A276228 - OEIS

A276228

a(n+3) = -a(n+2) - 2*a(n+1) + a(n) with a(0)=3, a(1)=-1, a(2)=-3.

3, -1, -3, 8, -3, -16, 30, -1, -75, 107, 42, -331, 354, 350, -1389, 1043, 2085, -5560, 2433, 10772, -21198, 2087, 51081, -76453, -23622, 227609, -256818, -222022, 963267, -776041, -1372515, 3887864, -1918875, -7229368, 14954982, -2415121, -34724211, 54509435, 12523866

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OFFSET

0,1

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..4925

Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.

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

FORMULA

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

Let (b, c, d) be the three roots of x^3 = 2*x^2 + x + 1, then a(n) = b^(-n) + c^(-n) + d^(-n) = A276225(-n).

a(2*n) = -3*a(2*n-2) - 6*a(2*n-4) + a(2*n-6).

a(n) = 2*A276229(n) + 2*A276229(n+1) + 3*A276229(n+2).

MATHEMATICA

LinearRecurrence[{-1, -2, 1}, {3, -1, -3}, 50]

CoefficientList[Series[(3 + 2 x + 2 x^2)/(1 + x + 2 x^2 - x^3), {x, 0, 38}], x] (* Michael De Vlieger, Aug 25 2016 *)

nxt[{a_, b_, c_}]:={b, c, a-2b-c}; NestList[nxt, {3, -1, -3}, 40][[All, 1]] (* Harvey P. Dale, Dec 19 2022 *)

PROG

(Magma) I:=[3, -1, -3]; [n le 3 select I[n] else -Self(n-1)- 2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 25 2016

(PARI) Vec((3+2*x+2*x^2)/(1+x+2*x^2-x^3) + O(x^99)) \\ Altug Alkan, Aug 25 2016

CROSSREFS

Cf. A276225, A276229.

Sequence in context: A046544 A011088 A352794 * A188938 A156368 A240665

Adjacent sequences: A276225 A276226 A276227 * A276229 A276230 A276231

KEYWORD

sign,easy

AUTHOR

G. C. Greubel, Aug 24 2016

EXTENSIONS

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

STATUS

approved