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A122983
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Binomial transform of aeration of A081294.
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15
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1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581, 1195743, 3587227, 10761681, 32285041, 96855123, 290565367, 871696101, 2615088301, 7845264903, 23535794707, 70607384121, 211822152361, 635466457083
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Binomial transform is A063376.
A122983 = (1,1,3,7,1,1,3,7,...) mod 10. - M. F. Hasler, Feb 25 2008
Equals row sums of triangle A158301. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 15 2009]
Contribution from Toby Gottfried (toby(AT)gottfriedville.net), Apr 18 2010: (Start)
a(n) = the number of ternary sequences of length n where the numbers of
(0's, 1's) are both even.
A015518 covers the (odd, even) and (even, odd) cases, and A081251 covers (odd, odd). (End)
This sequence also describes the number of moves of the k-th disk solving (non-optimally) the [RED ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi (MToH) puzzle. The sequence A183119 is the partial sums of the sequence in question (obviously describing the total number of moves associated with the specific solution algorithm). For other MToH-related sequences, Cf. A183111 - A183125.
Let B=[1,sqrt(2),0; sqrt(2),1,sqrt(2); 0,sqrt(2),1] be a 3 X 3 matrix. Then a(n)=[B^n]_(1,1), n=0,1,2,.... - L. Edson Jeffery, Dec 21 2011
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LINKS
| M. F. Hasler, Table of n, a(n) for n=0,...,199.
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FORMULA
| G.f.: (1-2*x-x^2)/((1-x)*(1+x)*(1-3*x)); a(n)=3^n/4+(-1)^n/4+1/2;
E.g.f.: cosh(x)^2*exp(x); - Paul Barry (pbarry(AT)wit.ie), Jun 14 2007
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MAPLE
| BB := n->if n=0 then 1; elif n=1 then 1; else (3*BB(n-2)+2*BB(n-1)) fi: L:=[]: for k from 0 to 22 do L:=[op(L), ceil(BB(k)/2)]: od: L; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 19 2007, corrected by M. F. Hasler, Feb 25 2008
A122983 := n -> ceil(3^n/4); 'A122983(n)' $ n=0..22; # - M. F. Hasler, Feb 25 2008
a[ -1]:=1:a[0]:=1:a[1]:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]-2 od: seq(a[n], n=-1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 28 2008
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PROG
| (PARI) A122983(n)=3^n\4+1 \\ - M. F. Hasler, Feb 25 2008
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CROSSREFS
| Cf. a(j+1) = A137822(2^j) and these are the record values of A137822.
A158301 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 15 2009]
Sequence in context: A056779 A183113 A102877 * A005355 A025235 A129366
Adjacent sequences: A122980 A122981 A122982 * A122984 A122985 A122986
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 22 2006
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EXTENSIONS
| Extended and corrected (existing Maple code) by M. F. Hasler (www.univ-ag.fr/~mhasler), Feb 25 2008
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