A247560 a(n) = 3*a(n-1) - 4*a(n-2) with a(0) = a(1) = 1.

1, 1, -1, -7, -17, -23, -1, 89, 271, 457, 287, -967, -4049, -8279, -8641, 7193, 56143, 139657, 194399, 24569, -703889, -2209943, -3814273, -2603047, 7447951, 32756041, 68476319, 74404793, -50690897, -449691863, -1146312001, -1640168551, -335257649, 5554901257

Offset: 0

Author

Michael Somos, Sep 19 2014

Keywords

• sign
• easy

Examples

			G.f. = 1 + x - x^2 - 7*x^3 - 17*x^4 - 23*x^5 - x^6 + 89*x^7 + 271*x^8 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2, pp. 251-267. See Theorem 2, p. 259.
• M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
• Index entries for linear recurrences with constant coefficients, signature (3,-4).

Crossrefs

Cf. A087168, A247487, A247564, A247565.

Programs

Haskell

a247560 n = a247560_list !! n
a247560_list = 1 : 1 : zipWith (-) (map (* 3) $ tail a247560_list)
                                   (map (* 4) a247560_list)
-- Reinhard Zumkeller, Sep 20 2014

Magma

I:=[1, 1]; [n le 2 select I[n] else 3*Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 04 2018

Maple

A247560:=n->simplify((1/14*I)*sqrt(7)*((3/2+(1/2*I)*sqrt(7))^n-(3/2-(1/2*I)*sqrt(7))^n)+1/2*((3/2+(1/2*I)*sqrt(7))^n+(3/2-(1/2*I)*sqrt(7))^n)): seq(A247560(n), n=0..40); # Wesley Ivan Hurt, Oct 02 2014

Mathematica

a[ n_] := Re[ (1 - 1/Sqrt[-7]) (3 + Sqrt[-7])^n / 2^n];
LinearRecurrence[{3,-4},{1,1},40] (* Harvey P. Dale, Jun 13 2017 *)

PARI

{a(n) = real( (1 + quadgen(-7))^n )};

Sage

[((1-1/sqrt(-7))*(3+sqrt(-7))^n/2^n).real() for n in range(34)] # Peter Luschny, Oct 02 2014 (after Somos)

Formula

G.f.: (1 - 2*x) / (1 - 3*x + 4*x^2).

a(n) = 3*a(n-1) - 4*a(n-2) for all n in Z.

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

a(n) = (-1)^n * A087168(n) for all n in Z.

A247565(n) = 2^n + a(n) for all n in Z.

a(n) = A247487(2*n + 1) = A247564(2*n + 1) for all n in Z.