A106345 -id:A106345 - cp's OEIS frontend

A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1

Offset: 0

Paul Barry, Apr 29 2005

A skew version of Sierpinski’s triangle A047999. - Johannes W. Meijer, Jun 05 2011
Row sums are A002487(n+1). Diagonal sums are A106345. Inverse is A106346.
Triangle formed by reading T triangle mod 2 with T := A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167. - Philippe Deléham, Dec 18 2008

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 0, 1;

G. C. Greubel, Rows n = 0..50, flattened
Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (1.6) p. 2.

Cf. A047999, A002487.
Cf. A106345 (diagonal sums), A106346 (inverse).
Cf. A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167.

Flat(List([0..15], n-> List([0..n], k-> (Binomial(k,n-k) mod 2) ))); # G. C. Greubel, Feb 07 2020

[ Binomial(k,n-k) mod 2: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 07 2020

seq(seq(`mod`(binomial(k, n-k), 2), k = 0..n), n = 0..15); # G. C. Greubel, Feb 07 2020

Table[Mod[Binomial[k, n-k], 2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 18 2017 *)

T(n,k) = binomial(k,n-k)%2;
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 07 2020

[[ mod(binomial(k,n-k), 2) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Feb 07 2020

A281185 a(0)=0, a(1)=1, a(2)=0; thereafter, a(2n) = a(n) + a(n+1) for n >= 2, a(2n+1) = a(n) for n >= 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 2, 2, 1, 2, 1, 3, 1, 2, 2, 3, 0, 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, 7, 3, 6, 4, 5, 2, 5, 3, 5, 2, 4, 3, 4, 1, 5, 3, 6, 2, 5, 4, 5, 1, 7, 4, 6, 3, 4, 3, 5, 1, 6, 4, 7, 2, 7, 5, 5, 2, 6, 3, 8, 3, 5, 5, 8, 0, 13, 8, 12, 5

Offset: 0

Melissa Dennison, Apr 12 2017

A "bow" sequence. The bow sequences are a family of recursive sequences defined to have the flipped recursion from the Stern sequence A002487 (called bow for the opposite end of the boat from the stern). The bow sequences require two initial conditions: a(1)=alpha, a(2)=beta. We also define a(0)=0, although it does not enter into the recursion.

The bow sequences then follow the recursion a(2n) = a(n) + a(n+1) for n at least 2, and a(2n+1) = a(n). This particular bow sequence has initial conditions a(1)=0, a(2)=1 and (along with the sequence A106345 with initial conditions a(1)=1, a(2)=0) is of particular importance when studying the general bow sequences.

			a(3) = a(1) = 1, a(4) = a(2) + a(3) = 0 + 1 = 1, a(5) = a(2) = 0.

Rémy Sigrist, Table of n, a(n) for n = 0..25000
M. Dennison, A Sequence Related to the Stern Sequence, Ph.D. dissertation, University of Illinois at Urbana-Champaign, 2010.
Melissa Dennison, On Properties of the General Bow Sequence, J. Int. Seq., Vol. 22 (2019), Article 19.2.7.

Cf. A002487, A106345.

f:=proc(n) option remember;
if n=0 then 0
elif n=1 then 1
elif n=2 then 0
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=0..150)]; # N. J. A. Sloane, Apr 26 2017

b[0]=0; b[1]=1; b[2]=0; b[n_?EvenQ]:=b[n]=b[n/2]+b[n/2+1]; b[n_?OddQ]:=b[n]=b[(n-1)/2]

Edited by N. J. A. Sloane, Apr 26 2017

cp's OEIS Frontend

A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

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