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A104435 Number of ways to split 1, 2, 3, ..., 2n into 2 arithmetic progressions each with n terms.

%I A104435 #31 Feb 13 2025 08:26:49
%S A104435 1,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A104435 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A104435 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A104435 Number of ways to split 1, 2, 3, ..., 2n into 2 arithmetic progressions each with n terms.
%C A104435 The common difference in an arithmetic progression must be a positive integer. - _David A. Corneth_, Apr 14 2024
%H A104435 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A104435 a(1) = 1, a(2) = 3, a(n) = 2 for n >= 3. Proof of the latter: if the common difference in an arithmetic progression, starting with a number at least 1, is at least 3 then the largest term in that arithmetic progression is at least 1 + 3*(n-1) = 3*n - 2. But 3*n - 2 > 2*n for n > 2. - _David A. Corneth_, Apr 14 2024
%F A104435 G.f.: x*(1 + 2*x - x^2)/(1 - x). - _Stefano Spezia_, Apr 14 2024
%e A104435 From _R. J. Mathar_, Apr 14 2024: (Start)
%e A104435 a(2)=3 offers 3 ways of splitting (1,2,3,4): {(1,2),(3,4)}, {(1,3),(2,4)}, {(1,4),(2,3)}.
%e A104435 a(n)=2 for n>=3 because there are at least the two ways of splitting (1,2,..,2n) into the even and odd numbers. (End)
%o A104435 (PARI) a(n) = if(n <= 2, [1,3][n], 2) \\ _David A. Corneth_, Apr 14 2024
%Y A104435 Cf. A104429, A104430, A104431, A104432, A104433, A104434, A104436, A104437, A104438, A104439, A104440, A104441, A104442, A104443.
%K A104435 nonn,easy
%O A104435 1,2
%A A104435 _Jonas Wallgren_, Mar 17 2005
%E A104435 More terms from _David A. Corneth_, Apr 14 2024