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 A286311 a(n) = 2*a(n-1) - a(n-2) + a(n-4), n>3, a(0)=0, a(1)=a(2)=1, a(3)=3. 3
 0, 1, 1, 3, 5, 8, 12, 19, 31, 51, 83, 134, 216, 349, 565, 915, 1481, 2396, 3876, 6271, 10147, 16419, 26567, 42986, 69552, 112537, 182089, 294627, 476717, 771344, 1248060, 2019403, 3267463, 5286867, 8554331, 13841198, 22395528, 36236725, 58632253, 94868979 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Difference table for a(n): 0, 1, 1, 3, 5, 8, 12, 19, 31, 51, 83, 134, 216, ... 1, 0, 2, 2, 3, 4, 7, 12, 20, 32, 51, 82, 133, ... -1, 2, 0, 1, 1, 3, 5, 8, 12, 19, 31, 51, 83, ... 3, -2, 1, 0, 2, 2, 3, 4, 7, 12, 20, 32, 51, ... etc. The pair a(n) = 0, 1, 1, 3, 5, 8, 12, 19, 31, 51, ... and b(n) = 0, 2, 2, 3, 4, 7, 12, 20, 32, 51, ... is interesting. a(n) and b(n) are autosequences of the first kind (see Link). a(n) and b(n) have the same first trisection: 3*A001076(n). a(n) + b(n) = A022086(n) = 3*A000045(n) (Fibonacci). b(n) - a(n) = 0, 1, 1, 0, -1, -1, 0, ... = A128834(n). a(n+6) - a(n) = b(n+6) - b(n) = 6*Fib(n+3). a(n) - a(n) mod 9 = 9*A004699(n) = b(n) - b(n) mod 9. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 OEIS Wiki, Autosequence Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1). FORMULA a(n) = 2*a(n-1) - a(n-2) + a(n-4). Valid for b(n). G.f.: x*(1 - x + 2*x^2) / ((1 - x + x^2)*(1 - x - x^2)). - Colin Barker, May 06 2017 MATHEMATICA LinearRecurrence[{2, -1, 0, 1}, {0, 1, 1, 3}, 40] (* or *) CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x + x^2) (1 - x - x^2)), {x, 0, 39}], x] (* Michael De Vlieger, May 07 2017 *) PROG (PARI) concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x + x^2)*(1 - x - x^2)) + O(x^60))) \\ Colin Barker, May 06 2017 (Magma) I:=[0, 1, 1, 3]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018 CROSSREFS Cf. A000034, A000045, A001076, A004699, A022086, A128834. Sequence in context: A227635 A295058 A004398 * A256057 A055606 A295832 Adjacent sequences: A286308 A286309 A286310 * A286312 A286313 A286314 KEYWORD nonn,easy AUTHOR Paul Curtz, May 06 2017 EXTENSIONS More terms from Colin Barker, May 06 2017 STATUS approved

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Last modified April 14 05:31 EDT 2024. Contains 371655 sequences. (Running on oeis4.)