A260708 - OEIS

%I #40 Nov 18 2015 11:49:10

%S 0,1,3,6,10,16,21,29,36,46,55,68,78,93,105,122,136,156,171,193,210,

%T 234,253,280,300,329,351,382,406,440,465,501,528,566,595,636,666,709,

%U 741,786,820,868,903,953,990,1042,1081,1136,1176,1233,1275,1334,1378

%N a(2n) = n*(2*n+1), a(2n+7) = a(2n+1) + 12*n + 28, with a(1)=1, a(3)=6, a(5)=16.

%C Conjecture: this sequence is 0 followed by A264041.

%C After 3, if a(n) is prime then n == 1 (mod 6).

%C a(n) is a square for n = 0, 1, 5, 8, 145, 288, 1777, 6533, 9800, 168097, 332928, 2051425, 7539845, ...

%H Colin Barker, <a href="/A260708/b260708.txt">Table of n, a(n) for n = 0..1000</a>

%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 From _Colin Barker_, Nov 17 2015: (Start)

%F G.f.: x*(1 + 2*x + 2*x^2 + 2*x^3 + 3*x^4 + 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) for n>8. (End)

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

%F a(6*k+1) = 18*k^2 + 10*k + 1,

%F a(6*k+3) = 2*(9*k^2 + 11*k + 3),

%F a(6*k+5) = 2*(k + 1)*(9*k + 8), that is A178574.

%F a(n) = A260699(n) + A008615(n).

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

%e a(0) = 0*1 = 0,

%e a(1) = 1,

%e a(2) = 1*3 = 3,

%e a(3) = 6,

%e a(4) = 2*5 = 10,

%e a(5) = 16,

%e a(6) = 3*7 = 21,

%e a(7) = a(1) +12*0 +28 = 29, etc.

%t LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 1, 3, 6, 10, 16, 21, 29, 36}, 50] (* _Bruno Berselli_, Nov 18 2015 *)

%o (PARI) concat(0, Vec(-x*(x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^3*(x+1)^2*(x^2-x+1)*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, Nov 18 2015

%o (Sage) [n*(n+1)/2+(1-(-1)^n)*floor(n/6+1/3)/2 for n in (0..60)] # _Bruno Berselli_, Nov 18 2015

%Y Cf. A000217, A008615, A014105, A260160, A260699, A264041.

%K nonn,easy

%O 0,3

%A _Paul Curtz_, Nov 17 2015

%E Edited by _Bruno Berselli_, Nov 18 2015