login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A264080 a(n) = 6*F(n)*F(n+1) + (-1)^n, where F = A000045. 6
1, 5, 13, 35, 91, 239, 625, 1637, 4285, 11219, 29371, 76895, 201313, 527045, 1379821, 3612419, 9457435, 24759887, 64822225, 169706789, 444298141, 1163187635, 3045264763, 7972606655, 20872555201, 54645058949, 143062621645, 374542805987, 980565796315 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is prime for n = 1, 2, 5, 7, 14, 15, 29, 40, 49, 57, 70, 87, 105, 127, 175, 279, 362, 647, 727, ...

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

FORMULA

G.f.: (1+3*x+x^2) / ((1+x)*(1-3*x+x^2)). - Corrected by Colin Barker, Sep 28 2016

a(n) = -a(-n-1) = 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z.

a(n) = L(2*n+1) + F(n)*F(n+1) = A002878(n) + A001654(n). See similar identity for A061647.

a(n) = A001654(n+1) + 3*A001654(n) + A001654(n-1).

a(n) - a(n-1) = 2*A099016(n) with a(-1)=-1.

a(n) + a(n-1) = 2*A097134(n) for n>0.

Sum_{i>=0} 1/a(i) = 1.3232560865206157372628688449331...

a(n) = (2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Sep 28 2016

E.g.f.: (1/5)*exp(-x)*(-1 + 6*exp(5*x/2)*(cosh((sqrt(5)*x)/2) + sqrt(5)*sinh((sqrt(5)*x)/2))). - Stefano Spezia, Dec 09 2019

MAPLE

a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1, 5, 13>>)[1, 1]:

seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016

MATHEMATICA

Table[6 Fibonacci[n] Fibonacci[n + 1] + (-1)^n, {n, 0, 30}]

LinearRecurrence[{2, 2, -1}, {1, 5, 13}, 30] (* Harvey P. Dale, Jul 12 2019 *)

PROG

(Sage) [6*fibonacci(n)*fibonacci(n+1)+(-1)^n for n in (0..30)]

(Maxima) makelist(6*fib(n)*fib(n+1)+(-1)^n, n, 0, 30);

(MAGMA) [6*Fibonacci(n)*Fibonacci(n+1)+(-1)^n: n in [0..30]];

(PARI) for(n=0, 30, print1(6*fibonacci(n)*fibonacci(n+1)+(-1)^n", "));

(PARI) a(n) = round((2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ Colin Barker, Sep 28 2016

(PARI) Vec((1+3*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 28 2016

CROSSREFS

Cf. A000032, A000045; A001654, A002878.

Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n: A226205 (k=1); A236428 (k=2); A014742 (k=3); A061647 (k=4); A002878 (k=5).

Sequence in context: A192310 A167710 A229924 * A290588 A272149 A272560

Adjacent sequences:  A264077 A264078 A264079 * A264081 A264082 A264083

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Nov 03 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 2 21:30 EDT 2021. Contains 346429 sequences. (Running on oeis4.)