%I #34 Sep 08 2022 08:46:04

%S -17,13,-7,73,103,233,313,493,623,853,1033,1313,1543,1873,2153,2533,

%T 2863,3293,3673,4153,4583,5113,5593,6173,6703,7333,7913,8593,9223,

%U 9953,10633,11413,12143,12973,13753,14633,15463,16393,17273,18253,19183,20213,21193

%N 50k^2-40k-17 interleaved with 50k^2+10k+13 for k=>0.

%C The sequence (the fourth in the family) is present as a family of interleaved sequences (five in total) which are separated or factored out to give individual sequences. The first sequence is the parent having the formulas: 50*n^2-100*n+25 and 50*n^2-50*n+25 whose entries are all divisible by 25 and is identical to A178218. The fourth sequence has the formulas 50*n^2-40*n-17 and 50*n^2+10*n+13 and is part of a group where each of the sequences are new, except for the parent (in the factored form).

%H Vincenzo Librandi, <a href="/A217893/b217893.txt">Table of n, a(n) for n = 0..1000</a>

%H Eddie Gutierrez <a href="http://oddwheel.com/square_sequencesVIII.html">New Interleaved Sequences Part H</a> or Oddwheel.com, Section B1 Line 28 (square_sequencesVIII.html), Part H.

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

%F G.f.: (-17+47*x-33*x^2+53*x^3)/((1+x)*(1-x)^3).

%F a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).

%F a(n) = 1+(10*n*(5*n-8)-75*(-1)^n+3)/4. [_Bruno Berselli_, Oct 15 2012]

%t Flatten[Table[{50 n^2 - 40 n - 17, 50 n^2 + 10 n + 13}, {n, 0, 23}]] (* _Bruno Berselli_, Oct 23 2012 *)

%t CoefficientList[Series[(-17 + 47*x - 33*x^2 + 53*x^3)/((1+x)*(1-x)^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 15 2012 *)

%o (Magma) &cat[[50*k^2-40*k-17,50*k^2+10*k+13]: k in [0..23]]; // _Bruno Berselli_, Oct 23 2012

%o (PARI) vector(48, n, k=(n-1)\2; if(n%2, 50*k^2-40*k-17, 50*k^2+10*k+13)) \\ _Bruno Berselli_, Oct 23 2012

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

%K sign,easy

%O 0,1

%A _Eddie Gutierrez_, Oct 14 2012

%E Definition rewritten by _Bruno Berselli_, Nov 09 2012