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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A065423 (at least one even summand).

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

Programs

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

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 *)

PARI

a(n)=if(n%2,n-1,n/2)

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

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

Extensions

Explicated definition by Reinhard Zumkeller, Nov 27 2012

Title simplified by Sean A. Irvine, Feb 29 2020