

A152980


First differences of toothpick corner sequence A153006.


50



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
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OFFSET

0,2


COMMENTS

Rows of A152978 when written as a triangle converge to this sequence.  Omar E. Pol, Jul 19 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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)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


FORMULA

G.f.: (1 + x) * Prod_{ n >= 1} (1 + x^(2^n1) + 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).


EXAMPLE

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


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^(n1)); fi;
i:=floor(log(n)/log(2));
j:=n2^i;
if j=0 then RETURN(n/2+1); fi;
if j<2^i1 then RETURN(2*S(j)+S(j+1)); fi;
if j=2^i1 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^k1) + 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 (* JeanFrançois Alcover, Jul 21 2022 *)


CROSSREFS

Equals A151688 divided by 2.  N. J. A. Sloane, Jun 03 2009
For generating functions of the form Prod_{k>=c} (1+a*x^(2^k1)+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.
Sequence in context: A260167 A035540 A114863 * A170891 A035535 A154309
Adjacent sequences: A152977 A152978 A152979 * A152981 A152982 A152983


KEYWORD

nonn,look


AUTHOR

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


EXTENSIONS

More terms (based on A147646) from N. J. A. Sloane, May 01 2009
Offset changed by N. J. A. Sloane, May 18 2009


STATUS

approved



