A006921 Diagonals of Pascal's triangle mod 2 interpreted as binary numbers. (Formerly M2252)

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

0,3

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.

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

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

Prog

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

Crossrefs

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

Cf. A168081, A260022.

Cf. A257971 (first differences).

Sequence in context: A263018 A215622 A195820 * A292204 A292203 A295642

Adjacent sequences: A006918 A006919 A006920 * A006922 A006923 A006924

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved