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A106345
Diagonal sums of number triangle A106344.
3
1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 1, 2, 0, 3, 2, 3, 1, 3, 2, 2, 1, 2, 1, 3, 1, 2, 2, 3, 0, 5, 3, 5, 2, 4, 3, 4, 1, 5, 3, 4, 2, 3, 2, 3, 1, 3, 2, 4, 1, 4, 3, 3, 1, 4, 2, 5, 2, 3, 3, 5, 0, 8, 5, 8, 3, 7, 5, 6, 2, 7, 4, 7, 3, 5, 4, 6, 1, 8, 5, 7, 3, 6, 4, 5, 2, 5, 3, 5, 2, 4, 3, 4, 1, 5, 3, 6, 2, 5, 4, 5, 1, 7
OFFSET
0,7
COMMENTS
This is a "bow" sequence, a companion to A281185. - N. J. A. Sloane, Apr 26 2017
Number of ways of writing n=sum_i c_i*2^i with c_i in {0,2,3} [Anders]. - R. J. Mathar, Mar 01 2023
LINKS
K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 7.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
Melissa Dennison, On Properties of the General Bow Sequence, J. Int. Seq., Vol. 22 (2019), Article 19.2.7.
FORMULA
a(n) = Sum_{k=0..floor(n/2)} (binomial(k, n-2k) mod 2).
G.f. A(x) satisfies: A(x) = (1 + x^2 + x^3) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019
MAPLE
f:=proc(n) option remember;
if n=0 then 0
elif n=1 then 0
elif n=2 then 1
else
if n mod 2 = 0 then f(n/2)+f(1+n/2) else f((n-1)/2) fi;
fi;
end;
[seq(f(n), n=2..150)]; # (Note that with this recurrence, we list the values starting at n = 2. N. J. A. Sloane, Apr 26 2017
MATHEMATICA
Table[Sum[Mod[Binomial[k, n-2k], 2], {k, 0, n/2}], {n, 0, 102}] (* Jean-François Alcover, Nov 16 2019 *)
PROG
(Python)
a = [0]*(104*2)
a[1]=1
for n in range(1, 104):
a[2*n ]=a[n-1]
a[2*n+1]=a[n]+a[n+1]
print(str(a[n]), end=', ')
# Alex Ratushnyak, Jul 04 2012
CROSSREFS
Sequence in context: A327489 A257886 A144477 * A319395 A374462 A194636
KEYWORD
nonn,look,easy
AUTHOR
Paul Barry, Apr 29 2005
STATUS
approved