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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.

%I A073424 #29 Nov 11 2024 03:44:53
%S A073424 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,
%T A073424 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,
%U A073424 0,0,0,0,0,0,0,0,0,0,1
%N 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.
%C A073424 Parity of the sums of two powers of any even number.
%H A073424 Michael De Vlieger, <a href="/A073424/b073424.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150).
%H A073424 Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%F A073424 a(n) = parity [ (2k)^n + (2k)^m, n = 0, 1, 2, 3 ..., m = 0, 1, 2, 3, ... n ]
%F A073424 T(n,0) = 1- 0^n, T(n,k) = 0 for k>0. - _Philippe Deléham_, Feb 11 2012
%F A073424 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
%F A073424 For n>0, a(n) = 1 if and only if n is in A000217. - _Chai Wah Wu_, Nov 09 2024
%e A073424 a(3) = 1 because (2k)^2 + (2k)^0 = 4k^2 + 1 is odd.
%e A073424 Triangle begins :
%e A073424   0
%e A073424   1, 0
%e A073424   1, 0, 0
%e A073424   1, 0, 0, 0
%e A073424   1, 0, 0, 0, 0
%e A073424   1, 0, 0, 0, 0, 0
%e A073424   1, 0, 0, 0, 0, 0, 0
%e A073424   1, 0, 0, 0, 0, 0, 0, 0
%e A073424   1, 0, 0, 0, 0, 0, 0, 0, 0
%e A073424   1, 0, 0, 0, 0, 0, 0, 0, 0, 0 - _Philippe Deléham_, Feb 11 2012
%p A073424 0, seq(op([1,0$n]), n=1..20); # _Robert Israel_, Mar 01 2016
%t A073424 Array[If[# == 1, {0}, PadRight[{1}, #]] &, 14] // Flatten (* or *)
%t A073424 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 *)
%o A073424 (Python)
%o A073424 from math import isqrt
%o A073424 def A073424(n): return int((k:=n<<1)==(m:=isqrt(k))*(m+1)) if n else 0 # _Chai Wah Wu_, Nov 09 2024
%Y A073424 Cf. A000035, A000217, A023531, A010054, A073423.
%K A073424 easy,nonn,tabl
%O A073424 0,1
%A A073424 _Jeremy Gardiner_, Jul 30 2002