A260699 - OEIS

%I #52 Sep 22 2025 16:01:22

%S 0,1,2,6,9,15,20,28,34,45,53,66,76,91,102,120,133,153,168,190,206,231,

%T 249,276,296,325,346,378,401,435,460,496,522,561,589,630,660,703,734,

%U 780,813,861,896,946,982,1035,1073

%N 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 Sequence extended to left:

%C ..., 36, 29, 21, 16, 10, 6, 3, 1, 0, 0, 1, 2, 6, 9, 15, 20, 28, 34, ...,

%C where 0, 1, 3, 6, 10, 16, 21, 29, 36, 46, ... is A260708.

%C After 2, if a(n) is prime then n == 4 (mod 6).

%C a(n) is a square for n = 0, 1, 4, 49, 52, 192, 1681, 4948, 57121, 60388, 221952, 1940449, 5710372, ...

%H <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 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 a(n) = a(n-1) +  a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9).

%F a(2*k+1) = A000217(2*k+1) by definition; for even indices:

%F a(6*k)   = 2*k*(9*k + 1),

%F a(6*k+2) = 2*(9*k^2 + 7*k + 1),

%F a(6*k+4) = 18*k^2 + 26*k + 9.

%F a(n) = n*(n + 1)/2 - (1 + (-1)^n)*floor(n/6 + 2/3)/2. [_Bruno Berselli_, Nov 18 2015]

%e a(0) = 0,

%e a(1) = 1*1 = 1,

%e a(2) = 2,

%e a(3) = 2*3 = 6,

%e a(4) = 9,

%e a(5) = 3*5 = 15,

%e a(6) = a(0) + 12*0 + 20 = 20, etc.

%t 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 (Magma) [n*(n+1)/2-(1+(-1)^n)*Floor(n/6+2/3)/2: n in [0..50]]; // _Bruno Berselli_, Nov 18 2015

%o (SageMath) [n*(n+1)/2-(1+(-1)^n)*floor(n/6+2/3)/2 for n in (0..50)] # _Bruno Berselli_, Nov 18 2015

%Y Cf. A000217, A264041, A260708.

%K nonn,easy

%O 0,3

%A _Paul Curtz_, Nov 16 2015

%E Edited by _Bruno Berselli_, Nov 17 2015