A030451 a(2*n) = n, a(2*n+1) = n+2.

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

Offset: 0

Author

Daniel Smith (2true(AT)gte.net)

Keywords

• nonn
• easy

Comments

Previous name was: Once started, this mixes the natural numbers and the natural numbers shifted by 1.

Smallest number of integer-sided squares needed to tile a 2 X n rectangle. a(5) = 4:

..._...

| | |_|

|_|___||. - _Alois P. Heinz, Jun 12 2013

Links

• Index entries for two-way infinite sequences.
• Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Crossrefs

Cf. A168361 (first differences), A198442 (partial sums).

Cf. A008619, A110570.

Row m=2 of A113881, A219158.

Essentially the same as A028242.

Programs

Maple

a:= n-> iquo(n, 2, 'r') +[0, 2][r+1]:
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 12 2013

Mathematica

Riffle[# + 1, #] &@ Range[0, 37] (* or *)
Table[3/4 - (-1)^n 3/4 + n/2, {n, 0, 72}] (* or *)
CoefficientList[Series[(2 x - x^2)/((1 - x) (1 - x^2)), {x, 0, 72}], x] (* Michael De Vlieger, Apr 25 2016 *)

PARI

a(n)=n\2+2*(n%2)

Formula

a(n) = 3/4 -(-1)^n*3/4 +n/2.

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

a(2n) = n, a(2n+1) = n+2.

a(n+2) = a(n)+1.

a(n) = -a(-3-n).

a(n) = A110570(n,2) for n>1. - Reinhard Zumkeller, Jul 28 2005

a(n) = (n+1)-a(n-1) with n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010

a(n) = Sum_{k=1..n} (-1)^(n+k)*(k+1). - Arkadiusz Wesolowski, Nov 23 2012

a(n+1) = (a(0) + a(1) + ... + a(n))/a(n) for n>0. This formula with different initial conditions produces A008619. - Ivan Neretin, Apr 25 2016

E.g.f.: (x*exp(x) + 3*sinh(x))/2. - Ilya Gutkovskiy, Apr 25 2016

Sum_{n>=1} (-1)^n/a(n) = 1. - Amiram Eldar, Oct 04 2022

Extensions

New name (using existing formula) from Joerg Arndt, Apr 26 2016