A186423 Partial sums of A186421.

0, 1, 3, 4, 8, 11, 17, 20, 28, 33, 43, 48, 60, 67, 81, 88, 104, 113, 131, 140, 160, 171, 193, 204, 228, 241, 267, 280, 308, 323, 353, 368, 400, 417, 451, 468, 504, 523, 561, 580, 620, 641, 683, 704, 748, 771, 817, 840, 888, 913, 963, 988, 1040, 1067, 1121, 1148, 1204, 1233, 1291, 1320

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

A062717 is the subsequence of even terms.

A186424 is the subsequence of odd terms.

Sequence in context: A051207 A217755 A133363 * A156056 A236453 A099108

Adjacent sequences: A186420 A186421 A186422 * A186424 A186425 A186426

Keyword

nonn,easy

Author

Reinhard Zumkeller, Feb 21 2011

Extensions

More terms added by G. C. Greubel, Oct 09 2019

Status

approved