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A034472 3^n + 1. 73
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; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Companion numbers to A003462.

Numbers n for which the expression 3^n/(n-1) is an integer. - Paolo P. Lava, May 29 2006

a(n) = A024101(n)/A024023(n). - Reinhard Zumkeller, Feb 14 2009

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

REFERENCES

Encyclopedia of Combinatorial Structures, Entry 454.

P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, pp. 35-36, 53.

D. Suprijanto, I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2211 - 2217; http://dx.doi.org/10.12988/ams.2014.4139.

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 454

K. Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 99-106.

Eric Weisstein's World of Mathematics, Lucas Sequence

Index entries for sequences related to linear recurrences with constant coefficients, signature (4,-3).

FORMULA

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). - R. J. Mathar, Sep 05 2008

E.g.f.: e^x+e^(3*x). - Mohammad K. Azarian, Jan 02 2009

EXAMPLE

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 + ...

MAPLE

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

MATHEMATICA

Table[3^n + 1, {n, 0, 24}]

a=2; lst={a}; Do[a=a*3-2; AppendTo[lst, a], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)

PROG

(PARI) {a(n) = 3^n + 1};

(Sage) [lucas_number2(n, 4, 3) for n in xrange(0, 27)] # Zerinvary Lajos, Jul 08 2008

(Sage) [sigma(3, n)for n in xrange(0, 26)] # Zerinvary Lajos, Jun 04 2009

CROSSREFS

Cf. A003462, A000204, A007051, A000051, A052539, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600 - A074624, A034524, A178248, A228081.

Sequence in context: A149821 A149822 * A094388 A187256 A148110 A149823

Adjacent sequences:  A034469 A034470 A034471 * A034473 A034474 A034475

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Additional comments from Rick L. Shepherd, Feb 13 2002

STATUS

approved

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Last modified October 25 02:31 EDT 2014. Contains 248517 sequences.