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A086901 a(1) = a(2) = 1; for n>2, a(n) = 4*a(n-1) + 3*a(n-2).

%I A086901 #55 Oct 29 2024 12:41:16
%S A086901 1,1,7,31,145,673,3127,14527,67489,313537,1456615,6767071,31438129,
%T A086901 146053729,678529303,3152278399,14644701505,68035641217,316076669383,
%U A086901 1468413601183,6821884412881,31692778455073,147236767058935
%N A086901 a(1) = a(2) = 1; for n>2, a(n) = 4*a(n-1) + 3*a(n-2).
%H A086901 Reinhard Zumkeller, <a href="/A086901/b086901.txt">Table of n, a(n) for n = 1..1000</a>
%H A086901 Sela Fried and Toufik Mansour, <a href="https://arxiv.org/abs/2312.08273">Staircase graph words</a>, arXiv:2312.08273 [math.CO], 2023.
%H A086901 Lucyna Trojnar-Spelina and Iwona Włoch, <a href="https://doi.org/10.1007/s40995-019-00757-7">On Generalized Pell and Pell-Lucas Numbers</a>, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
%F A086901 a(n) = ((c + 5)*b^n - (b + 5)*c^n)/14, where b = 2 + sqrt(7), c = 2 - sqrt(7).
%F A086901 From _Ralf Stephan_, Feb 01 2004: (Start)
%F A086901 G.f.: x*(1-3*x)/(1 - 4*x - 3*x^2).
%F A086901 a(n) = A015530(n) - 3*A015530(n-1) = 1 + 6*Sum_{k=0..n-2} A015530(k). (End)
%F A086901 a(n+1) = Sum_{k=0..n} 3^(n-k)*A122542(n,k), n>=0. - _Philippe Deléham_, Oct 27 2006
%F A086901 a(n) = upper left term in the 2 X 2 matrix [1,2; 3,3]^(n-1). - _Gary W. Adamson_, Mar 02 2008
%F A086901 G.f.: G(0)*(1-3*x)/(2-4*x), where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 16 2013
%F A086901 E.g.f.: exp(2*x)*( cosh(sqrt(7)*x) - (1/sqrt(7))*sinh(sqrt(7)*x) ). - _G. C. Greubel_, Oct 28 2024
%e A086901 a(3) = 4*1 + 3*1 = 7;
%e A086901 a(4) = 4*7 + 3*1 = 31.
%t A086901 a[n_]:=(MatrixPower[{{3,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%t A086901 Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{3,4},#]}]&, {1,1},40]][[1]]  (* _Harvey P. Dale_, Mar 23 2011 *)
%t A086901 LinearRecurrence[{4,3}, {1,1}, 41] (* _G. C. Greubel_, Oct 28 2024 *)
%o A086901 (PARI) A086901(n)=if(n<3,1,4*A086901(n-1)+3*A086901(n-2)) \\ _Michael B. Porter_, Apr 04 2010
%o A086901 (Haskell)
%o A086901 a086901 n = a086901_list !! (n-1)
%o A086901 a086901_list = 1 : 1 : zipWith (+)
%o A086901                (map (* 3) a086901_list) (map (* 4) $ tail a086901_list)
%o A086901 -- _Reinhard Zumkeller_, Feb 13 2015
%o A086901 (Magma) [n le 2 select 1 else 4*Self(n-1) +3*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Oct 28 2024
%o A086901 (SageMath)
%o A086901 A086901=BinaryRecurrenceSequence(4,3,1,1)
%o A086901 [A086901(n) for n in range(41)] # _G. C. Greubel_, Oct 28 2024
%Y A086901 Cf. A015530, A102900, A122542.
%K A086901 easy,nonn
%O A086901 1,3
%A A086901 Rick Powers (rick.powers(AT)mnsu.edu), Sep 18 2003
%E A086901 More terms from _Ray Chandler_, Sep 19 2003