OFFSET
0,1
COMMENTS
The second sequence of "less twisted numbers"; this sequence, A130750 and A130755 form a "suite en trio" (cf. reference, p. 130).
Sequence equals its third differences:
2.....5.....9....16....31....63...128...257...513..1024...
...3.....4.....7....15....32....65...129...256...511...
......1.....3.....8....17....33....64...127...255...
..........2.....5.....9....16....31....63...128...
REFERENCES
P. Curtz, Exercise Book, manuscript, 1995.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).
FORMULA
G.f.: (2 - x) / ((1 - 2*x)*(1 - x + x^2)).
a(0) = 2; a(1) = 5; a(2) = 9; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(n) = 2^(n+1) + A128834(n).
a(0) = 2; for n > 0, a(n) = 2*a(n-1) + A057079(n+1).
E.g.f.: 2*(sqrt(3)*exp(2*x) + sin(sqrt(3)*x/2)*exp(x/2))/sqrt(3). - Ilya Gutkovskiy, Jun 20 2016
a(n) = 2^(n+1) + (2*sin((Pi*n)/3))/sqrt(3). - Colin Barker, Jan 20 2017
MATHEMATICA
a[n_] := 2^(n+1) + 2*Sin[n*Pi/3]/Sqrt[3]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 13 2012 *)
LinearRecurrence[{3, -3, 2}, {2, 5, 9}, 40] (* Harvey P. Dale, Jun 21 2017 *)
PROG
(Magma) m:=31; S:=[ [2, 3, 1][(n-1) mod 3 +1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; /* Klaus Brockhaus, Aug 03 2007 */
(PARI) {m=31; v=vector(m); v[1]=2; v[2]=5; v[3]=9; for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} \\ Klaus Brockhaus, Aug 03 2007
(PARI) {for(n=0, 30, print1(2^(n+1)+[0, 1, 1, 0, -1, -1][n%6+1], ", "))} \\ Klaus Brockhaus, Aug 03 2007
(PARI) Vec((2-x) / ((1-2*x)*(1-x+x^2)) + O(x^40)) \\ Colin Barker, Jan 20 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Jul 13 2007
EXTENSIONS
Edited and extended by Klaus Brockhaus, Aug 03 2007
STATUS
approved