A006921 Diagonals of Pascal's triangle mod 2 interpreted as binary numbers.

1, 1, 3, 2, 7, 5, 13, 8, 29, 21, 55, 34, 115, 81, 209, 128, 465, 337, 883, 546, 1847, 1301, 3357, 2056, 7437, 5381, 14087, 8706, 29443, 20737, 53505, 32768, 119041, 86273, 226051, 139778, 472839, 333061, 859405, 526344, 1903901, 1377557, 3606327

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..1000
• B. R. Hodgson, Letter to N. J. A. Sloane, Oct. 1991
• B. R. Hodgson, On some number sequences related to the parity of binomial coefficients, Fib. Quart., 30 (1992), 35-47.

Crossrefs

Cf. A011973, A000079, A047999 (Sierpiński), A007318, A101624.

Cf. A168081, A260022.

Cf. A257971 (first differences).

Programs

Haskell

a006921 = sum . zipWith (*)
                a000079_list . map (flip mod 2) . reverse . a011973_row
-- Reinhard Zumkeller, Jul 14 2015

Maple

b2:=(n,k)->binomial(n,k) mod 2;
H:=n->add(b2(n-r,r)*2^( floor(n/2)-r ), r=0..floor(n/2));
[seq(H(n),n=0..30)]; # N. J. A. Sloane, Jul 14 2015

Python

def A006921(n): return sum(int(not r & ~(n-r))*2**(n//2-r) for r in range(n//2+1)) # Chai Wah Wu, Jun 20 2022

Formula

a(2*n) = A260022(n); a(2*n+1) = A168081(n+1). - Reinhard Zumkeller, Jul 14 2015

a(n) = Sum_{r=0..n/2} binomial(n-r,r){mod 2} * 2^(floor(n/2)-r). - _N. J. A. Sloane, Jul 14 2015