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
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).
Programs
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GAP
List([1..30], n-> n^6 -1); # G. C. Greubel, Aug 08 2019
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Haskell
a123866 = (subtract 1) . (^ 6) -- Reinhard Zumkeller, Mar 11 2014
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Magma
[n^6 - 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011
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Maple
A123866:=n->n^6 - 1; seq(A123866(n), n=1..40); # Wesley Ivan Hurt, Feb 26 2014
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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 *) -
Maxima
A123866(n):=n^6 -1 $ makelist(A123866(n),n,1,30); /* Martin Ettl, Nov 05 2012 */
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PARI
a(n)=n^6-1 \\ Charles R Greathouse IV, Oct 07 2015
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Sage
[n^6 -1 for n in (1..30)] # G. C. Greubel, Aug 08 2019
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
Comments