A139701 Binomial transform of [1, 100, 100, 100, ...].

1, 101, 301, 701, 1501, 3101, 6301, 12701, 25501, 51101, 102301, 204701, 409501, 819101, 1638301, 3276701, 6553501, 13107101, 26214301, 52428701, 104857501, 209715101, 419430301, 838860701, 1677721501, 3355443101, 6710886301, 13421772701, 26843545501

Offset: 1

Author

Gary W. Adamson, Apr 29 2008

Keywords

• nonn
• easy

Comments

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 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(3) = 301 = (1, 2, 1) dot (1, 100, 100) = (1 + 200 + 100).
		

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. A007318, A099003, A139634, A139635, A139697, A139698, A139700, A139701.

Programs

Magma

[100*2^(n-1)-99 : n in [1..30]]; // Wesley Ivan Hurt, Aug 16 2016

Maple

a:=proc(n) options operator, arrow: 100*2^(n-1)-99 end proc: seq(a(n), n=1.. 30); # Emeric Deutsch, May 03 2008

Mathematica

100*2^(Range[30] - 1) - 99 (* Wesley Ivan Hurt, Aug 16 2016 *)
LinearRecurrence[{3, -2}, {1, 101}, 40] (* Vincenzo Librandi, Aug 17 2016 *)

PARI

Vec(x*(98*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014

Formula

A007318 * [1, 100, 100, 100, ...].

a(n) = 100*2^(n-1)-99. - Emeric Deutsch, May 03 2008

a(n) = 2*a(n-1)+99 for n > 1. [Vincenzo Librandi, Nov 24 2010]

a(n) = 3*a(n-1) - 2*a(n-2) for n > 2. G.f.: x*(98*x+1) / ((x-1)*(2*x-1)). - Colin Barker, Mar 11 2014

Extensions

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014