A081685 a(n) = 6^n - 5^n - 4^n - 3^n + 3*2^n.

1, 0, -2, 24, 382, 3480, 26398, 183624, 1217662, 7844280, 49595998, 309603624, 1915345342, 11771312280, 71987479198, 438579414024, 2664184199422, 16146411375480, 97676153291998, 590010215086824, 3559688013155902, 21455704981601880, 129219894496730398, 777738831236334024

Offset: 0

Author

Paul Barry, Mar 30 2003

Keywords

• easy
• sign
• changed

Comments

Original name was: A sum of decreasing powers.

Links

• Zhuorui He, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (20,-155,580,-1044,720).

Crossrefs

Binomial transform of A081684.

Programs

Mathematica

Table[6^n-5^n-4^n-3^n+3*2^n,{n,0,30}] (* or *) LinearRecurrence[{20,-155,580,-1044,720},{1,0,-2,24,382},30] (* Harvey P. Dale, Sep 15 2014 *)

Formula

a(n) = 6^n - 5^n - 4^n - 3^n + 3*2^n.

G.f.:(-1-636*x^4+516*x^3-153*x^2+20*x)/((6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)) [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009]

a(0)=1, a(1)=0, a(2)=-2, a(3)=24, a(4)=382, a(n) = 20*a(n-1) - 155*a(n-2) + 580*a(n-3) - 1044*a(n-4) + 720*a(n-5). - Harvey P. Dale, Sep 15 2014

E.g.f.: exp(2*x)*(exp(4*x) - exp(3*x) - exp(2*x) - exp(x) + 3). - Elmo R. Oliveira, Sep 12 2024

Extensions

a(23) from Elmo R. Oliveira, Sep 12 2024

New name from formula from Zhuorui He, Sep 05 2025