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A256278 a(0)=1, a(1)=2, a(n) = 31*a(n-1) - 29*a(n-2).

%I A256278 #48 Aug 31 2025 15:49:00
%S A256278 1,2,33,965,28958,869713,26121321,784539274,23563199185,707707535789,
%T A256278 21255600833094,638400107288033,19173990901769297,575880114843495250,
%U A256278 17296237823997043137,519482849213446974997,15602377428720941973934,468608697663159238917041
%N A256278 a(0)=1, a(1)=2, a(n) = 31*a(n-1) - 29*a(n-2).
%C A256278 The sequence A084330 is a(0)=0, a(1)=1, a(n)=31a(n-1)-29a(n-2), and the ratio A084330(n+1)/a(n) converges to phi^7 (~29.034441853748633...), where phi is the golden ratio (A001622).
%C A256278 The continued fraction for phi^7 is {29,{29}}, and 29 occurs in the following approximations for n*phi^7: A248786 (29*n+floor(n/29)+0^n-0^(n mod 29)) for A004922 (floor(n*phi^7)), A249079 (29*n+floor(n/29)+0^(1-floor((14+(n mod 29))/29))) for A004942 (round(n*phi^7)), and A248739 (29*n+ceiling(n/29)) for A004962 (ceiling(n*phi^7)).
%H A256278 Karl V. Keller, Jr., <a href="/A256278/b256278.txt">Table of n, a(n) for n = 0..1000</a>
%H A256278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A256278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatioConjugate.html">Golden Ratio Conjugate</a>
%H A256278 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (31,-29).
%F A256278 G.f.: (1-29*x)/(29*x^2-31*x+1). - _Vincenzo Librandi_, Jun 03 2015
%F A256278 E.g.f.: exp(31*x/2)*(65*cosh(13*sqrt(5)*x/2) - 27*sqrt(5)*sinh(13*sqrt(5)*x/2))/65. - _Stefano Spezia_, Aug 31 2025
%e A256278 For n=3, 31*a(2)-29*a(1) = 31*(33)-29*(2) = 1023-58 = 965.
%p A256278 a:= n-> (<<0|1>, <-29|31>>^n. <<1, 2>>)[1,1]:
%p A256278 seq(a(n), n=0..23);  # _Alois P. Heinz_, Dec 22 2023
%t A256278 LinearRecurrence[{31, -29}, {1, 2}, 50] (* or *) CoefficientList[Series[(1 - 29 x)/(29 x^2 - 31 x + 1), {x, 0, 33}], x] (* _Vincenzo Librandi_, Jun 03 2015 *)
%o A256278 (Python)
%o A256278 print(1, end=', ')
%o A256278 print(2, end=', ')
%o A256278 an = [1,2]
%o A256278 for n in range(2,26):
%o A256278   print(31*an[n-1]-29*an[n-2], end=', ')
%o A256278   an.append(31*an[n-1]-29*an[n-2])
%o A256278 (Magma) I:=[1,2]; [n le 2 select I[n] else 31*Self(n-1)-29*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 03 2015
%Y A256278 Cf. A001622, A195819 (29*n), A084330, A004922, A004942, A004962, A248786, A249079, A248739.
%K A256278 nonn,easy,changed
%O A256278 0,2
%A A256278 _Karl V. Keller, Jr._, Jun 02 2015