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A055273 a(n) = 3*a(n-1) - a(n-2); a(0) = 1, a(1) = 8. 3
1, 8, 23, 61, 160, 419, 1097, 2872, 7519, 19685, 51536, 134923, 353233, 924776, 2421095, 6338509, 16594432, 43444787, 113739929, 297775000, 779585071, 2040980213, 5343355568, 13989086491, 36623903905, 95882625224, 251023971767, 657189290077, 1720543898464 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

LINKS

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

Tanya Khovanova, Recursive Sequences

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

FORMULA

a(n) = (8*(((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n) - (((3 + sqrt(5))/2)^(n - 1) - ((3 - sqrt(5))/2)^(n - 1)))/sqrt(5).

G.f.: (1 + 5*x)/(1 - 3*x + x^2).

From Rigoberto Florez, Dec 24 2018: (Start)

a(n) = F(2n+2) + 5*F(2n),

a(n) = 3*F(2n+2) - F(2n-3), where F(n) is the n-th Fibonacci number. (End)

E.g.f.: (1/5)*exp(3*x/2)*(5*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2)). - Franck Maminirina Ramaharo, Dec 26 2018

MAPLE

seq(coeff(series((1+5*x)/(1-3*x+x^2), x, n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Dec 29 2018

MATHEMATICA

Table[3Fibonacci[2n+2]-Fibonacci[2n-3], {n, 0, 20}] (* Rigoberto Florez, Dec 24 2018 *)

LinearRecurrence[{3, -1}, {1, 8}, 30] (* Vincenzo Librandi, Dec 25 2018 *)

PROG

(MAGMA) [Fibonacci(2*n+2) + 5*Fibonacci(2*n): n in [0..30]]; // Vincenzo Librandi, Dec 25 2018

(GAP) a:=[1, 8];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 29 2018

CROSSREFS

Cf. A055267.

Sequence in context: A048467 A002765 A048770 * A296323 A241353 A059209

Adjacent sequences:  A055270 A055271 A055272 * A055274 A055275 A055276

KEYWORD

easy,nonn

AUTHOR

Barry E. Williams, May 28 2000

STATUS

approved

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Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)