A047624 -id:A047624 - cp's OEIS frontend

A140081 Period 4: repeat [0, 1, 1, 2].

0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 03 2008

Also fix e = 4; then a(n) = minimal Hamming distance between the binary representation of n and the binary representation of any multiple k*e (0 <= k <= n/e) which is a child of n.

A number m is a child of n if the binary representation of n has a 1 in every position where the binary representation of m has a 1.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A140201. - Reinhard Zumkeller, Feb 21 2010

a140081 n = div (mod n 4 + mod n 2) 2
a140081_list = cycle [0, 1, 1, 2]  -- Reinhard Zumkeller, Aug 15 2015

A140081:=n->floor((3*(n mod 4)+1)/4); seq(A140081(n), n=0..100); # Wesley Ivan Hurt, Mar 27 2014

PadLeft[{}, 100, {0,1,1,2}] (* Harvey P. Dale, Aug 19 2011 *)
Table[Floor[(3 Mod[n, 4] + 1)/4], {n, 0, 100}] (* Wesley Ivan Hurt, Mar 27 2014 *)

a(n)=n%4-n%4\2 \\ Jaume Oliver Lafont, Aug 28 2009

a(n) = 1 + a(n - 1 - a(n-1)) + 2*a(a(n-1)) - 2*a(n-1), a(0)=0. - Ramasamy Chandramouli, Jan 31 2010
a(n) = A047624(n+2) - A047624(n+1) - 1. - Reinhard Zumkeller, Feb 21 2010
a(n) = 1 - cos(Pi*n/2)/2 - sin(Pi*n/2)/2 - (-1)^n/2. - R. J. Mathar, Oct 08 2011
a(n) = ((n mod 4) + (n mod 2))/2. - Gary Detlefs, Apr 21 2012
From Colin Barker, Jan 13 2013: (Start)
a(n) = a(n-4).
G.f.: -x*(2*x^2+x+1) / ((x-1)*(x+1)*(x^2+1)). (End)
a(n) = floor((3*(n mod 4) + 1)/4). - Wesley Ivan Hurt, Mar 27 2014
From Wesley Ivan Hurt, Apr 22 2015: (Start)
a(n) = floor(1/2 + (n mod 4)/2).
a(n) = 1 - (-1)^n/2 - (-1)^(n/2 - 1/4 + (-1)^n/4)/2. (End)
a(n) = n - floor(n/2) - 2*floor(n/4). - Ridouane Oudra, Oct 30 2019

A140201 Partial sums of A140081.

Original entry on oeis.org

0, 1, 2, 4, 4, 5, 6, 8, 8, 9, 10, 12, 12, 13, 14, 16, 16, 17, 18, 20, 20, 21, 22, 24, 24, 25, 26, 28, 28, 29, 30, 32, 32, 33, 34, 36, 36, 37, 38, 40, 40, 41, 42, 44, 44, 45, 46, 48, 48, 49, 50, 52, 52, 53, 54, 56, 56, 57, 58, 60, 60, 61, 62, 64, 64, 65, 66, 68, 68, 69, 70, 72, 72, 73, 74

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 09 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A002265, A004524, A004526, A042948, A047624, A057353, A121262.

I:=[0, 1, 2, 4, 4]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..80]]; // Vincenzo Librandi, Sep 17 2012

A140201:=n->(4*n+1-I^(2*n)+(-I)^(1+n)+I^(1+n))/4: seq(A140201(n), n=0..100); # Wesley Ivan Hurt, Jun 04 2016

Accumulate[PadRight[{}, 68, {0, 1, 1, 2}]] (* Harvey P. Dale, Aug 19 2011 *)

a(n) = A047624(n+1) - A042948(A004526(n)). - Reinhard Zumkeller, Feb 21 2010
a(n) = A002265(n+1) + A057353(n+1). - Reinhard Zumkeller, Feb 26 2011
From Bruno Berselli, Jan 27 2011: (Start)
G.f.: x*(1+x+2*x^2)/((1+x)*(1+x^2)*(1-x)^2).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
a(n) = n + A121262(n+1). (End)
a(n) = n when n+1 is not a multiple of 4, and a(n) = n+1 when n+1 is a multiple of 4. - Dennis P. Walsh, Aug 06 2012
a(n) = A004524(n+1) + A004526(n+1). - Arkadiusz Wesolowski, Sep 17 2012
a(n) = (4*n+1-i^(2*n)+(-i)^(1+n)+i^(1+n))/4 where i=sqrt(-1). - Wesley Ivan Hurt, Jun 04 2016
a(n) = n+1-(sign((n+1) mod 4) mod 3). - Wesley Ivan Hurt, Sep 26 2017

A173562 a(n) = n^2 + floor(n/4).

Original entry on oeis.org

0, 1, 4, 9, 17, 26, 37, 50, 66, 83, 102, 123, 147, 172, 199, 228, 260, 293, 328, 365, 405, 446, 489, 534, 582, 631, 682, 735, 791, 848, 907, 968, 1032, 1097, 1164, 1233, 1305, 1378, 1453, 1530, 1610, 1691, 1774, 1859, 1947, 2036, 2127, 2220, 2316, 2413, 2512

Offset: 0

Reinhard Zumkeller, Feb 21 2010

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A000290, A002265, A002378, A035608, A047624, A057353.

[Floor((n + 1/8)^2) : n in [0..80]]; // Wesley Ivan Hurt, Jun 04 2016

A173562:=n->floor((n + 1/8)^2): seq(A173562(n), n=0..80); # Wesley Ivan Hurt, Jun 04 2016

Table[n^2+Floor[n/4],{n,0,50}] (* or *) LinearRecurrence[{2,-1,0,1,-2,1}, {0,1,4,9,17,26}, 50] (* Harvey P. Dale, Nov 25 2011 *)

a(n)=n^2+n\4 \\ Charles R Greathouse IV, Oct 16 2015

def A173562(n): return n**2+(n>>2) # Chai Wah Wu, Feb 02 2023

a(n) = A002378(n)-A057353(n) = A035608(n)-A002265(n+2) = A000290(n)+A002265(n);
a(n+1) - a(n) = A047624(n+2).
a(n) = floor((n + 1/8)^2).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>5.
G.f.: x*(1+2*x+2*x^2+3*x^3)/((1+x)*(x^2+1)*(1-x)^3). - R. J. Mathar, Feb 27 2010
a(n) = (8*n^2+2*n-3+i^(2*n)+(1+i)*i^(-n)+(1-i)*i^n)/8 where i=sqrt(-1). - Wesley Ivan Hurt, Jun 04 2016

A265888 a(n) = n + floor(n/4)*(-1)^(n mod 4).

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 7, 6, 10, 7, 12, 9, 15, 10, 17, 12, 20, 13, 22, 15, 25, 16, 27, 18, 30, 19, 32, 21, 35, 22, 37, 24, 40, 25, 42, 27, 45, 28, 47, 30, 50, 31, 52, 33, 55, 34, 57, 36, 60, 37, 62, 39, 65, 40, 67, 42, 70, 43, 72, 45, 75, 46, 77, 48, 80, 49, 82, 51, 85, 52, 87

Offset: 0

Bruno Berselli, Dec 18 2015

This sequence does not include the numbers of the type 3*A047202(n)+2.

a(n) = n + floor(n/4)*(-1)^(n mod 2). - Chai Wah Wu, Jan 29 2023

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A047202, A047624.
Cf. A064455: n+floor(n/2)*(-1)^(n mod 2).
Cf. A265667: n+floor(n/3)*(-1)^(n mod 3).
Cf. A265734: n+floor(n/5)*(-1)^(n mod 5).

[n+Floor(n/4)*(-1)^(n mod 4): n in [0..70]];

Table[n + Floor[n/4] (-1)^Mod[n, 4], {n, 0, 70}]
LinearRecurrence[{0, 1, 0, 1, 0, -1}, {0, 1, 2, 3, 5, 4}, 80]

x='x+O('x^100); concat(0, Vec(x*(1+2*x+2*x^2+3*x^3)/((1+x^2)*(1- x^2)^2))) \\ Altug Alkan, Dec 22 2015

def A265888(n): return n+(-(n>>2) if n&1 else n>>2) # Chai Wah Wu, Jan 29 2023

[n+floor(n/4)*(-1)^mod(n, 4) for n in (0..70)]

G.f.: x*(1 + 2*x + 2*x^2 + 3*x^3)/((1 + x^2)*(1 - x^2)^2).
a(n) = a(n-2) + a(n-4) - a(n-6) for n>5.
a(n+1) + a(n) = A047624(n+1).
a(4*k+r) = (4+(-1)^r)*k + r mod 3, where r = 0..3.

cp's OEIS Frontend

A140081 Period 4: repeat [0, 1, 1, 2].

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A140201 Partial sums of A140081.

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A173562 a(n) = n^2 + floor(n/4).

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A265888 a(n) = n + floor(n/4)*(-1)^(n mod 4).

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