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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 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-François Alcover, Jul 02 2013 *) CROSSREFS Cf. A000045. Sequence in context: A323895 A274508 A347112 * 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 31 15:11 EST 2023. Contains 359976 sequences. (Running on oeis4.)