A153006 -id:A153006 - cp's OEIS frontend

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

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

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

			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

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

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.

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);

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 *)

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).

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

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

A153007 Triangular number A000217(n) minus toothpick number A153006(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 0, 3, 8, 11, 13, 15, 13, 5, 0, 7, 20, 31, 41, 51, 57, 57, 59, 69, 79, 82, 81, 74, 51, 17, 0, 15, 44, 71, 97, 123, 145, 161, 179, 205, 231, 250, 265, 274, 267, 249, 247, 273, 307, 334, 357, 374, 375, 364, 363, 376, 380, 364, 332, 270, 163, 49, 0, 31, 92, 151

Offset: 0

Omar E. Pol, Dec 19 2008, May 27 2009

nonn

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
OEIS (Plot 2), Triangular numbers (A000217) and toothpick numbers (A153006) vs n [From _Omar E. Pol_, Apr 29 2009]

Cf. A000217, A000225, A000396, A000668, A006516, A139250, A139251, A152980, A153006.

a(n) = A000217(n)-A153006(n).

More terms from R. J. Mathar, Jul 13 2009

A159795 a(n) = 4*A153006(n).

Original entry on oeis.org

0, 4, 12, 24, 36, 52, 80, 112, 132, 148, 176, 212, 252, 312, 400, 480, 516, 532, 560, 596, 636, 696, 784, 868, 924, 984, 1076, 1188, 1328, 1536, 1792, 1984, 2052, 2068, 2096, 2132, 2172, 2232, 2320, 2404, 2460, 2520, 2612, 2724, 2864, 3072, 3328, 3524, 3612

Offset: 0

Omar E. Pol, May 02 2009

nonn

For the first differences see A147646. - Omar E. Pol, Jul 24 2009

It appears that a(n) is also the total path length of a toothpick structure as A139250 after n-th stage which is constructed following a special rule: toothpicks of the new generation have length 4 when are placed on the square grid (note that every toothpick has four components of length 1), but after every stage, one (or two) of the four components of every toothpick of the new generation is removed, if such component contains a endpoint of the toothpick and if such endpoint is touching the midpoint or the endpoint of another toothpick. The truncated endpoints of the toothpicks remain exposed forever. Note that there are three sizes of toothpicks in the structure: toothpicks of length 4, 3 and 2. a(n) is also the total number of components in the structure after n-th stage. a(n) is also the number of grid points that are covered after n-th stage, except the central point of the structure. The toothpick sequence A139250 gives the number of toothpicks after n-th stage. - Omar E. Pol, Oct 24 2011

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. A139250, A139251, A152980, A153006, A159793, A159794.

Cf. A147646, A152980. - Omar E. Pol, Jul 24 2009

More terms from Omar E. Pol, Jul 24 2009

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

Original entry on oeis.org

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

A170926 Total number of squares and rectangles at the n-th stage in the corner toothpick structure (see A152890, A153006).

Original entry on oeis.org

0, 0, 1, 3, 4, 5, 10, 17, 20, 21, 25, 30, 33, 40, 58, 77, 84, 85, 89, 94, 97, 104, 121, 138, 145, 151, 164, 177, 190, 222, 278, 325, 340, 341, 345, 350, 353, 360, 377, 394, 401, 407, 420, 433, 446, 478, 533, 578, 593, 599, 612, 625, 638, 669, 720, 761, 781, 806, 845, 884, 942

Offset: 0

Omar E. Pol, Jan 18 2010

nonn

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

Partial sums of A168131.

a(2^k) = (4^k-4)/3 for k >= 2. - N. J. A. Sloane, Feb 13 2010.

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

A153009 Primes in toothpick sequence A153006.

Original entry on oeis.org

3, 13, 37, 53, 149, 269, 601, 653, 881, 941, 2053, 2069, 2137, 2417, 2477, 2657, 2713, 5281, 9697, 10009, 14561, 14713, 16033, 16693, 19489, 20149, 21617, 22091, 22741, 32789, 32909, 33377, 33529, 34273

Offset: 1

Omar E. Pol, Dec 20 2008

nonn

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. A139250, A139253, A153006.

More terms from Nathaniel Johnston, Nov 19 2010

A159793 a(n) = A153006(n)*2.

Original entry on oeis.org

0, 2, 6, 12, 18, 26, 40, 56, 66, 74, 88, 106, 126, 156, 200, 240, 258, 266, 280, 298, 318, 348, 392, 434, 462, 492, 538, 594, 664, 768, 896, 992, 1026, 1034, 1048, 1066, 1086, 1116, 1160, 1202, 1230, 1260, 1306, 1362, 1432, 1536, 1664, 1762, 1806, 1836, 1882, 1938

Offset: 0

Omar E. Pol, May 02 2009

nonn

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

Toothpick sequence: A139250.

Cf. A139251, A152980, A153006, A159794, A159795.

lista(nn) = concat(0, 2*Vec(x*((1+x)/(1-x))*prod(k=1, nn, 1+x^(2^k-1)+2*x^(2^k)) + O(x^nn))); \\ Jinyuan Wang, Mar 04 2020

More terms from Jinyuan Wang, Mar 04 2020

A159794 a(n) = A153006(n)*3.

Original entry on oeis.org

0, 3, 9, 18, 27, 39, 60, 84, 99, 111, 132, 159, 189, 234, 300, 360, 387, 399, 420, 447, 477, 522, 588, 651, 693, 738, 807, 891, 996, 1152, 1344, 1488, 1539, 1551, 1572, 1599, 1629, 1674, 1740, 1803, 1845, 1890, 1959, 2043, 2148, 2304, 2496, 2643

Offset: 0

Omar E. Pol, May 02 2009

nonn

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

Toothpick sequence: A139250.

Cf. A139251, A152980, A153006, A159793, A159795.

More terms from Max Alekseyev, Dec 13 2011

A160166 Toothpick number A153006(n) minus generalized pentagonal number A001318(n).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 5, 6, 7, 2, 4, 2, 6, 8, 23, 28, 29, 16, 14, 4, 4, -2, 9, 7, 9, -1, 9, 10, 31, 54, 103, 120, 121, 92, 82, 56, 48, 26, 29, 11, 5, -21, -19, -34, -21, -14, 27, 29, 27, -7, -9, -32, -23, -24

Offset: 0

Omar E. Pol, May 23 2009

sign

Nathaniel Johnston, Table of n, a(n) for n = 0..999
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. A001318, A139250, A153006, A153007, A159790, A160165.

a(n) = A153006(n) - A001318(n).

a(11) corrected and more terms from Nathaniel Johnston, Mar 22 2011

A162777 a(n) = A153003(n) - A153006(n).

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 5, 3, 1, 3, 5, 5, 7, 13, 15, 7, 1, 3, 5, 5, 7, 13, 15, 9, 7, 13, 17, 19, 29, 43, 39, 15, 1, 3, 5, 5, 7, 13, 15, 9, 7, 13, 17, 19, 29, 43, 39, 17, 7, 13, 17, 19, 29, 43, 41, 27, 29, 45, 55, 69, 103, 127, 95, 31

Offset: 0

Omar E. Pol, Jul 23 2009

The main entry for this sequence is the toothpick sequence A139250.

			If written as a triangle:
0;
0;
1,1;
1,3,5,3;
1,3,5,5,7,13,15,7;
1,3,5,5,7,13,15,9,7,13,17,19,29,43,39,15;
...
Rows converge to A162779.

Jinyuan Wang, Rows n = 0..14 of triangle, flattened
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. A139250, A152980, A153003, A153004, A153006, A162779.

Observation: It appears that a(2^i) = 1, i > 0 and a(2^i-1) = 2^(i-1) - 1, i > 0.

More terms from Jinyuan Wang, Mar 15 2020

cp's OEIS Frontend

A152980 First differences of toothpick corner sequence A153006.

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A153007 Triangular number A000217(n) minus toothpick number A153006(n).

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A159795 a(n) = 4*A153006(n).

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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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Maple

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A170926 Total number of squares and rectangles at the n-th stage in the corner toothpick structure (see A152890, A153006).

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A153009 Primes in toothpick sequence A153006.

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A159793 a(n) = A153006(n)*2.

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PARI

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A159794 a(n) = A153006(n)*3.

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A160166 Toothpick number A153006(n) minus generalized pentagonal number A001318(n).

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