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 A084128 a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2. 13
 1, 2, 12, 56, 272, 1312, 6336, 30592, 147712, 713216, 3443712, 16627712, 80285696, 387653632, 1871757312, 9037643776, 43637604352, 210700992512, 1017354387456, 4912221519872, 23718303629312, 114522100596736, 552961616904192, 2669934870003712 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Original name was: Generalized Fibonacci sequence. Binomial transform of A084058. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (4,4). FORMULA G.f.: (1-2*x)/(1-4*x-4*x^2). a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2. a(n) = (2 + 2*sqrt(2))^n/2 + (2 - 2*sqrt(2))^n/2. E.g.f.: exp(2*x)*cosh(2*x*sqrt(2)). From Johannes W. Meijer, Aug 01 2010: (Start) Lim_{k->infinity} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2). Lim_{n->infinity} A084128(n)/A057087(n-1) = sqrt(2). (End) a(n) = Sum_{k=0..n} A201730(n,k)*7^k. - Philippe Deléham, Dec 06 2011 G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k-2)/(x*(4*k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013 a(n) = 2^(n-1)*A002203(n). - Vladimir Reshetnikov, Oct 07 2016 MAPLE a:=proc(n) option remember; if n=0 then 1 elif n=1 then 2 else 4*a(n-1)+4*a(n-2); fi; end: seq(a(n), n=0..40); # Wesley Ivan Hurt, Jan 31 2017 MATHEMATICA CoefficientList[Series[(2 z - 1)/(4 z^2 + 4 z - 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *) Table[2^(n-1) LucasL[n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *) PROG (PARI) a(n)=if(n<0, 0, polsym(4+4*x-x^2, n)[n+1]/2) (Sage) [lucas_number2(n, 4, -4)/2 for n in xrange(0, 23)] # Zerinvary Lajos, May 14 2009 CROSSREFS Cf. A057087, A001541. Equals 2^n * A001333(n). Appears in A086346, A086347 and A086348. - Johannes W. Meijer, Aug 01 2010 Sequence in context: A181298 A247121 A078543 * A044047 A105487 A098453 Adjacent sequences:  A084125 A084126 A084127 * A084129 A084130 A084131 KEYWORD nonn,easy AUTHOR Paul Barry, May 16 2003 STATUS approved

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