A164604 - OEIS

%I #19 Apr 10 2024 09:46:31

%S 1,19,113,659,3841,22387,130481,760499,4432513,25834579,150574961,

%T 877615187,5115116161,29813081779,173763374513,1012767165299,

%U 5902839617281,34404270538387,200522783613041,1168732431139859

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

%C Binomial transform of A164603. Third binomial transform of A164702. Inverse binomial transform of A164605.

%C From _Klaus Purath_, Mar 14 2024: (Start)

%C For any two consecutive terms (a(n), a(n+1)) = (x,y): x^2 - 6xy + y^2 = 248 = A028884(13). In general, the following applies to all recursive sequences (t) with constant coefficients (6,-1) and t(0) = 1 and two consecutive terms (x,y): x^2 - 6xy + y^2 = A028884(t(1)-6). This includes and interprets the Feb 04 2014 comment on A001541 by Colin Barker as well as the Mar 17 2021 comment on A054489 by John O. Oladokun.

%C By analogy to this, for three consecutive terms (x,y,z) of any recursive sequence (t) of form (6,-1) with t(0) = 1: y^2 - xz = A028884(t(1)-6). (End)

%H G. C. Greubel, <a href="/A164604/b164604.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..155 from Vincenzo Librandi)

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).

%F a(n) = 6*a(n-1) - a(n-2) for n > 1; a(0) = 1, a(1) = 19.

%F G.f.: (1+13*x)/(1-6*x+x^2).

%F E.g.f.: exp(3*x)*( cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x) ). - _G. C. Greubel_, Aug 11 2017

%t LinearRecurrence[{6,-1}, [1,19}, 50] (* _G. C. Greubel_, Aug 11 2017 *)

%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(3+2*r)^n+(1-4*r)*(3-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 23 2009

%o (PARI) Vec((1+13*x)/(1-6*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 12 2011

%Y Cf. A164603, A164702, A164605.

%K nonn,easy

%O 0,2

%A Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 23 2009