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

Author

Omar E. Pol, May 27 2009

Keywords

• easy
• nonn

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

Examples

			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
		

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 sequences related to toothpick sequences
• Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Crossrefs

Cf. A002450, A007583, A034299, A139250.

Programs

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 *)

PARI

{a(n) = (2^(2*n + 3) + 3*n + 1)/9}; /* Michael Somos, Oct 16 2020 */

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

Extensions

More terms from Emeric Deutsch, Jun 20 2009