A216865 16k^2-32k+8 interleaved with 16k^2-16k+8 for k>=0.

8, 8, -8, 8, 8, 40, 56, 104, 136, 200, 248, 328, 392, 488, 568, 680, 776, 904, 1016, 1160, 1288, 1448, 1592, 1768, 1928, 2120, 2296, 2504, 2696, 2920, 3128, 3368, 3592, 3848, 4088, 4360, 4616, 4904, 5176, 5480, 5768, 6088, 6392, 6728, 7048, 7400, 7736

Offset: 0

Author

Eddie Gutierrez, Sep 18 2012

Keywords

• sign
• easy

Comments

The sequence (the first 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, A216844, A216875, A216876.

Programs

Magma

&cat[[16*k^2-32*k+8, 16*k^2-16*k+8]: k in [0..23]]; // Bruno Berselli, Oct 01 2012

Mathematica

Flatten[Table[{16 n^2 - 32 n + 8, 16 n^2 - 16 n + 8}, {n, 0, 23}]] (* Bruno Berselli, Sep 30 2012 *)

PARI

vector(47, n, k=(n-1)\2; if(n%2, 16*k^2-32*k+8, 16*k^2-16*k+8)) \\ Bruno Berselli, Oct 01 2012

Formula

G.f.: 8*(1-x-3*x^2+5*x^3)/((1+x)*(1-x)^3). [Bruno Berselli, Sep 30 2012]

a(n) = 2*(2*n*(n-4)-3*(-1)^n+7). [Bruno Berselli, Sep 30 2012]

a(n) = 8*A178218(n-3) with A178218(-3)=1, A178218(-2)=1, A178218(-1)=-1, A178218(0)=1. [Bruno Berselli, Oct 01 2012]

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2012