|
%I #32 Jul 21 2022 10:41:14
%S 1,2,3,3,4,7,8,5,4,7,9,10,15,22,20,9,4,7,9,10,15,22,21,14,15,23,28,35,
%T 52,64,48,17,4,7,9,10,15,22,21,14,15,23,28,35,52,64,49,22,15,23,28,35,
%U 52,65,56,43,53,74,91,122,168,176,112,33,4,7,9,10,15,22,21,14,15,23,28,35,52
%N First differences of toothpick corner sequence A153006.
%C Rows of A152978 when written as a triangle converge to this sequence. - _Omar E. Pol_, Jul 19 2009
%H N. J. A. Sloane, <a href="/A152980/b152980.txt">Table of n, a(n) for n = 0..16384</a>
%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%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/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F G.f.: (1 + x) * Prod_{ n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)). - _N. J. A. Sloane_, May 20 2009, corrected May 21 2009
%F For formula see A147646 (or, better, see the Maple code below).
%e Triangle begins:
%e .1;
%e .2;
%e .3,3;
%e .4,7,8,5;
%e .4,7,9,10,15,22,20,9;
%e .4,7,9,10,15,22,21,14,15,23,28,35,52,64,48,17;
%e ....
%e Rows converge to A153001. - _N. J. A. Sloane_, Jun 07 2009
%p Maple code from _N. J. A. Sloane_, May 18 2009. First define old version with offset 1:
%p S:=proc(n) option remember; local i,j;
%p if n <= 0 then RETURN(0); fi;
%p if n <= 2 then RETURN(2^(n-1)); fi;
%p i:=floor(log(n)/log(2));
%p j:=n-2^i;
%p if j=0 then RETURN(n/2+1); fi;
%p if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;
%p if j=2^i-1 then RETURN(2*S(j)+S(j+1)-1); fi;
%p -1;
%p end;
%p # Now change the offset:
%p T:=n->S(n+1);
%p G := (1 + x) * mul(1 + x^(2^k-1) + 2*x^(2^k),k=1..20);
%t nmax = 78;
%t G = x*((1 + x)/(1 - x)) * Product[ (1 + x^(2^n - 1) + 2*x^(2^n)), {n, 1, Log2[nmax] // Ceiling}];
%t CoefficientList[G + O[x]^nmax, x] // Differences (* _Jean-François Alcover_, Jul 21 2022 *)
%Y Equals A151688 divided by 2. - _N. J. A. Sloane_, Jun 03 2009
%Y For generating functions of the form Prod_{k>=c} (1+a*x^(2^k-1)+b*x^2^k)) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
%Y Equals A147646/4. - _N. J. A. Sloane_, May 01 2009
%Y Cf. A139250, A139251, A152968, A152978, A153006, A153001, A159785, A153004.
%K nonn,look
%O 0,2
%A _Omar E. Pol_, Dec 16 2008, Dec 19 2008, Jan 02 2009
%E More terms (based on A147646) from _N. J. A. Sloane_, May 01 2009
%E Offset changed by _N. J. A. Sloane_, May 18 2009
|
