

A098149


a(0)=1, a(1)=1, a(n)=3*a(n1)a(n2) for n>1.


9



1, 1, 4, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371
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OFFSET

0,3


COMMENTS

Sequence relates bisections of Lucas and Fibonacci numbers.
2*a(n) + A098150(n) = 8*(1)^(n+1)*A001519(n)  (1)^(n+1)*A005248(n+1). Apparently, if (z(n)) is any sequence of integers (not all zero) satisfying the formula z(n) = 2(z(n2)  z(n1)) + z(n3) then z(n+1)/z(n) > golden ratio phi + 1 = (3+sqrt(5))/2.
Pisano period lengths: 1, 3, 4, 6, 1, 12, 8, 6, 12, 3, 10, 12, 7, 24, 4, 12, 9, 12, 18, 6, ... .  R. J. Mathar, Aug 10 2012
[X(n) = (1)^n*(S(n, 3) + S(n1, 3)), Y(n) = X(n1)] gives all integer solutions (modulo sign flip between X and Y) of X^2 + Y^2 + 3*X*Y = +5, for n = oo..+oo, with Chebyshev S polynomials (A049310), with S(1, x) = 0, S(n, x) =  S(n2, x), for n >= 2, and S(n,x) = (1)^n*S(n, x). The present sequence is a(n) = X(n1), for n >= 0. See the formula section.
This binary indefinite quadratic form of discriminant 5, representing 5, has only this family of proper solutions (modulo sign flip), and no improper ones.
This comment is inspired by a paper by Robert K. Moniot (private communication) See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2  3*x*y = 1 (special Markov solutions). (End)


LINKS



FORMULA

a(n+1) = Sum_{k, 0<=k<=n}(5)^k*Binomial(n+k, nk) = Sum_{k, 0<=k<=n}(5)^k*A085478(n, k).  Philippe Deléham, Nov 28 2006
a(n) = (1/2)*[(3/2)(1/2)*sqrt(5)]^n+(1/2)*[(3/2)(1/2)*sqrt(5)]^n*sqrt(5)(1/2)*[(3/2)+(1/2)*sqrt(5)]^n*sqrt(5)(1/2)*[(3/2)+(1/2)*sqrt(5)]^n, with n>=0.  Paolo P. Lava, Nov 19 2008
a(n) = (1)^n*(S(n1, 3) + S(n2, 3)) = (1)^n*S(2*(n1), sqrt(5)), for n >= 0, with Chebyshev S polynomials (A049310), with S(1, x) = 0 and S(2, x) = 1. S(n, 3) = A001906(n+1) = F(2*(n+1)), with F = A000045.  Wolfdieter Lang, Oct 12 2020


MATHEMATICA

a[0] = a[1] = 1; a[n_] := a[n] = 3a[n  2]  a[n  1]; Table[ a[n], {n, 0, 27}] (* Robert G. Wilson v, Sep 01 2004 *)
LinearRecurrence[{3, 1}, {1, 1}, 30] (* Harvey P. Dale, Apr 19 2014 *)
CoefficientList[Series[(1 + 4 x)/(1 + 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)


CROSSREFS



KEYWORD

easy,sign


AUTHOR



EXTENSIONS



STATUS

approved



