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
G. C. Greubel, Table of n, a(n) for n = 0..1000
Péter Burcsi, Gabriele Fici, Zsuzsanna Lipták, Rajeev Raman, Joe Sawada, Generating a Gray code for prefix normal words in amortized polylogarithmic time per word, arXiv:2003.03222 [cs.DS], 2020.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
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
From Colin Barker, Apr 23 2012: (Start)
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)). (End)
E.g.f.: 3*exp(2*x) - (3+x)*exp(x). - G. C. Greubel, Jul 26 2019
Let Prod_{i=0..n-1} (1+x^{2^i}+x^{2*2^i}) =
Sum_{j=0..d} b_j x^j, where d=2^{n+1}-2. Then
a(n)=Sum_{j=0..d-1} b_j/b_{j+1} (proved). - R. P. Stanley, Aug 27 2019
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 *)
Table[3*2^n -(n+3), {n, 0, 30}] (* G. C. Greubel, Jul 26 2019 *)
PROG
(PARI) vector(30, n, n--; 3*2^n -(n+3)) \\ G. C. Greubel, Jul 26 2019
(Magma) [3*2^n -(n+3): n in [0..30]]; // G. C. Greubel, Jul 26 2019
(Sage) [3*2^n -(n+3) for n in (0..30)] # G. C. Greubel, Jul 26 2019
(GAP) List([0..30], n-> 3*2^n -(n+3)); # G. C. Greubel, Jul 26 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, May 30 2004
EXTENSIONS
Edited by Robert G. Wilson v, Jun 05 2004
STATUS
approved