A152980 First differences of toothpick corner sequence A153006.

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, 52, 64, 48, 17, 4, 7, 9, 10, 15, 22, 21, 14, 15, 23, 28, 35, 52, 64, 49, 22, 15, 23, 28, 35, 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

Offset: 0

Author

Omar E. Pol, Dec 16 2008, Dec 19 2008, Jan 02 2009

Keywords

• nonn
• look

Comments

Rows of A152978 when written as a triangle converge to this sequence. - Omar E. Pol, Jul 19 2009

Examples

			Triangle begins:
.1;
.2;
.3,3;
.4,7,8,5;
.4,7,9,10,15,22,20,9;
.4,7,9,10,15,22,21,14,15,23,28,35,52,64,48,17;
....
Rows converge to A153001. - _N. J. A. Sloane_, Jun 07 2009
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..16384
• David Applegate, The movie version
• David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences

Crossrefs

Equals A151688 divided by 2. - N. J. A. Sloane, Jun 03 2009

For generating functions of the form Product_{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.

Equals A147646/4. - N. J. A. Sloane, May 01 2009

Cf. A139250, A139251, A152968, A152978, A153006, A153001, A159785, A153004.

Programs

Maple

Maple code from N. J. A. Sloane, May 18 2009. First define old version with offset 1:
S:=proc(n) option remember; local i,j;
if n <= 0 then RETURN(0); fi;
if n <= 2 then RETURN(2^(n-1)); fi;
i:=floor(log(n)/log(2));
j:=n-2^i;
if j=0 then RETURN(n/2+1); fi;
if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;
if j=2^i-1 then RETURN(2*S(j)+S(j+1)-1); fi;
-1;
end;
# Now change the offset:
T:=n->S(n+1);
G := (1 + x) * mul(1 + x^(2^k-1) + 2*x^(2^k),k=1..20);

Mathematica

nmax = 78;
G = x*((1 + x)/(1 - x)) * Product[ (1 + x^(2^n - 1) + 2*x^(2^n)), {n, 1, Log2[nmax] // Ceiling}];
CoefficientList[G + O[x]^nmax, x] // Differences (* Jean-François Alcover, Jul 21 2022 *)

Formula

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

For formula see A147646 (or, better, see the Maple code below).

Extensions

More terms (based on A147646) from N. J. A. Sloane, May 01 2009

Offset changed by N. J. A. Sloane, May 18 2009