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A182193 Sequence of row differences related to table A182355.

%I A182193 #41 Sep 08 2022 08:45:55
%S A182193 -1,1,19,125,743,4345,25339,147701,860879,5017585,29244643,170450285,
%T A182193 993457079,5790292201,33748296139,196699484645,1146448611743,
%U A182193 6681992185825,38945504503219,226991034833501,1323000704497799,7711013192153305,44943078448422043
%N A182193 Sequence of row differences related to table A182355.
%C A182193 Sequence of row differences in table A182355. If A182355(k + 1, 0) - A182355(k, 0) = -1, a(n) = A182355(k + 1, n) - A182355(k, n).
%C A182193 If p is a prime of the form 8r = +/- 3, a(p) = 5 mod p; if p is a prime of the form 8r = +/- 1, a(p) = 1 mod p.
%H A182193 Vincenzo Librandi, <a href="/A182193/b182193.txt">Table of n, a(n) for n = 0..200</a>
%H A182193 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F A182193 a(n) = 6*a(n-1) - a(n-2) + 12.
%F A182193 a(0)=-1, a(1)=1, a(2)=19, a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3). - _Harvey P. Dale_, Feb 09 2014
%F A182193 From _Colin Barker_, Mar 05 2016: (Start)
%F A182193 a(n) = -3 + (1/4)*( (4-sqrt(2))*(3+2*sqrt(2))^n + (4+sqrt(2))*(3-2*sqrt(2))^n ).
%F A182193 G.f.: -(1-8*x-5*x^2) / ((1-x)*(1-6*x+x^2)).
%F A182193 (End)
%F A182193 a(n) = A002203(2*n) - A000129(2*n) - 3. - _G. C. Greubel_, May 24 2021
%p A182193 Pell:= proc(n) option remember;
%p A182193     if n<2 then n
%p A182193   else 2*Pell(n-1) + Pell(n-2)
%p A182193     fi; end:
%p A182193 seq(Pell(2*n) + 2*Pell(2*n-1) - 3, n=0..40); # _G. C. Greubel_, May 24 2021
%t A182193 LinearRecurrence[{7,-7,1},{-1,1,19},30] (* _Harvey P. Dale_, Feb 09 2014 *)
%o A182193 (Magma) I:=[-1,1]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)+12: n in [1..30]]; // _Vincenzo Librandi_, Feb 10 2014
%o A182193 (PARI) Vec(-(1-8*x-5*x^2)/((1-x)*(1-6*x+x^2)) + O(x^30)) \\ _Colin Barker_, Mar 05 2016
%o A182193 (Sage) [lucas_number2(2*n,2,-1) - lucas_number1(2*n,2,-1) - 3 for n in (0..40)] # _G. C. Greubel_, May 24 2021
%Y A182193 Cf. A000129, A002203, A182188, A182189, A182191, A182355.
%K A182193 sign,easy
%O A182193 0,3
%A A182193 _Kenneth J Ramsey_, Apr 17 2012
%E A182193 More terms from _Harvey P. Dale_, Feb 09 2014