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A129346 a(2n) = A100525(n), a(2n+1) = A001653(n+1); a Pellian-related sequence.

%I A129346 #13 Jun 22 2022 09:28:03
%S A129346 4,5,22,29,128,169,746,985,4348,5741,25342,33461,147704,195025,860882,
%T A129346 1136689,5017588,6625109,29244646,38613965,170450288,225058681,
%U A129346 993457082,1311738121,5790292204,7645370045,33748296142,44560482149,196699484648,259717522849
%N A129346 a(2n) = A100525(n), a(2n+1) = A001653(n+1); a Pellian-related sequence.
%C A129346 Summation of -a(n) and A129345 returns twice Pell numbers A000129 (apart from signs; starting from 2nd term of A000129).
%H A129346 Colin Barker, <a href="/A129346/b129346.txt">Table of n, a(n) for n = 0..1000</a>
%H A129346 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F A129346 O.g.f.: (4 + 5*x - 2*x^2 - x^3) / ((x^2 - 2*x - 1)*(x^2 + 2*x - 1)).
%F A129346 From _Colin Barker_, May 26 2016: (Start)
%F A129346 a(n) = (-(-1-sqrt(2))^(1+n)+(-1+sqrt(2))^(1+n)+(1-sqrt(2))^n*(-4+3*sqrt(2))+(1+sqrt(2))^n*(4+3*sqrt(2)))/(2*sqrt(2)).
%F A129346 a(n) = 6*a(n-2)-a(n-4) for n>3. (End)
%F A129346 E.g.f.: 2*cosh(sqrt(2)*x)*(sinh(x) + 2*cosh(x)) + (sinh(sqrt(2)*x)*(5*sinh(x) + 3*cosh(x)))/sqrt(2). - _Ilya Gutkovskiy_, May 26 2016
%t A129346 LinearRecurrence[{0,6,0,-1},{4,5,22,29},30] (* _Harvey P. Dale_, Apr 08 2018 *)
%o A129346 (PARI) Vec((4+5*x-2*x^2-x^3)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^40)) \\ _Colin Barker_, May 26 2016
%Y A129346 Cf. A129345, A000129, A001542, A038761.
%K A129346 nonn,easy
%O A129346 0,1
%A A129346 _Creighton Dement_, Apr 10 2007