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A182188 A sequence of row differences for table A182119.

%I A182188 #39 May 24 2021 00:23:55
%S A182188 1,-1,-11,-69,-407,-2377,-13859,-80781,-470831,-2744209,-15994427,
%T A182188 -93222357,-543339719,-3166815961,-18457556051,-107578520349,
%U A182188 -627013566047,-3654502875937,-21300003689579
%N A182188 A sequence of row differences for table A182119.
%C A182188 This is a list of row differences corresponding to a difference of 1 in table A182119, column 0. If A181119(k+1,0) - A182119(k,0) = 1, then a(n) = A182119(k+1,n) - A182119(k,n).
%C A182188 If p is a prime of the form 8*n +- 3, then a(p) == 3 (mod p). If p is a prime of the form 8*n +- 1, then a(p) == -1 (mod p).
%H A182188 G. C. Greubel, <a href="/A182188/b182188.txt">Table of n, a(n) for n = 0..1000</a>
%H A182188 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F A182188 a(n) = 6*a(n-1) - a(n-2) - 4. [corrected by _Klaus Purath_, Mar 19 2021]
%F A182188 a(n) = -(A182189(n-1) + 2*A182190(n-1)).
%F A182188 a(n) = 2 - A182189(n).
%F A182188 From _Klaus Purath_, Mar 19 2021: (Start)
%F A182188 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3).
%F A182188 a(n) = (-1)*Sum_{i=1..2*n-1} A001333(i) for n > 0.
%F A182188 a(n) = 1 - A001542(n) for n > 0.
%F A182188 a(n) = 1 - 2*A001109(n) for n > 0.
%F A182188 a(n) = (-1)*A005409(2*n) for n > 0. (End)
%F A182188 G.f.: (1 - 8*x + 3*x^2)/((1-x)*(1-6*x+x^2)). - _Chai Wah Wu_, Apr 08 2021
%F A182188 a(n) = 1 - Pell(2*n), where Pell(n) = A000129(n). - _G. C. Greubel_, May 24 2021
%t A182188 m = 13;n = 3; c = 0;
%t A182188 list3 = Reap[While[c < 22, t = 6 n - m - 4; Sow[t];m = n; n = t;c++]][[2,1]]
%t A182188 Table[1 -Fibonacci[2*n, 2], {n,0,40}] (* _G. C. Greubel_, May 24 2021 *)
%o A182188 (Sage) [1 - lucas_number1(2*n,2,-1) for n in (0..40)] # _G. C. Greubel_, May 24 2021
%Y A182188 Cf. A000129, A001109, A001333, A001542, A005409, A182189, A182190, A182119.
%K A182188 sign,easy
%O A182188 0,3
%A A182188 _Kenneth J Ramsey_, Apr 17 2012