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A164544 a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

%I A164544 #28 Sep 08 2022 08:45:47
%S A164544 1,7,21,91,329,1295,4893,18851,71953,275863,1055397,4041835,15471449,
%T A164544 59235743,226771629,868193459,3323788321,12724930855,48716379957,
%U A164544 186507275899,714029211497,2733609354287,10465423189053,40066111858115
%N A164544 a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
%C A164544 Binomial transform of A164640. Inverse binomial transform of A164545.
%H A164544 Vincenzo Librandi, <a href="/A164544/b164544.txt">Table of n, a(n) for n = 0..178</a>
%H A164544 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,7).
%F A164544 a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
%F A164544 a(n) = ((2+3*sqrt(2))*(1+2*sqrt(2))^n + (2-3*sqrt(2))*(1-2*sqrt(2))^n)/4.
%F A164544 G.f.: (1+5*x)/(1-2*x-7*x^2).
%F A164544 a(n)/a(n-1) ~ 1 + 2*sqrt(2). - _Kyle MacLean Smith_, Dec 15 2019
%F A164544 E.g.f.: exp(x)*cosh(2*sqrt(2)*x) + 3*exp(x)*sinh(2*sqrt(2)*x)/sqrt(2). - _Stefano Spezia_, Dec 16 2019
%F A164544 From _G. C. Greubel_, Jul 18 2021: (Start)
%F A164544 a(n) = (i*sqrt(7))^(n-1)*(i*sqrt(7)*ChebyshevU(n, -i/sqrt(7)) + 5*ChebyshevU(n-1, -i/sqrt(7))).
%F A164544 a(n) = Sum_{j=0..floor(n/2)} binomial(n-k, k)*((7*n -12*k)/(n-k))*7^k*2^(n-2*k-1). (End)
%t A164544 LinearRecurrence[{2,7},{1,7},40] (* _Harvey P. Dale_, Jul 15 2012 *)
%o A164544 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((2+3*r)*(1+2*r)^n+(2-3*r)*(1-2*r)^n)/4: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 19 2009
%o A164544 (Sage) [(i*sqrt(7))^(n-1)*(i*sqrt(7)*chebyshev_U(n, -i/sqrt(7)) + 5*chebyshev_U(n-1, -i/sqrt(7))) for n in (0..40)] # _G. C. Greubel_, Jul 18 2021
%Y A164544 Cf. A164545, A164640.
%K A164544 nonn,easy
%O A164544 0,2
%A A164544 Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
%E A164544 Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 19 2009