A164599 - OEIS

%I #24 Sep 07 2024 15:40:32

%S 1,15,163,1577,14417,127719,1110467,9543745,81420481,691330719,

%T 5851867459,49433600633,417032638289,3515077706295,29610553888547,

%U 249339102243793,2099051398651393,17667781775661231,148693529122641763

%N a(n) = 14*a(n-1) - 47*a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

%C Binomial transform of A164598. Seventh binomial transform of A164587. Inverse binomial transform of A081185 without initial term 0.

%C This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 -2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n =  (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 -2)*x^2) and have a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - _G. C. Greubel_, Mar 11 2021

%H G. C. Greubel, <a href="/A164599/b164599.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from Vincenzo Librandi)

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-47).

%F a(n) = ((1+4*sqrt(2))*(7+sqrt(2))^n + (1-4*sqrt(2))*(7-sqrt(2))^n)/2.

%F G.f.: (1+x)/(1-14*x+47*x^2).

%F E.g.f.: exp(7*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - _G. C. Greubel_, Aug 11 2017

%F From _G. C. Greubel_, Mar 11 2021: (Start)

%F a(n) = A147958(n) + 8*A081184(n).

%F a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*6^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

%p m:=30; S:=series( (1+x)/(1-14*x+47*x^2), x, m+1):

%p seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 11 2021

%t LinearRecurrence[{14,-47}, {1,15}, 30] (* _G. C. Greubel_, Aug 11 2017 *)

%o (Magma) [ n le 2 select 14*n-13 else 14*Self(n-1)-47*Self(n-2): n in [1..30] ];

%o (PARI) my(x='x+O('x^30)); Vec((1+x)/(1-14*x+47*x^2)) \\ _G. C. Greubel_, Aug 11 2017

%o (Sage)

%o def A164599_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1+x)/(1-14*x+47*x^2) ).list()

%o A164599_list(30) # _G. C. Greubel_, Mar 11 2021

%Y Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), this sequence (m=6), A081185 (m=7), A164600 (m=8).

%Y Cf. A002203, A081184, A081185, A147958, A164587.

%K nonn

%O 0,2

%A _Klaus Brockhaus_, Aug 17 2009