OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).
FORMULA
From R. J. Mathar, Feb 28 2011: (Start)
G.f.: x*(1 + 2*x + 2*x^3 + x^4)/( (1+x^2)*(1+x)^2*(1-x)^3 ).
a(n) = (6*n*(n+1) + 3 + (-1)^n*(2*n+1) - 4*A087960(n))/16. (End)
E.g.f.: ((2 + 5*x + 3*x^2)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x) + 2*sin(x) - 2*cos(x))/8. - G. C. Greubel, Oct 09 2019
MAPLE
A087960 := proc(n) op((n mod 4)+1, [1, -1, -1, 1]) ; end proc:
A186423 := proc(n) 3*n*(n+1)/8 +3/16 +(-1)^n*(2*n+1)/16 -A087960(n)/4 ; end proc: # R. J. Mathar, Feb 28 2011
MATHEMATICA
CoefficientList[Series[x(1+2x+2x^3+x^4)/((1-x)^3(1+x)^2(1+x^2)), {x, 0, 65}], x] (* Harvey P. Dale, Mar 13 2011 *)
Table[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial[n+1, 2])/16, {n, 0, 65}] (* G. C. Greubel, Oct 09 2019 *)
PROG
(Haskell)
a186423 n = a186423_list !! n
a186423_list = scanl1 (+) a186421_list
(PARI) vector(66, n, my(m=n-1); (6*m^2 +6*m +3 +(-1)^m*(2*m+1) -4*(-1)^binomial(m+1, 2))/16) \\ G. C. Greubel, Oct 09 2019
(Magma) [(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial(n+1, 2))/16: n in [0..65]]; // G. C. Greubel, Oct 09 2019
(Sage) [(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^binomial(n+1, 2))/16 for n in (0..65)] # G. C. Greubel, Oct 09 2019
(GAP) List([0..65], n-> (6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial(n+1, 2))/16); # G. C. Greubel, Oct 09 2019
(Python)
def A186423(n): return (6*n*(n+1)+3+(-2*n-1 if n&1 else 2*n+1)+(4 if n+1&2 else -4))>>4 # Chai Wah Wu, Jan 31 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Feb 21 2011
EXTENSIONS
More terms added by G. C. Greubel, Oct 09 2019
STATUS
approved