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A034472
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a(n) = 3^n + 1.
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106
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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, 2541865828330, 7625597484988
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OFFSET
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0,1
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COMMENTS
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Mahler exhibits this sequence with n>=2 as a proof that there exists an infinite number of x coprime to 3, such that x belongs to A005836 and x^2 belong to A125293. - Michel Marcus, Nov 12 2012
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REFERENCES
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Knuth, Donald E., Satisfiability, Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, pages 148 and 220, Problem 191.
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, pp. 35-36, 53.
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LINKS
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FORMULA
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a(n) = 3*a(n-1) - 2 = 4*a(n-1) - 3*a(n-2). (Lucas sequence, with A003462, associated to the pair (4, 3).)
G.f.: 2*(1-2*x)/((1-x)*(1-3*x)). 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). - R. J. Mathar, Sep 05 2008
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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).
G.f. = 2 + 4*x + 10*x^2 + 28*x^3 + 82*x^4 + 244*x^5 + 730*x^5 + ...
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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, Jun 19 2008
g:=1/(1-3*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)+1, n=0..31); # Zerinvary Lajos, Jan 09 2009
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MATHEMATICA
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Table[3^n + 1, {n, 0, 24}]
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PROG
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(PARI) a(n) = 3^n + 1
(PARI) Vec(2*(1-2*x)/((1-x)*(1-3*x)) + O(x^50)) \\ Altug Alkan, Nov 15 2015
(Sage) [lucas_number2(n, 4, 3) for n in range(27)] # Zerinvary Lajos, Jul 08 2008
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CROSSREFS
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Cf. A003462, A000204, A007051, A000051, A052539, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600 - A074624, A034524, A178248, A228081, A279396.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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