A216871 16k^2-16k-4 interleaved with 16k^2+4 for k>=0.

-4, 4, -4, 20, 28, 68, 92, 148, 188, 260, 316, 404, 476, 580, 668, 788, 892, 1028, 1148, 1300, 1436, 1604, 1756, 1940, 2108, 2308, 2492, 2708, 2908, 3140, 3356, 3604, 3836, 4100, 4348, 4628, 4892, 5188, 5468, 5780, 6076, 6404, 6716, 7060, 7388, 7748, 8092

Offset: 0

Author

Eddie Gutierrez, Sep 18 2012

Keywords

• sign
• easy

Comments

The sequence (the third in the family) is present as a family of single interleaved sequence of which are separated or factored out of the larger sequence to give individual sequences. The larger sequence produces four smaller interleaved sequences where one of them has the formula above and a second interleaved sequences having the formulas (16n^2-24n+1) and (16n^2-6n+5). This interleaved sequence is A214393. The fourth interleaved sequence in the group has the formulas (16n^2-8n-7) and (16n^2+2n+5) and it is A214405. There are a total of four sequences in this family.

Links

• Eddie Gutierrez New Interleaved Sequences Part C on oddwheel.com, Section B1 Line No. 23 (square_sequencesIII.html) Part C
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A178218, A214345, A214393, A214405, A216876.

Programs

Magma

&cat[[16*k^2-16*k-4, 16*k^2+4]: k in [0..23]]; // Bruno Berselli, Sep 27 2012

Mathematica

Flatten[Table[{16 n^2 - 16 n - 4, 16 n^2 + 4}, {n, 0, 23}]] (* Bruno Berselli, Sep 26 2012 *)
LinearRecurrence[{2,0,-2,1},{-4,4,-4,20},50] (* Harvey P. Dale, Dec 09 2015 *)

PARI

vector(47, n, k=(n-1)\2; if(n%2, 16*k^2-16*k-4, 16*k^2+4)) \\ Bruno Berselli, Sep 28 2012

Formula

Contribution from Bruno Berselli, Sep 27 2012: (Start)

G.f.: -4*(1-3*x+3*x^2-5*x^3)/((1+x)*(1-x)^3).

a(n) = 2*(2*n*(n-2)-3*(-1)^n+1).

a(n) = 4*A214345(n-3) with A214345(-3)=-1, A214345(-2)=1, A214345(-1)=-1. (End)

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2012