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A048967 Number of even entries in row n of Pascal's triangle (A007318). 17
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

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} ]

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

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

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Last modified February 26 00:08 EST 2020. Contains 332270 sequences. (Running on oeis4.)