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 A048736 Dana Scott's sequence: a(n) = (a(n-2) + a(n-1) * a(n-3)) / a(n-4), a(0) = a(1) = a(2) = a(3) = 1. 28
 1, 1, 1, 1, 2, 3, 5, 13, 22, 41, 111, 191, 361, 982, 1693, 3205, 8723, 15042, 28481, 77521, 133681, 253121, 688962, 1188083, 2249605, 6123133, 10559062, 19993321, 54419231, 93843471, 177690281, 483649942, 834032173, 1579219205, 4298430243, 7412446082, 14035282561, 38202222241, 65877982561 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The recursion has the Laurent property. If a(0), ..., a(3) are variables, then a(n) is a Laurent polynomial (a rational function with a monic monomial denominator). - Michael Somos, Feb 05 2012 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..3165 (first 501 terms from T. D. Noe) Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633. Hal Canary, The Dana Scott Recurrence [From Jaume Oliver Lafont, Sep 25 2009] S. Fomin and A. Zelevinsky, The Laurent phenomenon, arXiv:math/0104241 [math.CO], 2001. S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144. David Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), pp. 40-42. D. Gale, Tracking the Automatic Ant And Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998, p. 4. Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016. Eric Weisstein's World of Mathematics, Laurent Polynomial Index entries for linear recurrences with constant coefficients, signature (0,0,10,0,0,-10,0,0,1) FORMULA a(n) = 9*a(n-3) - a(n-6) - 3 - ( ceiling(n/3) - floor(n/3) ), with a(0) = a(1) = a(2) = a(3) = 1, a(4) = 2, a(5) = 3. - Michael Somos From Jaume Oliver Lafont, Sep 17 2009: (Start) a(n) = 10*a(n-3) - 10*a(n-6) + a(n-9). G.f.: (1 + x + x^2 - 9*x^3 - 8*x^4 - 7*x^5 + 5*x^6 + 3*x^7 + 2*x^8)/(1 - 10*x^3 + 10*x^6 - x^9)). (End) a(n) = a(3-n) for all n in Z. - Michael Somos, Feb 05 2012 EXAMPLE G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 13*x^6 + 22*x^7 + 41*x^8 + 111*x^9 + ... MATHEMATICA RecurrenceTable[{a == a == a == a == 1, a[n] == (a[n - 2] + a[n - 1]a[n - 3])/a[n - 4]}, a[n], {n, 40}] (* or *) LinearRecurrence[{0, 0, 10, 0, 0, -10, 0, 0, 1}, {1, 1, 1, 1, 2, 3, 5, 13, 22}, 41] (* Harvey P. Dale, Oct 22 2011 *) PROG (Haskell) a048736 n = a048736_list !! n a048736_list = 1 : 1 : 1 : 1 :    zipWith div      (zipWith (+)        (zipWith (*) (drop 3 a048736_list)                     (drop 1 a048736_list))        (drop 2 a048736_list))      a048736_list -- Reinhard Zumkeller, Jun 26 2011 (PARI) Vec((1+x+x^2-9*x^3-8*x^4-7*x^5+5*x^6+3*x^7+2*x^8) / (1-10*x^3+10*x^6-x^9)+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011 (MAGMA) I:=[1, 1, 1, 1]; [n le 4 select I[n] else (Self(n-2) + Self(n-1)*Self(n-3)) / Self(n-4): n in [1..30]]; // G. C. Greubel, Feb 20 2018 CROSSREFS Cf. A006720, A006721, A006722, A006723, A092420, A072881. Cf. A192241, A192242 (primes and where they occur). Cf. A276531. Sequence in context: A177374 A142881 A163159 * A235621 A193300 A215310 Adjacent sequences:  A048733 A048734 A048735 * A048737 A048738 A048739 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Michael Somos STATUS approved

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Last modified October 23 13:38 EDT 2019. Contains 328345 sequences. (Running on oeis4.)