A123866 a(n) = n^6 - 1.

0, 63, 728, 4095, 15624, 46655, 117648, 262143, 531440, 999999, 1771560, 2985983, 4826808, 7529535, 11390624, 16777215, 24137568, 34012223, 47045880, 63999999, 85766120, 113379903, 148035888, 191102975, 244140624, 308915775, 387420488

Offset

1,2

Comments

a(n) mod 7 = 0 iff n mod 7 > 0: a(A008589(n))=6; a(A047304(n)) = 0; a(n) mod 7 = 6*(1-A082784(n)).

a(n) = A005563(n-1)*A059826(n) = A068601(n)*A001093(n). - Reinhard Zumkeller, Feb 02 2007

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

G.f.: x^2*(63 + 287*x + 322*x^2 + 42*x^3 + 7*x^4 - x^5)/(1-x)^7. - Colin Barker, May 08 2012

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(1)=0, a(2)=63, a(3)=728, a(4)=4095, a(5)=15624, a(6)=46655, a(7)=117648. - Harvey P. Dale, Nov 18 2012

Sum_{n>=2} 1/a(n) = 11/12 - Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/6. - Vaclav Kotesovec, Feb 14 2015

E.g.f.: 1 + (-1 + x + 31*x^2 + 90*x^3 + 65*x^4 + 15*x^5 + x^6)*exp(x). - G. C. Greubel, Aug 08 2019

Product_{n>=2} (1 + 1/a(n)) = 6*Pi^2*sech(sqrt(3)*Pi/2)^2. - Amiram Eldar, Jan 20 2021

Maple

A123866:=n->n^6 - 1; seq(A123866(n), n=1..40); # Wesley Ivan Hurt, Feb 26 2014

Mathematica

Table[n^6-1, {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 63, 728, 4095, 15624, 46655, 117648}, 30] (* Harvey P. Dale, Nov 18 2012 *)

Prog

(Magma) [n^6 - 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011
(Maxima) A123866(n):=n^6 -1 $ makelist(A123866(n), n, 1, 30); /* Martin Ettl, Nov 05 2012 */
(Haskell)
a123866 = (subtract 1) . (^ 6)  -- Reinhard Zumkeller, Mar 11 2014
(PARI) a(n)=n^6-1 \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n^6 -1 for n in (1..30)] # G. C. Greubel, Aug 08 2019
(GAP) List([1..30], n-> n^6 -1); # G. C. Greubel, Aug 08 2019

Crossrefs

Cf. A024004, A001014, A005563, A123865, A123867, A123868.

Sequence in context: A343978 A284953 A069091 * A024004 A284927 A321545

Adjacent sequences: A123863 A123864 A123865 * A123867 A123868 A123869

Keyword

nonn,easy

Author

Reinhard Zumkeller, Oct 16 2006

Status

approved