0,1

First 20 terms are primes with periodic second order differences (4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4) which continue onwards.

The number of prime and nonprime terms is equal at a(1809): 905, a(1811): 906, a(1817): 909, a(1819): 910, a(1821): 911, a(1823): 912, a(1825): 913, a(1827): 914, a(1829): 915, and at a(1837): 919 (i.e. 919 prime and 919 nonprime terms among the total 1838 terms).

Single longer blocks containing only primes include 20, 14, 12, and 11 terms (in range n=0...10^8) while the longest block of nonprime terms in this range has length 78.

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

G.f.: (-149x^4 + 296x^3 - 300x^2 + 296x - 149)/((x-1)^3*(x^2+1)).

a(n) = 2*floor(3*n*(n + 1)/4) + 149.

a(n) = A007310(n(n + 1)/2 + 50).

a(n) = (3*(n^2+n+99)+(-1)^binomial(n+1,2))/2. (Suggested by Michel Marcus)

a(n) - a(-n-1) = 0. - Altug Alkan, Feb 04 2016

a(n) = ((1/4+i/4)*((297-297*i)-i*(-i)^n+i^n)+(3*n)/2+(3*n^2)/2) where i is the imaginary unit. - Colin Barker, Feb 09 2016

Table[2*Floor[3*n*(n+1)/4] + 149, {n, 0, 10000}] (* Efficient. *)

Table[(3*(n^2+n+99)+Cos[Pi*n/2]-Sin[Pi*n/2])/2, {n, 0, 1000}](* or *)

LinearRecurrence[{3, -4, 4, -3, 1}, {149, 151, 157, 167, 179}, 1000]

CoefficientList[Series[(-149 x^4 + 296 x^3 - 300 x^2 + 296 x - 149) / ((x - 1)^3 (x^2 + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 05 2016 *)

(PARI) Vec((-149*x^4 + 296*x^3 - 300*x^2 + 296*x - 149)/((x-1)^3*(x^2+1)) + O(x^60)) \\ Michel Marcus, Feb 04 2016

(PARI) a(n) = ((1/4+I/4)*((297-297*I)-I*(-I)^n+I^n)+(3*n)/2+(3*n^2)/2) \\ Colin Barker, Feb 09 2016

Cf. A007310 (subsequence), A087960.

Sequence in context: A263554 A248412 A182874 * A190654 A308895 A100723

Adjacent sequences: A268425 A268426 A268427 * A268429 A268430 A268431

nonn,easy

Mikk Heidemaa, Feb 04 2016

approved