OFFSET
0,2
COMMENTS
This sequence is one of 104 sequences mentioned in the Lang's paper; see page 4. - Omar E. Pol, Jun 13 2012
Also 1 plus the total number of toothpicks of the first n toothpick structures of A139250 in which the number of exposed toothpicks that are orthogonals to the initial toothpick is equal to 4. - Omar E. Pol, Jun 16 2012
This is the sequence A(1,4;5,-4;-1,n) of the family of sequences [a,b:c,d:k] considered by Gary Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Nov 16 2013
LINKS
Hacène Belbachir and El-Mehdi Mehiri, Enumerating moves in the optimal solution of the Tower of Hanoi, arXiv:2210.08657 [math.CO], 2022.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).
FORMULA
a(n) = (3n + 1 + 2^(2n+3))/9. - Emeric Deutsch, Jun 20 2009
G.f.: ( -1+2*x ) / ( (-1+4*x)*(x-1)^2 ). - R. J. Mathar, Jun 28 2012
From Wolfdieter Lang, Nov 16 2013: (Start)
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3), n >= 2, a(-1)=0, a(0)=1, a(1)=4.
a(n) = 5*a(n-1) - 4*a(n-2) -1, n >= 2, a(0)=1, a(1)=4. (End)
a(n) = A034299(2*n). - Michael Somos, Oct 16 2020
EXAMPLE
G.f. = 1 + 4*x + 15*x^2 + 58*x^3 + 229*x^4 + 912*x^5 + 3643*x^6 + ... - Michael Somos, Oct 16 2020
MAPLE
a := proc (n) options operator, arrow: (1/3)*n+1/9+(1/9)*2^(2*n+3) end proc: seq(a(n), n = 0 .. 25); # Emeric Deutsch, Jun 20 2009
MATHEMATICA
LinearRecurrence[{6, -9, 4}, {1, 4, 15}, 30] (* Harvey P. Dale, Oct 04 2018 *)
PROG
(PARI) {a(n) = (2^(2*n + 3) + 3*n + 1)/9}; /* Michael Somos, Oct 16 2020 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, May 27 2009
EXTENSIONS
More terms from Emeric Deutsch, Jun 20 2009
STATUS
approved