A048967 Number of even entries in row n of Pascal's triangle (A007318).

0, 0, 1, 0, 3, 2, 3, 0, 7, 6, 7, 4, 9, 6, 7, 0, 15, 14, 15, 12, 17, 14, 15, 8, 21, 18, 19, 12, 21, 14, 15, 0, 31, 30, 31, 28, 33, 30, 31, 24, 37, 34, 35, 28, 37, 30, 31, 16, 45, 42, 43, 36, 45, 38, 39, 24, 49, 42, 43, 28, 45, 30, 31, 0, 63, 62, 63, 60, 65, 62, 63, 56, 69, 66, 67

Offset: 0

Author

Brian Galebach

Keywords

• easy
• nonn

Comments

In rows 2^k - 1 all entries are odd.

a(n) = 0 (all the entries in the row are odd) iff n = 2^m - 1 for some m >= 0 and then n belongs to sequence A000225. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

Also number of zeros in n-th row of Sierpiński's triangle (cf. A047999): a(n) = A023416(A001317(n)). - Reinhard Zumkeller, Nov 24 2012

Examples

			Row 4 is 1 4 6 4 1 with 3 even entries so a(4)=3.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

Crossrefs

Cf. A007318, A001316, A000120, A000225, A038573.

Cf. A249304.

Programs

Haskell

import Data.List (transpose)
a048967 n = a048967_list !! n
a048967_list = 0 : xs where
   xs = 0 : concat (transpose [zipWith (+) [1..] xs, map (* 2) xs])
-- Reinhard Zumkeller, Nov 14 2014, Nov 24 2012

Mathematica

Table[n + 1 - Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ], {n, 0, 100} ]
a[n_] := n + 1 - 2^DigitCount[n, 2, 1]; Array[a, 100, 0] (* Amiram Eldar, Jul 27 2023 *)

PARI

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n/2,2*a((n-1)/2)))

Python

def A048967(n): return n+1-(1<Chai Wah Wu, May 03 2023

Formula

a(n) = n+1 - A001316(n) = n+1 - 2^A000120(n) = n+1 - Sum_{k=0..n} (C(n, k) mod 2) = Sum_{k=0..n} ((1 - C(n, k)) mod 2).

a(2n) = a(n) + n, a(2n+1) = 2a(n). - Ralf Stephan, Oct 07 2003

a(n) = row sums in A219463 = A000120(A219843(n)). - Reinhard Zumkeller, Nov 30 2012

A249304(n+1) = a(n+1) + a(n). - Reinhard Zumkeller, Nov 14 2014

G.f.: 1/(1 - x)^2 - Product_{k>=0} (1 + 2*x^(2^k)). - Ilya Gutkovskiy, Jul 19 2019