A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

3, 3, 2, 6, 3, 11, 6, 18, 11, 27, 18, 38, 27, 51, 38, 66, 51, 83, 66, 102, 83, 123, 102, 146, 123, 171, 146, 198, 171, 227, 198, 258, 227, 291, 258, 326, 291, 363, 326, 402, 363, 443, 402, 486, 443, 531, 486, 578, 531, 627, 578, 678, 627, 731, 678, 786, 731

Offset: 0

Author

Paul Curtz, Jan 22 2016

Keywords

• nonn
• easy

Comments

Trisections:

3, 6, 6, 27, 27, 66, 66, ... = 3*(1, 2, 2, 9, 9, 22, 22, ... ). See A056105.

3, 3, 18, 18, 51, 51, 102, ... = 3*(1, 1, 6, 6, 17, 17, ... ). See A056109.

2, 11, 11, 38, 38, 83, 83, ... (== 2 (mod 9)).

The trisections also have the signature (1,2,-2,-1,1). The corresponding main sequence is 0, 0, 0, 0, 1, 1, 3, 3, ... = A161680(n) with each term duplicated.

Examples

			a(0) = (2+13)/5, a(1) = (13+2)/5, a(2) = (5+5)/5, a(3) = (29+1)/5, ... (using first formula).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000290, A007395, A010701, A022998, A056105, A056109, A059100, A161680, A174474, A193356, A255844, A261327.

Programs

Magma

&cat [[(n-1)^2+2, (n+1)^2+2]: n in [0..50]]; // Vincenzo Librandi, Jan 23 2016

Mathematica

Flatten[Table[{n^2 - 2 n + 3, n^2 + 2 n + 3}, {n, 0, 30}]] (* Vincenzo Librandi, Jan 23 2016 *)
CoefficientList[Series[(3 - 7 x^2 + 4 x^3 + 2 x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 56}], x] (* Michael De Vlieger, Jan 24 2016 *)

PARI

Vec((3-7*x^2+4*x^3+2*x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 22 2016

Formula

a(n) = (A261327(n+2) + A261327(n-3))/5.

a(n+1) = a(n) + (-1)^n * A022998(n), a(0)=3.

a(n+3) = a(n) + 3*A193356(n), a(0)=a(1)=3, a(2)=2.

a(n) = 3 + A174474(n).

a(2n) + a(2n+1) = A255844(n).

From Colin Barker, Jan 22 2016: (Start)

a(n) = (2*n^2 - 6*(-1)^n*n - 2*n + 3*(-1)^n + 21)/8.

a(n) = (n^2 - 4*n + 12)/4 for n even.

a(n) = (n^2 + 2*n + 9)/4 for n odd.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.

G.f.: (3 - 7*x^2 + 4*x^3 + 2*x^4) / ((1-x)^3*(1+x)^2).

(End)

Extensions

More terms from Colin Barker, Jan 22 2016