0, 1, 2, 3, 4, 5,  6, 7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...
+  +  +  +  +  +   +  +   +   +   +   +   +   +   +   +   +   +   +
0, 0, 0, 0, 0, 1, -1, 1, -1,  1,  2, -2,  2, -2,  2,  3, -3,  3, -3, ...
------------------------------------------------------------------------
0, 1, 2, 3, 4, 6,  5, 8,  7, 10, 12,  9, 14, 11, 16, 18, 13, 20, 15, ...
------------------------------------------------------------------------

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).
Index entries for permutations of the positive (or nonnegative) integers.

Cf. A001477.
Cf. A064455: n+floor(n/2)*(-1)^(n mod 2).
Cf. A265667: n+floor(n/3)*(-1)^(n mod 3).
Cf. A265888: n+floor(n/4)*(-1)^(n mod 4)

[n+Floor(n/5)*(-1)^(n mod 5): n in [0..80]];

Table[n + Floor[n/5] (-1)^Mod[n, 5], {n, 0, 80}]

[n+floor(n/5)*(-1)^mod(n, 5) for n in (0..80)]

G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 6*x^4 + 3*x^5 + 4*x^6 + x^7 + 2*x^8) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)^2).
a(n) = 2*a(n-5) - a(n-10).
a(5*k+r) = (5+(-1)^r)*k + r, where r=0..4.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*(1/(2*sqrt(2))-1/(3*sqrt(3))) + log(2)/6. - Amiram Eldar, Mar 30 2023

cp's OEIS Frontend

A064433 Number of iterations of A064455 to reach 2 (or 1 in the case of 1).

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A014682 The Collatz or 3x+1 function: a(n) = n/2 if n is even, otherwise (3n+1)/2.

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A052928 The even numbers repeated.

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A080512 a(n) = n if n is odd, a(n) = 3*n/2 if n is even.

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A123684 Alternate A016777(n) with A000027(n).

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A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3).

A098557 Expansion of e.g.f. (1/2)(1+x)log((1+x)/(1-x)).