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

0,5

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

Links

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

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

Example

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

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 *)

Prog

(PARI) a(n)=if(n<1, 0, if(n%2==0, a(n/2)+n/2, 2*a((n-1)/2)))
(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
(Python)
def A048967(n): return n+1-(1<<n.bit_count()) # Chai Wah Wu, May 03 2023

Crossrefs

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

Cf. A249304.

Sequence in context: A089306 A244996 A086099 * A324182 A166592 A103497

Adjacent sequences: A048964 A048965 A048966 * A048968 A048969 A048970

Keyword

easy,nonn

Author

Brian Galebach

Status

approved