%I #39 Oct 18 2022 03:31:22
%S 1,4,15,58,229,912,3643,14566,58257,233020,932071,3728274,14913085,
%T 59652328,238609299,954437182,3817748713,15270994836,61083979327,
%U 244335917290,977343669141,3909374676544,15637498706155,62549994824598
%N Partial sums of A007583.
%C This sequence is one of 104 sequences mentioned in the Lang's paper; see page 4. - _Omar E. Pol_, Jun 13 2012
%C 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
%C 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
%H Hacène Belbachir and El-Mehdi Mehiri, <a href="https://arxiv.org/abs/2210.08657">Enumerating moves in the optimal solution of the Tower of Hanoi</a>, arXiv:2210.08657 [math.CO], 2022.
%H Wolfdieter Lang, <a href="/A160156/a160156.pdf">Notes on certain inhomogeneous three term recurrences.</a>
%H N. J. A. Sloane, <a href="http://oeis.org/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,4).
%F a(n) = (3n + 1 + 2^(2n+3))/9. - _Emeric Deutsch_, Jun 20 2009
%F G.f.: ( -1+2*x ) / ( (-1+4*x)*(x-1)^2 ). - _R. J. Mathar_, Jun 28 2012
%F From _Wolfdieter Lang_, Nov 16 2013: (Start)
%F 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.
%F a(n) = 5*a(n-1) - 4*a(n-2) -1, n >= 2, a(0)=1, a(1)=4. (End)
%F a(n) = A034299(2*n). - _Michael Somos_, Oct 16 2020
%e 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
%p 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
%t LinearRecurrence[{6,-9,4},{1,4,15},30] (* _Harvey P. Dale_, Oct 04 2018 *)
%o (PARI) {a(n) = (2^(2*n + 3) + 3*n + 1)/9}; /* _Michael Somos_, Oct 16 2020 */
%Y Cf. A002450, A007583, A034299, A139250.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, May 27 2009
%E More terms from _Emeric Deutsch_, Jun 20 2009