A170926 -id:A170926 - cp's OEIS frontend

A168131 Number of squares and rectangles that are created at the n-th stage in the corner toothpick structure (see A152980, A153006).

0, 0, 1, 2, 1, 1, 5, 7, 3, 1, 4, 5, 3, 7, 18, 19, 7, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 56, 47, 15, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 55, 45, 15, 6, 13, 13, 13, 31, 51, 41, 20, 25, 39, 39, 58, 120, 160, 111, 31, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 55, 45, 15, 6

Offset: 0

Omar E. Pol, Jan 18 2010

nonn

Essentially the first differences of A170926. - Omar E. Pol, Feb 16 2013

			If written as a triangle:
0,
0,
1,2,
1,1,5,7,
3,1,4,5,3,7,18,19,
7,1,4,5,3,7,17,17,7,6,13,13,13,32,56,47,
15,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6,13,13,13,31,51,41,20,...
The rows (omitting the first term) converge to A170929.

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

Cf. A152980, A153006, A170926, A160124, A160125, A139250.

w := proc(n) option remember; local k,i;
if (n=0) then RETURN(0)
elif (n <= 3) then RETURN(n-1)
else
k:=floor(log(n)/log(2));
i:=n-2^k;
if (i=0) then RETURN(2^(k-1)-1)
elif (i<2^k-2) then RETURN(2*w(i)+w(i+1));
elif (i=2^k-2) then RETURN(2*w(i)+w(i+1)+1);
else RETURN(2*w(i)+w(i+1)+2);
fi;
fi;
end;
[seq(w(n),n=0..256)];

a[n_] := a[n] = Module[{k, i}, Which[n==0, 0, n <= 3, n-1, True, k = Floor[Log2[n]]; i = n-2^k; Which[i==0, 2^(k-1)-1, i < 2^k-2, 2*a[i]+a[i+1], i==2^k-2, 2*a[i]+a[i+1]+1, True, 2*a[i]+a[i+1]+2]]];
Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Sep 25 2022, after Maple code *)

See Maple program for recurrence.

Edited and extended by N. J. A. Sloane, Feb 01 2010

A194278 Total number of polygons after n-th stage in the D-toothpick structure of A194270.

Original entry on oeis.org

0, 0, 0, 0, 2, 8, 14, 16, 26, 38, 46, 48, 56, 72, 102

Offset: 0

Omar E. Pol, Aug 26 2011

The structure of the D-toothpick cellular automaton contains at least several tens of different types of polygons. For more information see A194276 and A194277.

			Consider the structure with toothpicks of length 2 and D-toothpicks of length sqrt(2). After 3 stages the number of polygons in the structure is equal to 0. After 4 stages there are 2 hexagons, each with area = 6, so a(4) = 2. After 5 stages there are new 6 polygons: 2 hexagons, each with area = 8 and also 2 octagons, each with area = 14, so a(5) = 2+6 = 8.

Cf. A160124, A170926, A161418, A161828, A161836, A194270, A194276, A194277.

A170940 4^n-2^n-2.

Original entry on oeis.org

0, 10, 54, 238, 990, 4030, 16254, 65278, 261630, 1047550, 4192254, 16773118, 67100670, 268419070, 1073709054, 4294901758, 17179738110, 68719214590, 274877382654, 1099510579198, 4398044413950, 17592181850110, 70368735789054, 281474959933438, 1125899873288190

Offset: 1

N. J. A. Sloane, Feb 13 2010

nonn

a(n) is also the number whose binary representation is the concatenation of n-1 1's, 0, n-1 1's and 0 (See example). [From Omar E. Pol, Mar 16 2010]

			Contribution from _Omar E. Pol_, Mar 16 2010: (Start)
n ...... a(n) written in base 2 ..... a(n)
1 ................ 0 ................ 0
2 ............... 1010 .............. 10
3 .............. 110110 ............. 54
4 ............. 11101110 ............ 238
5 ............ 1111011110 ........... 990
6 ........... 111110111110 .......... 4030
7 .......... 11111101111110 ......... 16254
8 ......... 1111111011111110 ........ 65278
9 ........ 111111110111111110 ....... 261630
10 ...... 11111111101111111110 ...... 1047550
(End)

Index entries for linear recurrences with constant coefficients, signature (7, -14, 8).

Cf. A170926.

Cf. A006516, A138148, A173521. [From Omar E. Pol, Mar 16 2010]

a(n)= 7*a(n-1) -14*a(n-2) +8*a(n-3) = 2*A129868(n-1). G.f.: 2*x^2*(-5+8*x)/((x-1) * (2*x-1) * (4*x-1)). [From R. J. Mathar, Feb 14 2010]

a(n) = 2*(A006516(n)-1) [From Omar E. Pol, Mar 16 2010]

cp's OEIS Frontend

A168131 Number of squares and rectangles that are created at the n-th stage in the corner toothpick structure (see A152980, A153006).

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A194278 Total number of polygons after n-th stage in the D-toothpick structure of A194270.

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A170940 4^n-2^n-2.

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