A024068 a(n) = 6^n - n^6.

1, 5, -28, -513, -2800, -7849, 0, 162287, 1417472, 9546255, 59466176, 361025495, 2173796352, 13055867207, 78356634560, 470173593951, 2821093130240, 16926635307167, 101559922656192, 609359692964615, 3656158376062976, 21936950554611735, 131621703728887232, 789730222905566927

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

6^n in the formula can be removed (for example) with the following Maple code: "with(gfun): rec1:={u1(0)=1,u1(n+1)=6*u1(n)}: rec2:={u2(n)=n^6}: poltorec(u1(n)-u2(n),[rec1,rec2],u1(n),u2(n)],a(n));". This yields a polynomial recurrence: {a(n+1)-5*n^6+6*n^5+15*n^4+20*n^3+15*n^2-6*a(n)+6*n+1, a(0) = 1} that can further be transformed into a linear recurrence with constant coefficients. - Georg Fischer, Feb 23 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (13,-63,161,-245,231,-133,43,-6).

Crossrefs

Cf. A024012, A024026, A058794, A024040, A024054. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

Programs

Magma

[6^n-n^6: n in [0..30]]; // Vincenzo Librandi, May 14 2011

Mathematica

Table[6^n-n^6,{n,0,20}] (* Harvey P. Dale, Jan 30 2019 *)

Formula

From Chai Wah Wu, Jan 26 2020: (Start)

a(n) = 13*a(n-1) - 63*a(n-2) + 161*a(n-3) - 245*a(n-4) + 231*a(n-5) - 133*a(n-6) + 43*a(n-7) - 6*a(n-8) for n > 7.

G.f.: (5*x^7 + 348*x^6 + 1734*x^5 + 1545*x^4 + 5*x^3 - 30*x^2 - 8*x + 1)/((x - 1)^7*(6*x - 1)). (End)

Extensions

More terms from Georg Fischer, Feb 23 2021