A182415 a(0) = 1, a(1) = 2; for n>1, a(n) = a(n-1) + a(n-2) + 4.

1, 2, 7, 13, 24, 41, 69, 114, 187, 305, 496, 805, 1305, 2114, 3423, 5541, 8968, 14513, 23485, 38002, 61491, 99497, 160992, 260493, 421489, 681986, 1103479, 1785469, 2888952, 4674425, 7563381, 12237810, 19801195, 32039009, 51840208, 83879221, 135719433, 219598658

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Formula

From Colin Barker, May 07 2012: (Start)

a(n) = 2*a(n-1) - a(n-3).

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

a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - 4. - G. C. Greubel, Jul 22 2019

Example

a(3) = 7 + 2 + 4 = 13.

Mathematica

With[{f=Fibonacci}, Table[F[n+3]+3*F[n+1]-4, {n, 0, 40}]] (* G. C. Greubel, Jul 22 2019 *)
RecurrenceTable[{a[0]==1, a[1]==2, a[n]==a[n-1]+a[n-2]+4}, a, {n, 40}] (* or *) LinearRecurrence[{2, 0, -1}, {1, 2, 7}, 40] (* Harvey P. Dale, Nov 24 2020 *)

Prog

(PARI) vector(40, n, n--; f=fibonacci; f(n+3) +3*f(n+1) -4 ) \\ G. C. Greubel, Jul 22 2019
(Magma) F:=Fibonacci; [F(n+3)+3*F(n+1)-4: n in [0..40]]; // G. C. Greubel, Jul 22 2019
(Sage) f=fibonacci; [f(n+3)+3*f(n+1)-4 for n in (0..40)] # G. C. Greubel, Jul 22 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+3)+3*F(n+1)-4); # G. C. Greubel, Jul 22 2019

Crossrefs

Cf. A000045, A014739, A022095 (first differences), A171516.

Sequence in context: A045377 A180470 A237443 * A275491 A360673 A183435

Adjacent sequences: A182412 A182413 A182414 * A182416 A182417 A182418

Keyword

nonn,easy

Author

Alex Ratushnyak, Apr 28 2012

Status

approved