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A160552
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a(0)=0, a(1)=1; a(2^i+j)=2*a(j)+a(j+1) for 0 <= j < 2^i.
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25
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0, 1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31, 5, 11, 17, 19, 21, 39, 49, 35, 21, 39, 53, 59, 81, 127, 129, 63, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| This recurrence is patterned after the one for A152980, but without the special cases.
Sequence viewed as triangle:
.0,
.1,
.1,3,
.1,3,5,7,
.1,3,5,7,5,11,17,15,
.1,3,5,7,5,11,17,15,5,11,17,19,21,39,49,31
The rows converge to A151548.
Also the sum of the terms in the k-th row (k >= 1) is 4^(k-1). Proof by induction. - N. J. A. Sloane, Jan 21 2010.
Comment from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 19 2009: If this sequence [1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, ...] is convolved with [1, 2, 2, 2, 2, 2, 2, ...) we obtain A139250, the toothpick sequence. Example: A139250(5) = 15 = (1, 2, 2, 2, 2) * (3, 1, 3, 1, 1).
Starting with 1 and convolved with [1, 2, 0, 0, 0,...] = A151548. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]
Refer to A162956 for the analogous triangle using N=3. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 20 2009]
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..16384
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
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FORMULA
| G.f.: x*(1+2*x)/(1+x) + (4*x^2/(1+2*x))*(mul(1+x^(2^k-1)+2*x^(2^k),k=1..oo)-1). - N. J. A. Sloane, May 23 2009, based on Gary Adamson's comment above and the known g.f. for A139250.
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EXAMPLE
| a(2) = a(2^1+0) = 2a(0)+a(1) = 1, a(3) = a(2^1+1) = 2a(1) + a(2) = 3 a(2^i) = 2a(0) + a(1) = 1
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MAPLE
| S:=proc(n) option remember; local i, j; if n <= 1 then RETURN(n); fi; i:=floor(log(n)/log(2)); j:=n-2^i; 2*S(j)+S(j+1); end; [from N. J. A. Sloane, May 18 2009]
H := x*(1+2*x)/(1+x) + (4*x^2/(1+2*x))*(mul(1+x^(2^k-1)+2*x^(2^k), k=1..20)-1); series(H, x, 120); # - N. J. A. Sloane, May 23 2009
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CROSSREFS
| For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.
Cf. A152980, A139250, A139251, A151548, A160570, A151568.
Cf. A162956, A170903.
Sequence in context: A016646 A182600 A179760 * A006257 A170898 A189442
Adjacent sequences: A160549 A160550 A160551 * A160553 A160554 A160555
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KEYWORD
| nonn,tabf
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AUTHOR
| David Applegate (david(AT)research.att.com), May 18 2009
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