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Search: id:A153006
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| A153006 |
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Toothpick sequence starting at the outside corner of an infinite square from which protudes a half toothpick. |
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37
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| 0, 1, 3, 6, 9, 13, 20, 28, 33, 37, 44, 53, 63, 78, 100, 120, 129, 133, 140, 149, 159, 174, 196, 217, 231, 246, 269, 297, 332, 384, 448, 496, 513, 517, 524, 533, 543, 558, 580, 601, 615, 630, 653, 681, 716, 768, 832, 881, 903, 918, 941
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n) is the total number of integer toothpicks after n steps.
It appears that this sequence is related to triangular numbers, Mersenne primes and even perfect numbers. Conjecture: a(A000668(n))=A000217(A000668(n)). Conjecture: a(A000668(n))=A000396(n), assuming there are no odd perfect numbers.
The main entry for this sequence is A139250. But see also A152980 and A147646.
The Mersenne prime conjectures are true, but aren't really about Mersenne primes. a(2^i-1) = 2^i (2^i-1)/2 for all i (whether or not i or 2^i-1 is prime). This follows from the formulae for A139250(2^i-1) and A139250(2^i). [From David Applegate (david(AT)research.att.com), May 11 2009]
Then we can write a(A000225(k))=A006516(k), for k>0. See Plot 2 link, here. [From Omar E. Pol (info(AT)polprimos.com), May 23 2009]
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..16387
O. E. Pol, Illustration of initial terms
O. E. Pol, Illustration of A153006(31) = 496
OEIS (Plot 2), Triangular numbers (A000217) and toothpick numbers (A153006) vs n
O. E. Pol, OEIS, Plot 2, Ratio of A000217 to A153006 vs n [From Omar E. Pol (info(AT)polprimos.com), May 27 2009]
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FORMULA
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Comments from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2009. Equals A151550 convolved with [1, 2, 2, 2,...]. (This is equivalent to the observation that the g.f. is x((1+x)/(1-x)) * Prod_{ n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)).) Equivalently, equals A151555 convolved with A151575.
G.f.: x*((1 + x)/(1 - x)) * Prod_{ n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)). - N. J. A. Sloane, May 20, 2009
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EXAMPLE
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1) Put a toothpick on the table so to make a tee ====> a(1) = 1.
2) Put two other toothpicks ==========================> a(2) = 3.
3) Put three other toothpicks ========================> a(3) = 6.
And so on.
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MAPLE
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G:=x*((1 + x)/(1 - x)) * mul( (1 + x^(2^n-1) + 2*x^(2^n)), n=1..20); - N. J. A. Sloane, May 20, 2009
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CROSSREFS
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Cf. A000079, A000217, A000396, A000668, A139250, A139251, A139560, A152978, A152979, A152980, A153000, A153001, A153002.
Cf. A000225, A006516. [From Omar E. Pol (info(AT)polprimos.com), May 23 2009]
Cf. A153007, A078008, A151555.
Sequence in context: A117469 A073359 A137041 this_sequence A086838 A161669 A128261
Adjacent sequences: A153003 A153004 A153005 this_sequence A153007 A153008 A153009
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KEYWORD
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nonn
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AUTHOR
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Omar E. Pol (info(AT)polprimos.com), Dec 17 2008, Dec 19 2008, Apr 28 2009
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 19 2008
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