OFFSET
0,1
COMMENTS
Companion numbers to A003462.
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
a(n-1) is the number of n-digit base 3 numbers that have an even number of digits 0. - Yifan Xie, Jul 13 2024
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
Kurt 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 and 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
a(n) = 2*A007051(n). - R. J. Mathar, Apr 07 2022
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Additional comments from Rick L. Shepherd, Feb 13 2002
STATUS
approved