OFFSET
0,1
COMMENTS
For n = 0, 1, 2, 3, 4, 5, 6, 7, ...,
a(n-1) = -1, 7, 25, 26, 44, 133, 187, 125, ...
+ A177049(n) = 1, 5, 14, 55, 91, 68, 95, 253, ...
gives 0, 12, 39, 81, 135, 201, 282, 378, ...
which are increasing multiples of 3.
a(n) mod 9 = period 4: repeat 7,7,8,8.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).
FORMULA
G.f.: (7 +4*x -7*x^2 +46*x^3 -9*x^4 +8*x^5 +4*x^6 +2*x^7 -x^8)/((1-x)^3 * (1 + x^2)^3 ). - R. J. Mathar, Jan 16 2011
From G. C. Greubel, Feb 26 2020: (Start)
a(n) = (6 + i^n*(1 - i + (-1)^n*(1 + i)))*(9*n^2 + 27*n + 14)/16.
E.g.f.: ( 3*(14+36*x+9*x^2)*exp(x) + (14+36*x-9*x^2)*cos(x) + (14-36*x-9*x^2)*sin(x) )/8. (End)
Sum_{n>=0} 1/a(n) = 1 - (3 + 4*sqrt(3))*Pi/45. - Amiram Eldar, Aug 12 2022
MAPLE
m:=50; S:=series((7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 26 2020
MATHEMATICA
LinearRecurrence[{3, -6, 10, -12, 12, -10, 6, -3, 1}, {7, 25, 26, 44, 133, 187, 125, 161, 403}, 50] (* Harvey P. Dale, May 21 2015 *)
PROG
(PARI) Vec( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) +O('x^50) ) \\ G. C. Greubel, Feb 26 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) )); // G. C. Greubel, Feb 26 2020
(Sage)
def A178370_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (7+4*x-7*x^2+46*x^3-9*x^4+8*x^5+4*x^6+2*x^7-x^8)/((1-x)^3*(1 + x^2)^3 ) ).list()
A178370_list(50) # G. C. Greubel, Feb 26 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Dec 21 2010
EXTENSIONS
More terms from Jinyuan Wang, Feb 26 2020
STATUS
approved