A160156 Partial sums of A007583.

1, 4, 15, 58, 229, 912, 3643, 14566, 58257, 233020, 932071, 3728274, 14913085, 59652328, 238609299, 954437182, 3817748713, 15270994836, 61083979327, 244335917290, 977343669141, 3909374676544, 15637498706155, 62549994824598

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

Table of n, a(n) for n=0..23.

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 sequences related to toothpick sequences

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

Cf. A002450, A007583, A034299, A139250.

Sequence in context: A164589 A017950 A003126 * A102052 A128714 A007342

Adjacent sequences: A160153 A160154 A160155 * A160157 A160158 A160159

Keyword

easy,nonn

Author

Omar E. Pol, May 27 2009

Extensions

More terms from Emeric Deutsch, Jun 20 2009

Status

approved