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A227104 a(0)=-1, a(1)=3; a(n+2) = a(n+1) + a(n) + 2*A057078(n+1). 0
-1, 3, 2, 3, 7, 10, 15, 27, 42, 67, 111, 178, 287, 467, 754, 1219, 1975, 3194, 5167, 8363, 13530, 21891, 35423, 57314, 92735, 150051, 242786, 392835, 635623, 1028458, 1664079, 2692539, 4356618, 7049155, 11405775, 18454930, 29860703, 48315635, 78176338, 126491971 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.

a(n) and its differences:

.  -1,   3,   2,   3,   7,  10,  15,  27,  42,

.   4,  -1,   1,   4,   3,   5,  12,  15,  25,

.  -5,   2,   3,  -1,   2,   7,   3,  10,  19,

.   7,   1,  -4,   3,   5,  -4,   7,   9,   4,

.  -6,  -5,   7,   2,  -9,  11,   2,  -5,  15,

.   1,  12,  -5, -11,  20,  -9,  -7,  20,  -5,

.  11, -17,  -6,  31, -29,   2,  27, -25,   2,

. -28,  11,  37, -60,  31,  25, -52,  27,  29,

.  39,  26, -97,  91,  -6, -77,  79,   2, -81.

Main diagonal: -(-1)^floor(n/2)*A108411(n).

LINKS

Table of n, a(n) for n=0..39.

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

FORMULA

a(3n) = 2*F(3n)-1, a(3n+1) = 2*F(3n+1)+1, a(3n+2) = 2*F(3n+2), where F=A000045.

a(n+3) = a(n) + 4*F(n+1).

a(n) = A226328(n) + 1 for n>1.

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) and many others by telescoping the fundamental recurrence.

G.f.: -(1-3*x-3*x^2-2*x^3) / ( (1-x-x^2)*(1+x+x^2) ). [Bruno Berselli, Jul 02 2013]

a(n) = a(n-2) + 2*a(n-3) - a(n-4). [Bruno Berselli, Jul 02 2013]

EXAMPLE

a(6) = 2*F(6)-1 = 2*8-1 = 15; a(7) = 2*F(7)+1 = 2*13+1 = 27; a(8) = 2*F(8) = 2*21 = 42.

MATHEMATICA

a[n_] := (m = Mod[n, 3]; 2*Fibonacci[n] - (3*m - 1)*(m - 2)/2); Table[a[n], {n, 0, 39}]  (* Jean-Fran├žois Alcover, Jul 02 2013 *)

CROSSREFS

Cf. A000045.

Sequence in context: A215934 A323895 A274508 * A171721 A225695 A226469

Adjacent sequences:  A227101 A227102 A227103 * A227105 A227106 A227107

KEYWORD

sign,easy

AUTHOR

Paul Curtz, Jul 01 2013

EXTENSIONS

Edited by Bruno Berselli, Jul 02 2013

STATUS

approved

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Last modified January 18 13:49 EST 2020. Contains 331010 sequences. (Running on oeis4.)