A113531 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.

1, 21, 321, 2005, 7737, 22461, 54121, 114381, 219345, 390277, 654321, 1045221, 1604041, 2379885, 3430617, 4823581, 6636321, 8957301, 11886625, 15536757, 20033241, 25515421, 32137161, 40067565, 49491697, 60611301, 73645521

Offset: 0

Author

Jonathan Vos Post, Jan 12 2006

Keywords

• easy
• nonn

Comments

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 = (n^7 - 1)/(n-1). a(6) = 1 + 2*6 + 3*6^2 + 4*6^3 + 5*6^4 + 6*6^5 = 54121 is prime, the smallest prime in the sequence. The next is a(a(1)) = a(21) = 1 + 2*21 + 3*21^2 + 4*21^3 + 5*21^4 + 6*21^5 = 25515421. Then a(24) = 49491697.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Crossrefs

Cf. A000012, A005408, A056109, A056578, A056579.

Programs

Mathematica

With[{eq=Total[# n^(#-1)&/@Range[6]]},Table[eq,{n,0,30}]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,21,321,2005,7737,22461},30] (* Harvey P. Dale, Nov 02 2011 *)

PARI

for(n=0,50, print1(1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5, ", ")) \\ G. C. Greubel, Mar 15 2017

Formula

a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.

O.g.f.: 2064/(-1+x)^4+3/(-1+x)+2040/(-1+x)^5+132/(-1+x)^2+720/(-1+x)^6+872/(-1+x)^3 . - R. J. Mathar, Feb 26 2008

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), a(0)=1, a(1)=21, a(2)=321, a(3)=2005, a(4)=7737, a(5)=22461. - Harvey P. Dale, Nov 02 2011

Extensions

Corrected by R. J. Mathar, Feb 26 2008