A096053 a(n) = (3*9^n - 1)/2.

1, 13, 121, 1093, 9841, 88573, 797161, 7174453, 64570081, 581130733, 5230176601, 47071589413, 423644304721, 3812798742493, 34315188682441, 308836698141973, 2779530283277761, 25015772549499853, 225141952945498681

Offset: 0

Author

Benoit Cloitre, Jun 18 2004

Keywords

• nonn
• easy

Comments

Generalized NSW numbers. - Paul Barry, May 27 2005

Counts total area under elevated Schroeder paths of length 2n+2, where area under a horizontal step is weighted 3. Case r=4 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315. Fifth binomial transform of (1+8x)/(1-16x^2), A107906. - Paul Barry, May 27 2005

Primes in this sequence include: a(2) = 13, a(4) = 1093, a(7) = 797161. Semiprimes in this sequence include: a(3) = 121 = 11^2, a(5) = 9841 = 13 * 757, a(6) = 88573 = 23 * 3851, a(9) = 64570081 = 1871 * 34511, a(10) = 581130733 = 1597 * 363889, a(12) = 47071589413 = 47 * 1001523179, a(19) = 225141952945498681 = 13097927 * 17189128703.

Sum of divisors of 9^n. - Altug Alkan, Nov 10 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-9).

Crossrefs

Cf. A083420, A096045, A096046, A096047, A096054.

Cf. A107903, A138894 ((5*9^n-1)/4).

Programs

Magma

[(3*9^n-1)/2: n in [0..20]]; // Vincenzo Librandi, Nov 01 2011

Mathematica

Table[(3*9^n - 1)/2, {n, 0, 18}] (* L. Edson Jeffery, Feb 13 2015 *)

PARI

a(n)=(3*9^n-1)/2 \\ Charles R Greathouse IV, Sep 28 2015

PARI

vector(30, n, n--; sigma(9^n)) \\ Altug Alkan, Nov 10 2015

Formula

From Paul Barry, May 27 2005: (Start)

G.f.: (1+3*x)/(1-10*x+9*x^2);

a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*4^k;

a(n) = ((1+sqrt(4))*(5+2*sqrt(4))^n+(1-sqrt(4))*(5-2*sqrt(4))^n)/2. (End)

a(n-1) = (-9^n/3)*B(2n,1/3)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k)=B(k,0) is the k-th Bernoulli number.

a(n) = 10*a(n-1) - 9*a(n-2).

a(n) = 9*a(n-1) + 4. - Vincenzo Librandi, Nov 01 2011

a(n) = A000203(A001019(n)). - Altug Alkan, Nov 10 2015

a(n) = A320030(3^n-1). - Nathan M Epstein, Jan 02 2019

Extensions

Edited by N. J. A. Sloane, at the suggestion of Andrew S. Plewe, Jun 15 2007