A216865 - OEIS

%I #37 Sep 08 2022 08:46:03

%S 8,8,-8,8,8,40,56,104,136,200,248,328,392,488,568,680,776,904,1016,

%T 1160,1288,1448,1592,1768,1928,2120,2296,2504,2696,2920,3128,3368,

%U 3592,3848,4088,4360,4616,4904,5176,5480,5768,6088,6392,6728,7048,7400,7736

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

%C 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.

%H Eddie Gutierrez <a href="http://www.oddwheel.com/square_sequencesIII.html">New Interleaved Sequences Part C</a> on oddwheel.com, Section B1 Line No. 23 (square_sequencesIII.html) Part C

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

%F G.f.: 8*(1-x-3*x^2+5*x^3)/((1+x)*(1-x)^3). [_Bruno Berselli_, Sep 30 2012]

%F a(n) = 2*(2*n*(n-4)-3*(-1)^n+7). [_Bruno Berselli_, Sep 30 2012]

%F a(n) = 8*A178218(n-3) with A178218(-3)=1, A178218(-2)=1, A178218(-1)=-1, A178218(0)=1. [_Bruno Berselli_, Oct 01 2012]

%t Flatten[Table[{16 n^2 - 32 n + 8, 16 n^2 - 16 n + 8}, {n, 0, 23}]] (* _Bruno Berselli_, Sep 30 2012 *)

%o (Magma) &cat[[16*k^2-32*k+8, 16*k^2-16*k+8]: k in [0..23]]; // _Bruno Berselli_, Oct 01 2012

%o (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

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

%K sign,easy

%O 0,1

%A _Eddie Gutierrez_, Sep 18 2012

%E Definition rewritten by _Bruno Berselli_, Oct 25 2012