A011906 If b(n) is A011900(n) and c(n) is A001109(n), then a(n) = b(n)*c(n) = b(n) + (b(n)+1) + (b(n)+2) + ... + c(n).

1, 18, 525, 17340, 586177, 19896030, 675781821, 22956120408, 779829016225, 26491211221770, 899921240562957, 30570830315362260, 1038508305678375841, 35278711540581704598, 1198437683944896688125, 40711602541832856049200, 1382996048733983114022337

Offset: 1

Author

Mario Velucchi (mathchess(AT)velucchi.it)

Keywords

• nonn
• easy

Examples

			a(3) = 525 = 15*35 = 15 + 16 + ... + 35.
		

References

• Mario Velucchi "From the desk of ... Mario Velucchi" in 'Mathematics and Informatics quarterly' volume 7 - 2/1997, p. 81.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..650
• Index entries for linear recurrences with constant coefficients, signature (41,-246,246,-41,1).

Crossrefs

Cf. A001109, A011900, A029547, A091761.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1-23*x+33*x^2-3*x^3)/((1-x)*(1-34*x+x^2)*(1-6*x+x^2)) )); // G. C. Greubel, Oct 18 2024

Maple

A011900 := proc(n) coeftayl( (1-4*x+x^2)/((1-x)*(1-6*x+x^2)),x=0,n) ; end proc: A001109 := proc(n) coeftayl( x/(1-6*x+x^2),x=0,n) ; end proc: A011906 := proc(n) A001109(n)*A011900(n-1) ; end proc: seq(A011906(n),n=1..30) ; # R. J. Mathar, Apr 15 2010

Mathematica

LinearRecurrence[{41, -246, 246, -41, 1}, {1, 18, 525, 17340, 586177}, 20] (* Paul Cleary, Dec 05 2015 *)
CoefficientList[Series[(-1 + 23*x - 33*x^2 + 3*x^3)/((x - 1)*(x^2 - 34*x + 1)*(1 - 6*x + x^2)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Sep 16 2017 *)

SageMath

def A011906_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-23*x+33*x^2-3*x^3)/((1-x)*(1-34*x+x^2)*(1-6*x+x^2)) ).list()
a=A011906_list(30); a[1:] # G. C. Greubel, Oct 18 2024

Formula

From R. J. Mathar, Apr 15 2010: (Start)

G.f.: x*(1-23*x+33*x^2-3*x^3)/((1-x)*(1-34*x+x^2)*(1-6*x+x^2)).

a(n) = 41*a(n-1) -246*a(n-2) +246*a(n-3) -41*a(n-4) +a(n-5). (End)

Lim_{n -> infinity} a(n)/a(n-1) = A156164. - César Aguilera, Jul 17 2020

a(n) = (1/16)*(1 - A029547(n) + 41*A091761(n) + 8*A001109(n)). - G. C. Greubel, Oct 18 2024

Extensions

More terms from R. J. Mathar, Apr 15 2010