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Search: id:A034472
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| 2, 4, 10, 28, 82, 244, 730, 2188, 6562, 19684, 59050, 177148, 531442, 1594324, 4782970, 14348908, 43046722, 129140164, 387420490, 1162261468, 3486784402, 10460353204, 31381059610, 94143178828, 282429536482, 847288609444
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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Companion numbers to A003462.
Numbers n for which the expression 3^n/(n-1) is an integer. - Paolo P. Lava (ppl(AT)spl.at), May 29 2006
a(n) = A024101(n)/A024023(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 14 2009]
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REFERENCES
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P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, pp. 35-36, 53.
Encyclopedia of Combinatorial Structures, Entry 454.
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 454
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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a(n) = 3a(n-1) - 2 = 4a(n-1) - 3a(n-2). (Lucas sequence, with A003462, associated to the pair (4, 3).)
G.f.: 2(1-2x)/((1-x)(1-3x)). Inverse binomial transforms yields 2,2,4,8,16,... i.e., A000079 with the first entry changed to 2. Binomial transform yields A063376 without A063376(-1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 05 2008]
E.g.f.: e^x+e^(3*x). [From Mohammad K. Azarian (azarian(AT)evansville.edu), Jan 02 2009]
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EXAMPLE
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a(3)=28 because 4*a(2)-3*a(1)=4*10-3*4=28 (28 is also 3^3 + 1).
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MAPLE
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ZL:= [S, {S=Union(Sequence(Z), Sequence(Union(Z, Z, Z)))}, unlabeled]: seq(combstruct[count](ZL, size=n), n=0..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 19 2008
g:=1/(1-3*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)+1, n=0..31); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 09 2009]
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MATHEMATICA
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Table[3^n + 1, {n, 0, 24}]
a=2; lst={a}; Do[a=a*3-2; AppendTo[lst, a], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 25 2008]
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PROGRAM
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(PARI) a(n)=3^n+1
sage: [lucas_number2(n, 4, 3) for n in xrange(0, 27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2008
(Other) sage: [sigma(3, n)for n in xrange(0, 26)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]
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CROSSREFS
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Cf. A003462, A000204, A000051, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074600 - A074624.
Cf. A007051.
Sequence in context: A149820 A149821 A149822 this_sequence A094388 A148110 A149823
Adjacent sequences: A034469 A034470 A034471 this_sequence A034473 A034474 A034475
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Feb 13 2002
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