A073424 Triangle read by rows: T(m,n) = parity of 0^n + 0^m, n = 0,1,2,3 ..., m = 0,1,2,3, ... n.

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

Offset: 0

Author

Jeremy Gardiner, Jul 30 2002

Keywords

• easy
• nonn
• tabl

Comments

Parity of the sums of two powers of any even number.

Examples

			a(3) = 1 because (2k)^2 + (2k)^0 = 4k^2 + 1 is odd.
Triangle begins :
  0
  1, 0
  1, 0, 0
  1, 0, 0, 0
  1, 0, 0, 0, 0
  1, 0, 0, 0, 0, 0
  1, 0, 0, 0, 0, 0, 0
  1, 0, 0, 0, 0, 0, 0, 0
  1, 0, 0, 0, 0, 0, 0, 0, 0
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0 - _Philippe Deléham_, Feb 11 2012
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150).
• Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Crossrefs

Cf. A000035, A000217, A023531, A010054, A073423.

Programs

Maple

0, seq(op([1,0$n]), n=1..20); # Robert Israel, Mar 01 2016

Mathematica

Array[If[# == 1, {0}, PadRight[{1}, #]] &, 14] // Flatten (* or *)
Unprotect[Power]; Power[0, 0] = 1; Protect[Power]; Table[0^m + 0^n - 2 Boole[m == n == 0], {n, 0, 14}, {m, 0, n}] // Flatten (* Michael De Vlieger, Aug 22 2018 *)

Python

from math import isqrt
def A073424(n): return int((k:=n<<1)==(m:=isqrt(k))*(m+1)) if n else 0 # Chai Wah Wu, Nov 09 2024

Formula

a(n) = parity [ (2k)^n + (2k)^m, n = 0, 1, 2, 3 ..., m = 0, 1, 2, 3, ... n ]

T(n,0) = 1- 0^n, T(n,k) = 0 for k>0. - Philippe Deléham, Feb 11 2012

G.f.: Theta_2(0,sqrt(x))/(2*x^(1/8))-1, where Theta_2 is a Jacobi theta function. - Robert Israel, Mar 01 2016

For n>0, a(n) = 1 if and only if n is in A000217. - Chai Wah Wu, Nov 09 2024