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

1,2

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2).

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

Example

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

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 *)

Prog

(PARI) Vec(x*(98*x+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Mar 11 2014
(Magma) [100*2^(n-1)-99 : n in [1..30]]; // Wesley Ivan Hurt, Aug 16 2016

Crossrefs

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

Sequence in context: A142530 A033241 A140021 * A195294 A142578 A256048

Adjacent sequences: A139698 A139699 A139700 * A139702 A139703 A139704

Keyword

nonn,easy

Author

Gary W. Adamson, Apr 29 2008

Extensions

More terms from Emeric Deutsch, May 03 2008

More terms from Colin Barker, Mar 11 2014

Status

approved