A215495 a(4*n) = a(4*n+2) = a(2*n+1) = 2*n + 1.

1, 1, 1, 3, 3, 5, 3, 7, 5, 9, 5, 11, 7, 13, 7, 15, 9, 17, 9, 19, 11, 21, 11, 23, 13, 25, 13, 27, 15, 29, 15, 31, 17, 33, 17, 35, 19, 37, 19, 39, 21, 41, 21, 43, 23, 45, 23, 47, 25, 49, 25, 51, 27, 53, 27, 55, 29, 57, 29, 59, 31, 61, 31, 63, 33, 65, 33, 67, 35, 69, 35, 71, 37, 73, 37, 75, 39, 77, 39, 79, 41, 81, 41, 83, 43, 85, 43

Offset: 0

Author

Paul Curtz, Aug 13 2012

Keywords

• nonn
• easy

Comments

A214282(n) and -A214283(n) are companions. Separately or together, they have many links with the Catalan's numbers A000108(n). Examples:

A214282(n+1) - 2*A214282(n) = -1, -1, 1, 0, -2, -5, 5, 0, -14, -42, 42, 0, -132, ....

2*A214283(n) - A214283(n+1) = 1, 0, -1, -2, 2, 0, -5, -14, 14, 0, -42, -132, 132, ....

A214282(n) + A214283(n) = 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42,... (A126120).

The companion to a(n) is b(n) = -A214283(n)/(1,1,1,1,2,2,5,5,...) = 0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, ....

a(n) - b(n) = A056594(n).

Discovered as a(n) = A214282(n+1)/A000108([n/2]). See abs(A129996(n-2)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Crossrefs

Cf. A130823, A090192.

Programs

Magma

I:=[1,1,1,3,3,5]; [n le 6 select I[n] else Self(n-2) +Self(n-4) -Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 23 2018

Mathematica

a[n_?EvenQ] := n/2 + Boole[Mod[n, 4] == 0]; a[n_?OddQ] := n; Table[a[n], {n, 0, 86}] (* Jean-François Alcover, Aug 14 2012 *)
LinearRecurrence[{0,1,0,1,0,-1}, {1,1,1,3,3,5}, 50] (* G. C. Greubel, Apr 23 2018 *)

PARI

x='x+O('x^30); Vec(( 1+x+2*x^3+x^4+x^5 )/( (x^2+1)*(x-1)^2*(1+x)^2 )) \\ G. C. Greubel, Apr 23 2018

Formula

a(n+3) = (A185048(n+3)=2,2,4,2,... ) + 1.

a(n+2) - a(n) = 0, 2, 2, 2. (Period 4).

a(n) = 2*a(n-4) - a(n-8).

a(2*n) = A109613(n).

a(n+1) - a(n) = 2* (-1)^n * A059169(n).

G.f. : ( 1+x+2*x^3+x^4+x^5 ) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - Jean-François Alcover, Aug 14 2012