A113618 a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7.

1, 36, 1793, 24604, 167481, 756836, 2620201, 7526268, 18831569, 42374116, 87654321, 169343516, 309160393, 538155684, 899445401, 1451432956, 2271560481, 3460629668, 5147732449, 7495831836, 10708033241, 15034586596, 20780659593

Offset: 0

Author

Jonathan Vos Post, Jan 14 2006

Keywords

• easy
• nonn

Comments

1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*n^6 + 8*n^7 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 = (x^9 - 1)/(x-1).

Examples

			1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 = 18831569 = 173 * 199 * 547.
1 + 2*26 + 3*26^2 + 4*26^3 + 5*26^4 + 6*26^5 + 7*26^6 + 8*26^7 = 66490537361 is prime, the smallest prime in the sequence.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Crossrefs

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

Programs

Magma

[1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7: n in [1..43]] // Vincenzo Librandi, Dec 21 2010

Mathematica

Join[{1},Table[Total[Table[p*n^(p-1),{p,8}]],{n,30}]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,36,1793,24604,167481,756836,2620201,7526268},30] (* Harvey P. Dale, Jul 16 2014 *)

Formula

G.f.: (1+28*x+1533*x^2+11212*x^3+18907*x^4+7956*x^5+679*x^6+4*x^7)/(x-1)^8. - R. J. Mathar, Dec 21 2010

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=1, a(1)=36, a(2)=1793, a(3)=24604, a(4)=167481, a(5)=756836, a(6)=2620201, a(7)=7526268. - Harvey P. Dale, Jul 16 2014