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

0,3

Comments

A005843 interleaved with A109613.

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

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).

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

Example

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 ... .

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 *)

Prog

(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
(Maxima) makelist(n-(1-(-1)^n)*(n+%i^(n*(n+1)))/4, n, 0, 90); /* Bruno Berselli, Mar 04 2013 */
(Magma) [IsEven(n) select n else 2*Floor(n/4)+1: n in [0..100]]; // Vincenzo Librandi, Jul 13 2015
(Python)
def A186421(n): return (n>>1)|1 if n&1 else n # Chai Wah Wu, Jan 31 2023

Crossrefs

Cf. A005843, A109043, A109613.

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

Sequence in context: A234586 A338657 A347082 * A351752 A361189 A004560

Adjacent sequences: A186418 A186419 A186420 * A186422 A186423 A186424

Keyword

nonn,easy

Author

Reinhard Zumkeller, Feb 21 2011

Extensions

Edited by Bruno Berselli, Mar 04 2013

Status

approved