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A139251
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First differences of toothpicks numbers A139250.
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166
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0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 12, 12, 16, 28, 32, 16, 4, 8, 12, 12, 16, 28, 32, 20, 16, 28, 36, 40, 60, 88, 80, 32, 4, 8, 12, 12, 16, 28, 32, 20, 16, 28, 36, 40, 60, 88, 80, 36, 16, 28, 36, 40, 60, 88, 84, 56, 60, 92, 112, 140, 208, 256, 192, 64, 4, 8, 12, 12, 16, 28, 32, 20, 16, 28
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of toothpicks added to the toothpick structure at the n-th step (see A139250).
It appears that if n is equal to 1 plus a power of 2 with positive exponent then a(n) = 4. (For proof see the second Applegate link.)
It appears that there is a relation between this sequence, even superperfect numbers, Mersenne primes and even perfect numbers. Conjecture: The sum of the toothpicks added to the toothpick structure between the stage A061652(k) and the stage A000668(k) is equal to the k-th even perfect number, for k >= 1. For example: A000396(1)= 2+4 =6. A000396(2)= 4+4+8+12 =28. A000396(3)=16+4+8+12+12+16+28+32+20+16+28+36+40+60+88+80 =496. [From Omar E. Pol (info(AT)polprimos.com), May 04 2009]
Concerning this conjecture, see David Applegate's comments on the conjectures in A153006. - N. J. A. Sloane, May 14 2009
In the triangle (See example lines), the sum of row k is equal to A006516(k), for k >= 1. [From Omar E. Pol (info(AT)polprimos.com), May 15 2009]
Equals (1, 2, 2, 2,...) convolved with A160762: (1, 0, 2, -2, 2, 2, 2, -6,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2009]
Convolved with the Jacobsthal sequence A001045 = A160704: (1, 3, 9, 19, 41,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 24 2009]
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..65535
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
David Applegate, The movie version
O. E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures) [From Omar E. Pol (info(AT)polprimos.com), Nov 02 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
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FORMULA
| Recurrence from N. J. A. Sloane, Jul 20 2009: a(0) = 0; a(2^i)=2^i for all i; otherwise write n=2^i+j, 0<j<2^i, then a(n) = 2a(j)+a(j+1). Proof: This is a simplification of the following recurrence of David Applegate. QED
Recurrence from David Applegate, Apr 29 2009: (Start)
Write n=2^(i+1)+j, where 0<=j<2^(i+1). Then, for n > 3:
for j=0, a(n) = 2*a(n-2^i) (= n = 2^(i+1))
for 1<=j<=2^i-1, a(n) = a(n-2^i)
for j=2^i, a(n) = a(n-2^i)+4 (= 2^(i+1)+4)
for 2^i+1<=j<=2^(i+1)-2, a(n) = 2*a(n-2^i)+a(n-2^i+1)
for j=2^(i+1)-1, a(n) = 2*a(n-2^i)+a(n-2^i+1)-4
and a(n) = 2^(n-1) for n=1,2,3. (End)
G.f.: (x/(1+2*x)) * (1 + 2*x*Product(1+x^(2^k-1)+2*x^(2^k),k=0..oo)). - N. J. A. Sloane, May 20 2009, Jun 05 2009
With offset 0 (which would be more natural, but offset 1 is now entrenched): a(0) = 1, a(1) = 2; for i >= 1, a(2^i) = 4; otherwise write n = 2^i +j, 0 < j < 2^i, then a(n) = 2 * Sum_{ k >= 0 } 2^(wt(j+k)-k)*binomial(wt(j+k),k). - N. J. A. Sloane, Jun 03 2009
It appears that a(n) = A187221(n+1)/2. - Omar E. Pol, Mar 08 2011.
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EXAMPLE
| Triangle begins:
. 0;
. 1;
. 2,4;
. 4,4,8,12;
. 8,4,8,12,12,16,28,32;
.16,4,8,12,12,16,28,32,20,16,28,36,40,60,88,80;
.32,4,8,12,12,16,28,32,20,16,28,36,40,60,88,80,36,16,28,36,40,60,88,84,56,...
...
The row sums give A006516.
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MAPLE
| G := (x/(1+2*x)) * (1 + 2*x*mul(1+x^(2^k-1)+2*x^(2^k), k=0..20)). - N. J. A. Sloane, May 20 2009, Jun 05 2009
# From N. J. A. Sloane, Dec 25, 2009: A139250 is T, A139251 is a.
a:=[0, 1, 2, 4]; T:=[0, 1, 3, 7]; M:=10;
for k from 1 to M do
a:=[op(a), 2^(k+1)];
T:=[op(T), T[nops(T)]+a[nops(a)]];
for j from 1 to 2^(k+1)-1 do
a:=[op(a), 2*a[j+1]+a[j+2]];
T:=[op(T), T[nops(T)]+a[nops(a)]];
od: od: a; T;
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CROSSREFS
| Equals 2*A152968 and 4*A152978 (if we ignore the first couple of terms).
See A147646 for the limiting behavior of the rows. See also A006516.
Cf. A139250, A139252, A139253, A152980, A153000, A153001, A000396, A000668, A061652, A153006.
Cf. A006516, A153007, A159790, A001045, A160704, A160762, A160121, A147582.
Sequence in context: A170893 A194445 A160809 * A182635 A188346 A173531
Adjacent sequences: A139248 A139249 A139250 * A139252 A139253 A139254
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KEYWORD
| nonn,tabf
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AUTHOR
| Omar E. Pol (info(AT)polprimos.com), Apr 24 2008, Dec 16 2008, Apr 20 2009
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EXTENSIONS
| The layout of the triangle was adjusted by David Applegate, Apr 29 2009, to reveal that the columns become constant.
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