A216852 18k^2-36k+9 interleaved with 18k^2-18k+9 for k>=0.

9, 9, -9, 9, 9, 45, 63, 117, 153, 225, 279, 369, 441, 549, 639, 765, 873, 1017, 1143, 1305, 1449, 1629, 1791, 1989, 2169, 2385, 2583, 2817, 3033, 3285, 3519, 3789, 4041, 4329, 4599, 4905, 5193, 5517, 5823, 6165, 6489, 6849, 7191, 7569, 7929, 8325, 8703

Offset: 0

Author

Eddie Gutierrez, Sep 17 2012

Keywords

• sign
• easy

Comments

The sequence is present as a family of single interleaved sequence of which there are many which are separated or factored out to give individual sequences. The larger sequence produces two smaller interleaved sequences where one of them has the formulas above and the other interleaved sequence has the formulas (18n^2-24n+1) and (18n^2-6n+5). The latter interleaved sequence is A214493. There are three sequences in this family.

Links

• Eddie Gutierrez New Interleaved Sequences Part B on oddwheel.com, Section B1 Line No. 22 (square_sequencesII.html) Part B
• Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Crossrefs

Cf. A178218, A214345, A214393, A214405, A216844, A216865, A216875, A216876.

Programs

Magma

&cat[[18*k^2-36*k+9, 18*k^2-18*k+9]: k in [0..23]]; // Bruno Berselli, Oct 01 2012

Mathematica

Flatten[Table[{18 n^2 - 36 n + 9, 18 n^2 - 18 n + 9}, {n, 0, 23}]] (* Bruno Berselli, Oct 01 2012 *)
Flatten[Table[18n^2+9-{36n,18n},{n,0,50}]] (* or *) LinearRecurrence[ {2,0,-2,1},{9,9,-9,9},100] (* Harvey P. Dale, Apr 26 2014 *)

PARI

vector(47, n, k=(n-1)\2; if(n%2, 18*k^2-36*k+9, 18*k^2-18*k+9)) \\ Bruno Berselli, Oct 01 2012

Formula

From Bruno Berselli, Oct 01 2012: (Start)

G.f.: 9*(1-x-3*x^2+5*x^3)/((1+x)*(1-x)^3).

a(n) = (9/4)*(2*n*(n-4)-3*(-1)^n+7).

a(n) = 9*A178218(n-3) with A178218(-3)=1, A178218(-2)=1, A178218(-1)=-1, A178218(0)=1. (End)

a(0)=9, a(1)=9, a(2)=-9, a(3)=9, a(n)=2*a(n-1)-2*a(n-3)+a(n-4). - Harvey P. Dale, Apr 26 2014

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2012