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<x>:=PolynomialRing(Integers()); N<r>:=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
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