A106345 Diagonal sums of number triangle A106344.

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

Author

Paul Barry, Apr 29 2005

Keywords

• nonn
• look
• easy

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

• Rémy Sigrist, Table of n, a(n) for n = 0..25000
• 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.

Crossrefs

Cf. A002487, A281185.

Programs

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 *)

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

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