A182193 - OEIS

%I #41 Sep 08 2022 08:45:55

%S -1,1,19,125,743,4345,25339,147701,860879,5017585,29244643,170450285,

%T 993457079,5790292201,33748296139,196699484645,1146448611743,

%U 6681992185825,38945504503219,226991034833501,1323000704497799,7711013192153305,44943078448422043

%N Sequence of row differences related to table A182355.

%C 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 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 Vincenzo Librandi, <a href="/A182193/b182193.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).

%F a(n) = 6*a(n-1) - a(n-2) + 12.

%F 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 From _Colin Barker_, Mar 05 2016: (Start)

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

%F G.f.: -(1-8*x-5*x^2) / ((1-x)*(1-6*x+x^2)).

%F (End)

%F a(n) = A002203(2*n) - A000129(2*n) - 3. - _G. C. Greubel_, May 24 2021

%p Pell:= proc(n) option remember;

%p     if n<2 then n

%p   else 2*Pell(n-1) + Pell(n-2)

%p     fi; end:

%p seq(Pell(2*n) + 2*Pell(2*n-1) - 3, n=0..40); # _G. C. Greubel_, May 24 2021

%t LinearRecurrence[{7,-7,1},{-1,1,19},30] (* _Harvey P. Dale_, Feb 09 2014 *)

%o (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 (PARI) Vec(-(1-8*x-5*x^2)/((1-x)*(1-6*x+x^2)) + O(x^30)) \\ _Colin Barker_, Mar 05 2016

%o (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 Cf. A000129, A002203, A182188, A182189, A182191, A182355.

%K sign,easy

%O 0,3

%A _Kenneth J Ramsey_, Apr 17 2012

%E More terms from _Harvey P. Dale_, Feb 09 2014