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 A164604 a(n) = ((1+4*sqrt(2))*(3+2*sqrt(2))^n + (1-4*sqrt(2))*(3-2*sqrt(2))^n)/2. 3
 1, 19, 113, 659, 3841, 22387, 130481, 760499, 4432513, 25834579, 150574961, 877615187, 5115116161, 29813081779, 173763374513, 1012767165299, 5902839617281, 34404270538387, 200522783613041, 1168732431139859 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Binomial transform of A164603. Third binomial transform of A164702. Inverse binomial transform of A164605. From Klaus Purath, Mar 14 2024: (Start) For any two consecutive terms (a(n), a(n+1)) = (x,y): x^2 - 6xy + y^2 = 248 = A028884(13). In general, the following applies to all recursive sequences (t) with constant coefficients (6,-1) and t(0) = 1 and two consecutive terms (x,y): x^2 - 6xy + y^2 = A028884(t(1)-6). This includes and interprets the Feb 04 2014 comment on A001541 by Colin Barker as well as the Mar 17 2021 comment on A054489 by John O. Oladokun. By analogy to this, for three consecutive terms (x,y,z) of any recursive sequence (t) of form (6,-1) with t(0) = 1: y^2 - xz = A028884(t(1)-6). (End) LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..155 from Vincenzo Librandi) Index entries for linear recurrences with constant coefficients, signature (6,-1). FORMULA a(n) = 6*a(n-1) - a(n-2) for n > 1; a(0) = 1, a(1) = 19. G.f.: (1+13*x)/(1-6*x+x^2). E.g.f.: exp(3*x)*( cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x) ). - G. C. Greubel, Aug 11 2017 MATHEMATICA LinearRecurrence[{6, -1}, [1, 19}, 50] (* G. C. Greubel, Aug 11 2017 *) PROG (Magma) Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(3+2*r)^n+(1-4*r)*(3-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 23 2009 (PARI) Vec((1+13*x)/(1-6*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2011 CROSSREFS Cf. A164603, A164702, A164605. Sequence in context: A080442 A033655 A245753 * A142370 A084751 A041694 Adjacent sequences: A164601 A164602 A164603 * A164605 A164606 A164607 KEYWORD nonn,easy AUTHOR Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009 EXTENSIONS Edited and extended beyond a(5) by Klaus Brockhaus, Aug 23 2009 STATUS approved

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Last modified August 14 13:35 EDT 2024. Contains 375165 sequences. (Running on oeis4.)