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A095151 a(n+3) = 3*a(n+2)-2*a(n+1)+1 with a(0)=0, a(1)=2. 12
0, 2, 7, 18, 41, 88, 183, 374, 757, 1524, 3059, 6130, 12273, 24560, 49135, 98286, 196589, 393196, 786411, 1572842, 3145705, 6291432, 12582887, 25165798, 50331621, 100663268, 201326563, 402653154, 805306337, 1610612704, 3221225439 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A sequence generated from a Bell difference row matrix, companion to A095150.

A095150 uses the same recursion rule but the multiplier [1 1 1] instead of [1 0 0].

For n>0, (a(n)) is row 2 of the convolution array A213568. - Clark Kimberling, Jun 20 2012

For n>0, (a(n)) is row 2 of the convolution array A213568. - Clark Kimberling, Jun 20 2012

LINKS

Table of n, a(n) for n=0..30.

FORMULA

Let M = a 3 X 3 matrix having Bell triangle difference terms (A095149 is composed of differences of the Bell triangle A011971): (fill in the 3 X 3 matrix with zeros): [1 0 0 / 1 1 0 / 2 1 2] = M. Then M^n * [1 0 0] = [1 n a(n)].

a(n) = 3*2^n-n-3 = 2*a(n-1)+n+1 = A000295(n+2)-A000079(n). For n>0, a(n)= A077802(n). - Henry Bottomley, Oct 25 2004

a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). G.f.: x*(2-x)/((1-x)^2*(1-2*x)). - Colin Barker, Apr 23 2012

a(n) = A125128(n) + A000225(n). - Miquel Cerda, Aug 07 2016

a(n) = 2*A125128(n) - A000325(n) + 1. - Miquel Cerda, Aug 12 2016

a(n) = A125128(n) + A000325(n) + n - 1. - Miquel Cerda, Aug 27 2016

EXAMPLE

a(6) = 183 = 3*88 -2*41 + 1.

a(4) = 41 since M^4 * [1 0 0] = [1 4 41].

MAPLE

a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=2*a[n-1]+n od: seq(a[n], n=1..31); # Zerinvary Lajos, Feb 22 2008

MATHEMATICA

a[n_] := (MatrixPower[{{1, 0, 0}, {1, 1, 0}, {2, 1, 2}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 30}] (* Robert G. Wilson v, Jun 05 2004 *)

CROSSREFS

Cf. A095149, A095150, A011971, A000110.

Sequence in context: A192955 A055503 A077802 * A147611 A007991 A037294

Adjacent sequences:  A095148 A095149 A095150 * A095152 A095153 A095154

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson, May 30 2004

EXTENSIONS

Edited by Robert G. Wilson v, Jun 05 2004

STATUS

approved

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Last modified July 27 04:25 EDT 2017. Contains 289841 sequences.