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A081016 (Lucas(4n+3)+1)/5, or Fibonacci(2n+1)*Fibonacci(2n+2), or A081015(n)/5. 13
1, 6, 40, 273, 1870, 12816, 87841, 602070, 4126648, 28284465, 193864606, 1328767776, 9107509825, 62423800998, 427859097160, 2932589879121, 20100270056686, 137769300517680, 944284833567073, 6472224534451830 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1) is, together with b(n) := A089508(n), n>=1, the solution to a binomial problem; see A089508.

Numbers k such that 1 - 2 x + 5 x^2 is a square. - Artur Jasinski, Oct 26 2008

Also solution y of diophantine equation x^2+y^2+y^2+y^2+y^2=k^2 for which x=y-1. - Carmine Suriano, Jun 23 2010

REFERENCES

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 26.

LINKS

Table of n, a(n) for n=0..19.

Index entries for linear recurrences with constant coefficients, signature (8, -8, 1).

FORMULA

a(n) = 8a(n-1) - 8a(n-2) + a(n-3).

G.f.: (1-2*x)/((1-x)*(1-7*x+x^2)).

a(n) = F(1) + F(5) + F(9) +...+ F(4n+1) = F(2n)*F(2n+3) + 1.

a(n) = (1/5)+(2/5)*{[(7/2)-(3/2)*sqrt(5)]^n+[(7/2)+(3/2)*sqrt(5)]^n+(1/5)*sqrt(5)*{[(7/2)+(3/2) *sqrt(5)]^n-[(7/2)-(3/2)*sqrt(5)]^n}, with n>=0. - Paolo P. Lava, Dec 01 2008

a(n) = 7*a(n-1)-a(n-2)-1, n>=2. R. J. Mathar, Nov 07 2015

MAPLE

luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 25 do printf(`%d, `, (luc(4*n+3)+1)/5) od: # James A. Sellers, Mar 03 2003

MATHEMATICA

Join[{a=1, b=6}, Table[c=7*b-1*a-1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)

PROG

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, -8, 8]^n*[1; 6; 40])[1, 1] \\ Charles R Greathouse IV, Sep 28 2015

CROSSREFS

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A081015.

Partial sums of A033889. Bisection of A001654. Equals A003482 + 1.

Cf. A145995, A178898.

Sequence in context: A122074 A244253 A123357 * A083426 A122471 A178397

Adjacent sequences:  A081013 A081014 A081015 * A081017 A081018 A081019

KEYWORD

nonn,easy

AUTHOR

R. K. Guy, Mar 01 2003

STATUS

approved

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Last modified June 27 19:44 EDT 2017. Contains 288790 sequences.