A142150 The nonnegative integers interleaved with 0's.

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 0, 11, 0, 12, 0, 13, 0, 14, 0, 15, 0, 16, 0, 17, 0, 18, 0, 19, 0, 20, 0, 21, 0, 22, 0, 23, 0, 24, 0, 25, 0, 26, 0, 27, 0, 28, 0, 29, 0, 30, 0, 31, 0, 32, 0, 33, 0, 34, 0, 35, 0, 36, 0, 37, 0, 38, 0, 39, 0, 40, 0, 41, 0, 42, 0, 43, 0

Offset: 0

Author

Reinhard Zumkeller, Jul 15 2008

Keywords

• nonn
• easy

Comments

Number of vertical pairs in a wheel with n equal sections. - Wesley Ivan Hurt, Jan 22 2012

Number of even terms of n-th row in the triangles A162610 and A209297. - Reinhard Zumkeller, Jan 19 2013

Also the result of writing n-1 in base 2 and multiplying the last digit with the number with its last digit removed. See A115273 and A257844-A257850 for generalization to other bases. - M. F. Hasler, May 10 2015

Also follows the rule: a(n+1) is the number of terms that are identical with a(n) for a(0..n-1). - Marc Morgenegg, Jul 08 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Reinhard Zumkeller, Logical Convolutions
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A000004, A000007, A000027, A000217, A000326, A001057, A001477, A003817, A008805, A027656, A086099, A142149, A142151, A162610, A191967, A195034, A209297.

Programs

Haskell

a142150 = uncurry (*) . (`divMod` 2) . (+ 1)
a142150_list = scanl (+) 0 a001057_list
-- Reinhard Zumkeller, Apr 02 2012

Magma

[n*(1+(-1)^n)/4 : n in [0..100]]; // Wesley Ivan Hurt, Aug 21 2014

Magma

&cat[[n, 0]: n in [0..50]]; // Vincenzo Librandi, Oct 31 2016

Maple

A142150:=n->n*(1+(-1)^n)/4: seq(A142150(n), n=0..100); # Wesley Ivan Hurt, Aug 21 2014

Mathematica

Table[Mod[Floor[n^2/2], n], {n, 200}] (* Enrique Pérez Herrero, Jul 29 2009 *)
Riffle[Range[0, 50], 0] (* Paolo Xausa, Feb 08 2024 *)

PARI

a(n)=!bittest(n,0)*n>>1 \\ M. F. Hasler, May 10 2015

Python

def A142150(n): return (n+1>>1)*(n&1^1) # Chai Wah Wu, Jan 19 2023

Formula

a(n) = XOR{k AND (n-k): 0<=k<=n}.

a(n) = (n/2)*0^(n mod 2); a(2*n)=n and a(2*n+1)=0.

a(n) = floor(n^2/2) mod n. - Enrique Pérez Herrero, Jul 29 2009

a(n) = A027656(n-2). - Reinhard Zumkeller, Nov 05 2009

a(n) = Sum_{k=0..n} (k mod 2)*((n-k) mod 2). - Reinhard Zumkeller, Nov 05 2009

a(n+1) = A000217(n) mod A000027(n+1) = A000217(n) mod A001477(n+1). - Edgar Almeida Ribeiro (edgar.a.ribeiro(AT)gmail.com), May 19 2010

From Bruno Berselli, Oct 19 2010: (Start)

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

G.f.: x^2/(1-x^2)^2.

a(n) = 2*a(n-2)-a(n-4) for n > 3.

Sum_{i=0..n} a(i) = (2*n*(n+1)+(2*n+1)*(-1)^n-1)/16 (see A008805). (End)

a(n) = -a(-n) = A195034(n-1)-A195034(-n-1). - Bruno Berselli, Oct 12 2011

a(n) = A000326(n) - A191967(n). - Reinhard Zumkeller, Jul 07 2012

a(n) = Sum_{i=1..n} floor((2*i-n)/2). - Wesley Ivan Hurt, Aug 21 2014

a(n-1) = floor(n/2)*(n mod 2), where (n mod 2) is the parity of n, or remainder of division by 2. - M. F. Hasler, May 10 2015

a(n) = A158416(n) - 1. - Filip Zaludek, Oct 30 2016

E.g.f.: x*sinh(x)/2. - Ilya Gutkovskiy, Oct 30 2016

a(n) = A000007(a(n-1)) + a(n-2) for n > 1. - Nicolas Bělohoubek, Oct 06 2024