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

0,7

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

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.

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

Cf. A002487, A281185.

Sequence in context: A327489 A257886 A144477 * A319395 A374462 A194636

Adjacent sequences: A106342 A106343 A106344 * A106346 A106347 A106348

Keyword

nonn,look,easy

Author

Paul Barry, Apr 29 2005

Status

approved