OFFSET
0,5
COMMENTS
This sequence is similar to A005246 whose recursion is a(n) = (a(n-1)*a(n-2) + 1) / a(n-3). - Michael Somos, Feb 10 2017
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..4411
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. See (better) also. See Example 6.2.18.
Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-1).
FORMULA
G.f.: (1 + x + x^2 - 4*x^3 - 3*x^4 - 2*x^5) / (1 - 5*x^3 + x^6).
a(n) = a(3-n) for all n in Z.
a(n+3) + a(n-3) = 5*a(n) for all n in Z.
a(n+1) + a(n-1) = a(n) * (2 + [n mod 3 == 0]) for all n in Z.
a(n+3k)+a(n-3k) = A003501(k)*a(n) for n>=3k. - Bruno Berselli, Mar 25 2013
EXAMPLE
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 9*x^7 + 14*x^8 + 19*x^9 + ...
MATHEMATICA
a[ n_] := With[{m = If [n < 0, 3 - n, n]}, SeriesCoefficient[ (1 + x + x^2 - 4 x^3 - 3 x^4 - 2 x^5) / (1 - 5 x^3 + x^6), {x, 0, m}]]; (* Michael Somos, Jan 18 2015 *)
LinearRecurrence[{0, 0, 5, 0, 0, -1}, {1, 1, 1, 1, 2, 3}, 40] (* Harvey P. Dale, Nov 20 2016 *)
PROG
(PARI) {a(n) = if( n<0, n = 3-n); polcoeff( (1 + x + x^2 - 4*x^3 - 3*x^4 - 2*x^5) / (1 - 5*x^3 + x^6) + x * O(x^n), n)};
(Magma) [n le 3 select 1 else (Self(n)*Self(n-2)+1)/Self(n-3): n in [0..40]]; // Bruno Berselli, Mar 25 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Mar 25 2013
STATUS
approved