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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

REFERENCES

D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998, p. 4.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..3165 (first 501 terms from T. D. Noe)

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.

David Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), pp. 40-42.

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 two-way infinite sequences

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[0] == a[1] == a[2] == a[3] == 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

David Johnson-Davies

EXTENSIONS

More terms from Michael Somos

STATUS

approved

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Last modified November 19 13:16 EST 2018. Contains 317351 sequences. (Running on oeis4.)