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A260699 a(2n+6) = a(2n) + 12*n + 20, a(2n+1) = (n+1)*(2*n+1), with a(0)=0, a(2)=2, a(4)=9.

%I A260699 #51 Sep 08 2022 08:46:13
%S A260699 0,1,2,6,9,15,20,28,34,45,53,66,76,91,102,120,133,153,168,190,206,231,
%T A260699 249,276,296,325,346,378,401,435,460,496,522,561,589,630,660,703,734,
%U A260699 780,813,861,896,946,982,1035,1073
%N A260699 a(2n+6) = a(2n) + 12*n + 20, a(2n+1) = (n+1)*(2*n+1), with a(0)=0, a(2)=2, a(4)=9.
%C A260699 Sequence extended to left:
%C A260699 ..., 36, 29, 21, 16, 10, 6, 3, 1, 0, 0, 1, 2, 6, 9, 15, 20, 28, 34, ...,
%C A260699 where 0, 1, 3, 6, 10, 16, 21, 29, 36, 46, ... is A260708.
%C A260699 After 2, if a(n) is prime then n == 4 (mod 6).
%C A260699 a(n) is a square for n = 0, 1, 4, 49, 52, 192, 1681, 4948, 57121, 60388, 221952, 1940449, 5710372, ...
%H A260699 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,1,-1,-1,1).
%F A260699 G.f.: x*(1 + x + 3*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6)/((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)).
%F A260699 a(n) = a(n-1) +  a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9).
%F A260699 a(2*k+1) = A000217(2*k+1) by definition; for even indices:
%F A260699 a(6*k)   = 2*k*(9*k + 1),
%F A260699 a(6*k+2) = 2*(9*k^2 + 7*k + 1),
%F A260699 a(6*k+4) = 18*k^2 + 26*k + 9.
%F A260699 a(n) = n*(n + 1)/2 - (1 + (-1)^n)*floor(n/6 + 2/3)/2. [_Bruno Berselli_, Nov 18 2015]
%e A260699 a(0) = 0,
%e A260699 a(1) = 1*1 = 1,
%e A260699 a(2) = 2,
%e A260699 a(3) = 2*3 = 6,
%e A260699 a(4) = 9,
%e A260699 a(5) = 3*5 = 15,
%e A260699 a(6) = a(0) + 12*0 + 20 = 20, etc.
%t A260699 LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 1, 2, 6, 9, 15, 20, 28, 34}, 50] (* _Bruno Berselli_, Nov 18 2015 *)
%o A260699 (Magma) [n*(n+1)/2-(1+(-1)^n)*Floor(n/6+2/3)/2: n in [0..50]]; // _Bruno Berselli_, Nov 18 2015
%o A260699 (Sage) [n*(n+1)/2-(1+(-1)^n)*floor(n/6+2/3)/2 for n in (0..50)] # _Bruno Berselli_, Nov 18 2015
%Y A260699 Cf. A000217, A264041, A260708.
%K A260699 nonn,easy
%O A260699 0,3
%A A260699 _Paul Curtz_, Nov 16 2015
%E A260699 Edited by _Bruno Berselli_, Nov 17 2015