login
A206625
Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) in powers of x.
1
0, 1, 1, 1, 2, 5, 5, 13, 16, 37, 45, 109, 130, 313, 377, 905, 1088, 2617, 3145, 7561, 9090, 21853, 26269, 63157, 75920, 182525, 219413, 527509, 634114, 1524529, 1832625, 4405969, 5296384, 12733489, 15306833, 36800465, 44237570, 106355317
OFFSET
0,5
COMMENTS
This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).
REFERENCES
J. A. Sjogren, Cycles and spanning trees. Math. Comput. Modelling 15, No.9, 87-102 (1991).
FORMULA
G.f.: x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8).
a(n) = a(-n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8) for all n in Z.
a(2*n + 5) = A071100(n). a(2*n + 6) = A071101(n). a(n + 3) = A112835(n). a(2*n) = A138573(n).
EXAMPLE
G.f. = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 13*x^7 + 16*x^8 + 37*x^9 + ...
MATHEMATICA
CoefficientList[Series[x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)
PROG
(PARI) {a(n) = my(m = abs(n)); polcoeff( x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) + x * O(x^m), m)};
(PARI) {a(n) = my(m = abs(n), v); v = polroots( Pol([ 1, 2, 4, 2, 1])); sqrtint( round( prod( k=1, 4, v[k]^m - 1, 2^(m%2) / 20)))};
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ))); // G. C. Greubel, Aug 12 2018
CROSSREFS
Cf. A071100 (bisection), A071101 (bisection), A112835, A138573.
Sequence in context: A124201 A100953 A112835 * A176168 A308770 A222114
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Feb 10 2012
STATUS
approved