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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Companion numbers to A003462.

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

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

Examples

			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 + ...
		

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).

Crossrefs

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

Programs

Magma

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

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}]

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

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

Extensions

Additional comments from Rick L. Shepherd, Feb 13 2002