A139635 Binomial transform of [1, 11, 11, 11, ...].

1, 12, 34, 78, 166, 342, 694, 1398, 2806, 5622, 11254, 22518, 45046, 90102, 180214, 360438, 720886, 1441782, 2883574, 5767158, 11534326, 23068662, 46137334, 92274678, 184549366, 369098742, 738197494, 1476394998, 2952790006, 5905580022, 11811160054

Offset: 1

Author

Gary W. Adamson, Apr 29 2008

Keywords

• nonn
• easy

Comments

A007318 * [1, 11, 11, 11, ...].

The binomial transform of [1, c, c, c, ...] has the terms a(n) = 1 - c + c*2^(n-1) if the offset 1 is chosen. The o.g.f. of the a(n) is x*(1+(c-2)*x)/((2x-1)*(x-1)). This applies to A139634 with c=10, to A139635 with c=11, to A139697 with c=12, to A139698 with c=25 and to A099003, A139700, A139701 accordingly. - R. J. Mathar, May 11 2008

Examples

			a(4) = 78 = (1, 3, 3, 1) dot (1, 11, 11, 11) = (1 + 33 + 33 + 11).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A139634.

Programs

Maple

seq(11*2^(n-1)-10,n=1.. 25); # Emeric Deutsch, May 03 2008

Mathematica

a=1; lst={a}; k=11; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 17 2008 *)
CoefficientList[Series[(9 x + 1)/((x - 1) (2 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 13 2014 *)
LinearRecurrence[{3,-2},{1,12},40] (* Harvey P. Dale, Oct 26 2015 *)

PARI

Vec(x*(9*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

Formula

a(n) = 11*2^(n-1) - 10. - Emeric Deutsch, May 03 2008

a(n) = 2*a(n-1) + 10, with n > 1, a(1)=1. - Vincenzo Librandi, Nov 24 2010

From Colin Barker, Mar 11 2014: (Start)

a(n) = 3*a(n-1) - 2*a(n-2).

G.f.: x*(9*x+1) / ((x-1)*(2*x-1)). (End)

Extensions

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014