A029578 The natural numbers interleaved with the even numbers.

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

Offset

0,4

Comments

a(n) = number of ordered, length two, compositions of n with at least one odd summand - Len Smiley, Nov 25 2001

Also number of 0's in n-th row of triangle in A071037. - Hans Havermann, May 26 2002

For n > 2: a(n) = number of odd terms in row n-2 of triangle A265705. - Reinhard Zumkeller, Dec 15 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Formula

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

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

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

a(n) = floor((n+1)/2) + (n is odd)*floor((n+1)/2).

a(n) = (n - n mod 2)/(2 - n mod 2). - Reinhard Zumkeller, Jul 30 2002

a(n) = floor(n/2)*binomial(2, mod(n, 2)) - Paul Barry, May 25 2003

a(2*n) = n.

a(2*n-1) = 2*n-2.

a(-n) = -A065423(n+2).

a(n) = Sum_{k=0..floor((n-2)/2)} (C(n-k-2, k) mod 2)((1+(-1)^k)/2)*2^A000120(n-2k-2). - Paul Barry, Jan 06 2005

a(n) = Sum_{k=0..n-2} gcd(n-k-1, k+1). - Paul Barry, May 03 2005

For n>6: a(n) = floor(a(n-1)*a(n-2)/a(n-3)). - Reinhard Zumkeller, Mar 06 2011

E.g.f.: (1/4)*((x+2)*exp(-x) + (3*x-2)*exp(x)). - G. C. Greubel, Jan 22 2025

Mathematica

With[{nn=40}, Riffle[Range[0, nn], Range[0, 2nn, 2]]] (* or *) LinearRecurrence[ {0, 2, 0, -1}, {0, 0, 1, 2}, 80] (* Harvey P. Dale, Aug 23 2015 *)

Prog

(PARI) a(n)=if(n%2, n-1, n/2)
(Haskell)
import Data.List (transpose)
a029578 n =  (n - n `mod` 2) `div` (2 - n `mod` 2)
a029578_list = concat $ transpose [a001477_list, a005843_list]
-- Reinhard Zumkeller, Nov 27 2012
(Magma)
A029578:= func< n | (n + (n-2)*(n mod 2))/2 >;
[A029578(n): n in [0..80]]; // G. C. Greubel, Jan 22 2025
(Python)
def A029578(n): return (n + (n-2)*(n%2))//2
print([A029578(n) for n in range(81)]) # G. C. Greubel, Jan 22 2025

Crossrefs

Cf. A065423 (at least one even summand).

Cf. A001477, A005843, A009531, A071037, A211538 (partial sums), A265705.

Sequence in context: A321015 A365195 A365433 * A054345 A368582 A352956

Adjacent sequences: A029575 A029576 A029577 * A029579 A029580 A029581

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Explicated definition by Reinhard Zumkeller, Nov 27 2012

Title simplified by Sean A. Irvine, Feb 29 2020

Status

approved