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 A022098 Fibonacci sequence beginning 1, 8. 10
 1, 8, 9, 17, 26, 43, 69, 112, 181, 293, 474, 767, 1241, 2008, 3249, 5257, 8506, 13763, 22269, 36032, 58301, 94333, 152634, 246967, 399601, 646568, 1046169, 1692737, 2738906, 4431643, 7170549, 11602192, 18772741, 30374933, 49147674, 79522607, 128670281 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(8; n-1-k, k) with n >= 1, a(-1) = 7. These are the SW-NE diagonals in P(8; n, k), the (8, 1) Pascal triangle A093565. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Pisano period lengths: 1, 3, 8, 6, 4, 24, 16, 12, 24, 12, 10, 24, 28, 48, 8, 24, 36, 24, 18, 12, ... (is this the same as A106291?). - R. J. Mathar, Aug 10 2012 Also the sum of five consecutive Lucas numbers starting with L(-3). - Alonso del Arte, Sep 26 2013 The Pisano period lengths of this sequence are exactly the same as those of the Lucas sequence (A000032), given in A106291. - Klaus Purath, Apr 20 2019 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (1,1). FORMULA a(n) = a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=8, and a(-1):=7. G.f.: (1 + 7*x)/(1 - x - x^2). a(n) = ((1 + sqrt(5))^n - (1 - sqrt(5))^n)/(2^n*sqrt(5)) + 3.5*((1 + sqrt(5))^(n-1) - (1 - sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)) for n>0. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009 a(n) = 2^(-1-n)*((1 - sqrt(5))^n*(-15 + sqrt(5)) + (1 + sqrt(5))^n*(15 + sqrt(5)))/sqrt(5). - Herbert Kociemba, Dec 18 2011 a(n) = 7*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012 a(n) = 8*A000045(n) + A000045(n-1). - Paolo P. Lava, May 18 2015 a(n) = 9*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017 a(n) = Lucas(n+3) + Lucas(n-3) - 3*Lucas(n) for n>1. - Bruno Berselli, Dec 29 2016 MATHEMATICA LinearRecurrence[{1, 1}, {1, 8}, 40] (* Alonso del Arte, Sep 26 2013 *) CoefficientList[Series[(1 + 7 x)/(1 - x - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 27 2013 *) Table[LucasL[n + 3] + LucasL[n - 3] - 3 LucasL[n], {n, 2, 40}] (* Bruno Berselli, Dec 30 2016 *) PROG (MAGMA) a0:=1; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013 (PARI) a(n)=([0, 1; 1, 1]^n*[1; 8])[1, 1] \\ Charles R Greathouse IV, Oct 07 2016 CROSSREFS Cf. A000032, A000045. a(n) = A109754(7, n+1) = A101220(7, 0, n+1). Sequence in context: A175053 A274406 A261454 * A129659 A041130 A041307 Adjacent sequences:  A022095 A022096 A022097 * A022099 A022100 A022101 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 20 15:29 EDT 2019. Contains 328267 sequences. (Running on oeis4.)