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A098149 a(0)=-1, a(1)=-1, a(n)=-3*a(n-1)-a(n-2) for n>1. 7
-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 (list; graph; refs; listen; history; text; internal format)
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(n-2) - z(n-1)) + z(n-3) 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

REFERENCES

Stees, Ryan, "Sequences of Spiral Knot Determinants" (2016). Senior Honors Projects. Paper 84. James Madison Univ., May 2016; http://commons.lib.jmu.edu/cgi/viewcontent.cgi?article=1043&context=honors201019

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Seong Ju Kim, R. Stees, L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (-3,-1)

FORMULA

G.f.: -(1+4*x)/(1+3*x+x^2). - Philippe Deléham, Nov 19 2006

a(n) = (-1)^n*A002878(n-1). - R. J. Mathar, Jan 30 2011

-a(n+1) = Sum_{k, 0<=k<=n}(-5)^k*Binomial(n+k, n-k) = 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

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

Cf. A098150, A001519, A005248.

Sequence in context: A027968 A027970 A027972 * A002878 A124861 A110579

Adjacent sequences:  A098146 A098147 A098148 * A098150 A098151 A098152

KEYWORD

easy,sign

AUTHOR

Creighton Dement, Aug 29 2004

EXTENSIONS

Simpler definition from Philippe Deléham, Nov 19 2006

STATUS

approved

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Last modified August 18 17:58 EDT 2017. Contains 290732 sequences.