A186421 Even numbers interleaved with repeated odd numbers.

0, 1, 2, 1, 4, 3, 6, 3, 8, 5, 10, 5, 12, 7, 14, 7, 16, 9, 18, 9, 20, 11, 22, 11, 24, 13, 26, 13, 28, 15, 30, 15, 32, 17, 34, 17, 36, 19, 38, 19, 40, 21, 42, 21, 44, 23, 46, 23, 48, 25, 50, 25, 52, 27, 54, 27, 56, 29, 58, 29, 60, 31, 62, 31, 64, 33, 66, 33, 68, 35, 70, 35, 72, 37, 74, 37, 76, 39, 78, 39, 80, 41, 82, 41, 84, 43, 86, 43

Offset: 0

Author

Reinhard Zumkeller, Feb 21 2011

Keywords

• nonn
• easy

Comments

A005843 interleaved with A109613.

Row sum of superimposed binary filled triangle. - Craig Knecht, Aug 07 2015

Examples

			A005843: 0   2   4   6   8   10   12   14   16   18   20    22
A109613:   1   1   3   3   5    5    7    7    9    9    11    11
this   : 0 1 2 1 4 3 6 3 8 5 10 5 12 7 14 7 16 9 18 9 20 11 22 ... .
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Craig Knecht, Row sums of superimposed binary triangles.
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Crossrefs

Cf. A005843, A109043, A109613.

Cf. A186422 (first differences), A186423 (partial sums), A240828 (row sum of superimposed binary triangle).

Programs

Haskell

a186421 n = a186421_list !! n
a186421_list = interleave [0,2..] $ rep [1,3..] where
   interleave (x:xs) ys = x : interleave ys xs
   rep (x:xs) = x : x : rep xs

Magma

[IsEven(n) select n else 2*Floor(n/4)+1: n in [0..100]]; // Vincenzo Librandi, Jul 13 2015

Maple

A186421:=n->n-(1-(-1)^n)*(n+(-1)^(n*(n+1)/2))/4: seq(A186421(n), n=0..100); # Wesley Ivan Hurt, Aug 07 2015

Mathematica

Table[n - (1 - (-1)^n)*(n + I^(n (n + 1)))/4, {n, 0, 87}] (* or *)
CoefficientList[Series[x (1 + 2 x + 2 x^3 + x^4)/((1 + x^2) (x - 1)^2 (1 + x)^2), {x, 0, 87}], x] (* or *)
n = 88; Riffle[Range[0, n, 2], Flatten@ Transpose[{Range[1, n, 2], Range[1, n, 2]}]] (* Michael De Vlieger, Jul 14 2015 *)

Maxima

makelist(n-(1-(-1)^n)*(n+%i^(n*(n+1)))/4, n, 0, 90); /* Bruno Berselli, Mar 04 2013 */

Python

def A186421(n): return (n>>1)|1 if n&1 else n # Chai Wah Wu, Jan 31 2023

Formula

a(2*k) = 2*k, a(4*k+1) = a(4*k+3) = 2*k+1.

a(n) = n if n is even, else 2*floor(n/4)+1.

a(2*n-(2*k+1)) + a(2*n+2*k+1) = 2*n, 0 <= k < n.

a(n+2) = A109043(n) - a(n).

G.f.: x*(1+2*x+2*x^3+x^4) / ( (1+x^2)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Feb 23 2011

a(n) = n-(1-(-1)^n)*(n+i^(n(n+1)))/4, where i=sqrt(-1). - Bruno Berselli, Feb 23 2011

a(n) = a(n-2)+a(n-4)-a(n-6) for n>5. - Wesley Ivan Hurt, Aug 07 2015

E.g.f.: (x*cosh(x) + sin(x) + 2*x*sinh(x))/2. - Stefano Spezia, May 09 2021

Extensions

Edited by Bruno Berselli, Mar 04 2013