A178370 The trisection A178242(3n+2).

7, 25, 26, 44, 133, 187, 125, 161, 403, 493, 296, 350, 817, 943, 539, 611, 1375, 1537, 854, 944, 2077, 2275, 1241, 1349, 2923, 3157, 1700, 1826, 3913, 4183, 2231, 2375, 5047, 5353, 2834, 2996, 6325, 6667, 3509, 3689, 7747, 8125, 4256, 4454, 9313, 9727, 5075, 5291

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

a(n) = A060819(2+3*n)*(A060819(7+3*n) + A176672(n+2))/2. - corrected by G. C. Greubel, Feb 26 2020

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

Cf. A060819, A176672, A177049.

Sequence in context: A242322 A249437 A246792 * A034125 A196013 A272366

Adjacent sequences: A178367 A178368 A178369 * A178371 A178372 A178373

Keyword

nonn,easy

Author

Paul Curtz, Dec 21 2010

Extensions

More terms from Jinyuan Wang, Feb 26 2020

Status

approved