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A034472 a(n) = 3^n + 1. 98
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

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.

LINKS

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

T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066, eq 5.

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.

Burkard Polster, Special numbers in 3-coloring of Pascal's triangle, Mathologer video (2019).

Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.

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.

Eric Weisstein's World of Mathematics, Lucas Sequence

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

FORMULA

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

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

a(n) = A279396(n+3,3). - Wolfdieter Lang, Jan 10 2017

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}]

PROG

(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

(Sage) [sigma(3, n) for n in range(27)] # Zerinvary Lajos, Jun 04 2009

(Sage) [3^n+1 for n in range(30)] # Bruno Berselli, Jan 11 2017

(MAGMA) [3^n+1: n in [0..30]]; // Vincenzo Librandi, Jan 11 2017

CROSSREFS

Cf. A003462, A000204, A007051, A000051, A052539, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600 - A074624, A034524, A178248, A228081, A279396.

Sequence in context: A149820 A149821 A149822 * A094388 A187256 A148110

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 23 23:52 EDT 2020. Contains 337975 sequences. (Running on oeis4.)