%I #34 Sep 08 2022 08:46:03

%S 9,9,-9,9,9,45,63,117,153,225,279,369,441,549,639,765,873,1017,1143,

%T 1305,1449,1629,1791,1989,2169,2385,2583,2817,3033,3285,3519,3789,

%U 4041,4329,4599,4905,5193,5517,5823,6165,6489,6849,7191,7569,7929,8325,8703

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

%C 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.

%H Eddie Gutierrez <a href="http://www.oddwheel.com/square_sequencesII.html">New Interleaved Sequences Part B</a> on oddwheel.com, Section B1 Line No. 22 (square_sequencesII.html) Part B

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, -2, 1).

%F From _Bruno Berselli_, Oct 01 2012: (Start)

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

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

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

%F 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

%t Flatten[Table[{18 n^2 - 36 n + 9, 18 n^2 - 18 n + 9}, {n, 0, 23}]] (* _Bruno Berselli_, Oct 01 2012 *)

%t 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 *)

%o (Magma) &cat[[18*k^2-36*k+9, 18*k^2-18*k+9]: k in [0..23]]; // _Bruno Berselli_, Oct 01 2012

%o (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

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

%K sign,easy

%O 0,1

%A _Eddie Gutierrez_, Sep 17 2012

%E Definition rewritten by _Bruno Berselli_, Oct 25 2012