A265888 -id:A265888 - cp's OEIS frontend

A064455 a(2n) = 3n, a(2n-1) = n.

1, 3, 2, 6, 3, 9, 4, 12, 5, 15, 6, 18, 7, 21, 8, 24, 9, 27, 10, 30, 11, 33, 12, 36, 13, 39, 14, 42, 15, 45, 16, 48, 17, 51, 18, 54, 19, 57, 20, 60, 21, 63, 22, 66, 23, 69, 24, 72, 25, 75, 26, 78, 27, 81, 28, 84, 29, 87, 30, 90, 31, 93, 32, 96, 33, 99, 34, 102, 35, 105, 36, 108

Offset: 1

N. J. A. Sloane, Oct 02 2001

Also number of 1's in n-th row of triangle in A071030. - Hans Havermann, May 26 2002
Number of ON cells at generation n of 1-D CA defined by Rule 54. - N. J. A. Sloane, Aug 09 2014
a(n)*A098557(n) equals the second right hand column of A167556. - Johannes W. Meijer, Nov 12 2009
Given a(1) = 1, for all n > 1, a(n) is the least positive integer not equal to a(n-1) such that the arithmetic mean of the first n terms is an integer. The sequence of arithmetic means of the first 1, 2, 3, ..., terms is 1, 2, 2, 3, 3, 4, 4, ... (A004526 disregarding its first three terms). - Rick L. Shepherd, Aug 20 2013

			a(13) = a(2*7 - 1) = 7, a(14) = a(2*7) = 21.
a(8) = 8-9+10-11+12-13+14-15+16 = 12. - _Bruno Berselli_, Jun 05 2013

Harry J. Smith, Table of n, a(n) for n = 1..1000
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, 2016.
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Interleaving of A000027 and A008585 (without first term).

Cf. A004526, A052928, A064433, A071030, A080512, A098557, A123684, A137501, A167556, A225144, A265888.

maxarg := 75; for n := 1 to maxarg do if n mod 2 = 1 then write((n+1) div 2, " ") else write((n div 2)*3," "); end; end;

a:=[];;  for n in [1..75] do if n mod 2 = 0 then Add(a,3*n/2); else Add(a,(n+1)/2); fi; od; a; # Muniru A Asiru, Oct 28 2018

import Data.List (transpose)
a064455 n = n + if m == 0 then n' else - n'  where (n',m) = divMod n 2
a064455_list = concat $ transpose [[1 ..], [3, 6 ..]]
-- Reinhard Zumkeller, Oct 12 2013

[(1/2)*n*(-1)^n+n+(1/4)*(1-(-1)^n): n in [1..80]]; // Vincenzo Librandi, Aug 10 2014

A064455 := proc(n)
    if type(n,'even') then
        3*n/2 ;
    else
        (n+1)/2 ;
    end if;
end proc: # R. J. Mathar, Aug 03 2015

Table[ If[ EvenQ[n], 3n/2, (n + 1)/2], {n, 1, 70} ]

a(n) = { if (n%2, (n + 1)/2, 3*n/2) } \\ Harry J. Smith, Sep 14 2009

a(n)=if(n<3,2*n-1,((n-1)*(n-2))%(2*n-1)) \\ Jim Singh, Oct 14 2018

def A064455(n): return (3*n - (2*n-1)*(n%2))//2
print([A064455(n) for n in range(1,81)]) # G. C. Greubel, Jan 30 2025

a(n) = (1/2)*n*(-1)^n + n + (1/4)*(1 - (-1)^n). - Stephen Crowley, Aug 10 2009
G.f.: x*(1+3*x) / ( (1-x)^2*(1+x)^2 ). - R. J. Mathar, Mar 30 2011
From Jaroslav Krizek, Mar 22 2011: (Start)
a(n) = n - A123684(n-1) for odd n.
a(n) = n + a(n-1) for even n.
a(n) = A123684(n) + A137501(n).
Abs( a(n) - A123684(n) ) =  A052928(n). (End)
a(n) = Sum_{i=n..2*n} i*(-1)^i. - Bruno Berselli, Jun 05 2013
a(n) = n + floor(n/2)*(-1)^(n mod 2). - Bruno Berselli, Dec 14 2015
a(n) = (n^2-3*n+2) mod (2*n-1) for n>2. - Jim Singh, Oct 31 2018
E.g.f.: (1/2)*(x*cosh(x) + (1+3*x)*sinh(x)). - G. C. Greubel, Jan 30 2025

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

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

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A064455 a(2n) = 3n, a(2n-1) = n.

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

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

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Magma

Mathematica

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