

A106737


a(n) = Sum_{k=0..n} ({binomial(n+k,nk)*binomial(n,k)} mod 2).


12



1, 2, 2, 3, 2, 4, 3, 4, 2, 4, 4, 6, 3, 6, 4, 5, 2, 4, 4, 6, 4, 8, 6, 8, 3, 6, 6, 9, 4, 8, 5, 6, 2, 4, 4, 6, 4, 8, 6, 8, 4, 8, 8, 12, 6, 12, 8, 10, 3, 6, 6, 9, 6, 12, 9, 12, 4, 8, 8, 12, 5, 10, 6, 7, 2, 4, 4, 6, 4, 8, 6, 8, 4, 8, 8, 12, 6, 12, 8, 10, 4, 8, 8, 12, 8, 16, 12, 16, 6, 12, 12, 18, 8, 16, 10, 12
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OFFSET

0,2


COMMENTS

The formula (the recurrence, if confirmed to be equal to sum binomial formula) implies that this is the run length transform of the sequence 1,2,3,4,5,...  N. J. A. Sloane, Feb 05 2015. Note: That sequence should be considered as a successor function a(n) = n+1, starting from offset 0. See also A020725.  Antti Karttunen, Oct 15 2016
The recurrence formula is correct. See paper in links.  Chai Wah Wu, Oct 16 2016


LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.


FORMULA

a(0)=1, a(2n) = a(n), a(4n+1) = 2*a(2n), a(4n+3) = 2*a(2n+1)  a(n).
From Antti Karttunen, Oct 15 2016: (Start)
a(n) = A000005(A005940(1+n)). [Follows from the Run Length Transforminterpretation.]
For n > 1, a(n^2) is always even. [Based on RLTinterpretation. n^2 = 1 modulo 4 for all odd n and ((2^k)*n)^2 = 2^(2k) * (n^2), thus the last 1bit is always alone and contributes 2 to the product, making it even.]
(End)


MATHEMATICA

Table[Sum[Mod[#, 2] &[Binomial[n + k, n  k] Binomial[n, k]], {k, 0, n}], {n, 0, 95}] (* Michael De Vlieger, Oct 17 2016 *)


PROG

(PARI) a(n) = sum(k=0, n, (binomial(n+k, nk)*binomial(n, k)) % 2); \\ Michel Marcus, Dec 08 2013
(Python)
def A106737(n):
return sum(int(not (~(n+k) & (nk))  (~n & k)) for k in range(n+1)) # Chai Wah Wu, Feb 09 2016
(Scheme, two mathematically equal implementations, based on RLTinterpretation)
;; The first one implements the given recurrence and uses memoizationmacro definec:
(definec (A106737 n) (cond ((zero? n) 1) ((even? n) (A106737 (/ n 2))) ((= 1 (modulo n 4)) (* 2 (A106737 (/ ( n 1) 2)))) (else ( (* 2 (A106737 (/ ( n 1) 2))) (A106737 (/ ( n 3) 4))))))
;; This one applies the Run Length Transform explicitly to r > r+1 function:
(define (A106737 n) (foldleft (lambda (a r) (* a (+ 1 r))) 1 (bisect (reverse (binexp>runcount1list n)) ( 1 (modulo n 2))))) ;; See A227349 for the required other functions.
;; Antti Karttunen, Oct 15 2016


CROSSREFS

Row sums of triangle in A253084.
Cf. A000005, A005940, A020725, A227349, A277335 (positions of odd terms).
Cf. also A153013.
Sequence in context: A066241 A060025 A067399 * A280338 A062821 A296080
Adjacent sequences: A106734 A106735 A106736 * A106738 A106739 A106740


KEYWORD

nonn


AUTHOR

Benoit Cloitre, May 15 2005


STATUS

approved



