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%I #34 May 25 2023 07:32:01
%S 1,7,55,469,4201,38647,360415,3383989,31878001,300780487,2840172775,
%T 26828857909,253476581401,2395031249527,22630944493135,
%U 213846879174229,2020725695234401,19094743928789767,180435210107977495
%N a(n) = 14*a(n-1) - 43*a(n-2), n > 1; a(0)=1, a(1)=7.
%C Binomial transform of A145301. Inverse binomial transform of A152263. - _Philippe Deléham_, Dec 03 2008
%H Harvey P. Dale, <a href="/A152262/b152262.txt">Table of n, a(n) for n = 0..1000</a>
%H H. D. Nguyen and D. Taggart, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.391.2522&rep=rep1&type=pdf">Mining the OEIS: Ten Experimental Conjectures</a>, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-43).
%F a(n) = ((7 + sqrt(6))^n + (7 - sqrt(6))^n)/2.
%F From _Philippe Deléham_, Dec 03 2008: (Start)
%F G.f.: (1-7*x)/(1-14*x+43*x^2).
%F a(n) = Sum_{k=0..n} A098158(n,k)*7^(2k-n)*6^(n-k). (End)
%F a(n) = Sum_{k=0..n} A027907(n,2k)*6^k. - _J. Conrad_, Aug 24 2016
%F E.g.f.: cosh(sqrt(6)*x)*exp(7*x). - _Ilya Gutkovskiy_, Aug 24 2016
%F a(n) = m^n*(ChebyshevU(n, 7/m) - (7/m)*ChebyshevU(n-1, 7/m)), with m = sqrt(43). - _G. C. Greubel_, May 23 2023
%t LinearRecurrence[{14,-43},{1,7},30] (* _Harvey P. Dale_, Apr 26 2015 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r6>:=NumberField(x^2-6); S:=[ ((7+r6)^n+(7-r6)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Dec 03 2008
%o (Magma) [n le 2 select 7^(n-1) else 14*Self(n-1) -43*Self(n-2): n in [1..30]]; // _G. C. Greubel_, May 23 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A152262
%o if (n<2): return 7^n
%o else: return 14*a(n-1) - 43*a(n-2)
%o [a(n) for n in range(41)] # _G. C. Greubel_, May 23 2023
%Y Cf. A027907, A098158, A145301, A152263.
%K nonn,easy
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008
%E Name from _Philippe Deléham_, Dec 03 2008
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