A160128 - OEIS

%I #26 Feb 24 2021 02:48:18

%S 3,7,19,63,235,919,3651,14575,58267,233031,932083,3728287,14913099,

%T 59652343,238609315,954437199,3817748731,15270994855,61083979347,

%U 244335917311,977343669163,3909374676567,15637498706179

%N a(n) = number of grid points that are covered after (2^n)th stage of A139250.

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,4).

%F a(n) = A147614(A000079(n)).

%F a(n) = (1/9)*(2^(2*n+3) + 12*n + 19). [_Nathaniel Johnston_, Mar 29 2011]

%F It appears that a(n) = A139252(2^(n+1)). - _Omar E. Pol_, Sep 11 2012

%F a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). - _Paul Curtz_, May 07 2020

%F G.f.: (3 - 11*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)). - _Colin Barker_, May 13 2020

%o (PARI) Vec((3 - 11*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)) + O(x^40)) \\ _Colin Barker_, May 13 2020

%Y Cf. A000079, A007583, A139250, A139251, A139252, A139560, A147614.

%Y Cf. Same recurrence: A073724, A210985, A014825.

%K nonn,easy

%O 0,1

%A _Omar E. Pol_, May 09 2009

%E Terms after a(10) from _Nathaniel Johnston_, Mar 29 2011