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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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38
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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
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OFFSET
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0,4
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COMMENTS
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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
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). - Gary W. Adamson, May 19 2009
Starting with 1 and convolved with [1, 2, 0, 0, 0,...] = A151548. - Gary W. Adamson, Jun 04 2009
Refer to A162956 for the analogous triangle using N=3. - Gary W. Adamson, Jul 20 2009
It appears that the sums of two successive terms give the positive terms of A139251. - Omar E. Pol, Feb 18 2015
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LINKS
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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
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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)-1). - N. J. A. Sloane, May 23 2009, based on Gary W. Adamson's comment above and the known g.f. for A139250.
It appears that a(n) = A169708(n)/4, n >= 1. - Omar E. Pol, Feb 15 2015
It appears that a(n) = A139251(n) - a(n-1), n >= 1. - Omar E. Pol, Feb 18 2015
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EXAMPLE
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a(2) = a(2^1+0) = 2*a(0)+a(1) = 1, a(3) = a(2^1+1) = 2*a(1) + a(2) = 3*a(2^i) = 2*a(0) + a(1) = 1.
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MAPLE
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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; # 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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MATHEMATICA
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Nest[Join[#, 2 # + Append[Rest@#, 1]] &, {0}, 7] (* Ivan Neretin, Feb 09 2017 *)
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CROSSREFS
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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 * A256263 A006257 A170898
Adjacent sequences: A160549 A160550 A160551 * A160553 A160554 A160555
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KEYWORD
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nonn
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AUTHOR
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David Applegate, May 18 2009
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STATUS
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approved
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