A107906 -id:A107906 - cp's OEIS frontend

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

Benoit Cloitre, Jun 18 2004

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

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

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

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

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

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

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

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

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

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

A107903 Generalized NSW numbers.

Original entry on oeis.org

1, 10, 76, 568, 4240, 31648, 236224, 1763200, 13160704, 98232832, 733219840, 5472827392, 40849739776, 304906608640, 2275853910016, 16987204845568, 126794223124480, 946404965613568, 7064062832410624, 52726882796830720

Offset: 0

Paul Barry, May 27 2005

Counts total area under elevated Schroeder paths of length 2n+2, where horizontal steps can choose from three colors.
Case r=3 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315 and case r=4 gives NSW numbers A096053.
Fifth binomial transform of (1+8x)/(1-16x^2), A107906.
If p is an odd prime, a((p-1)/2) == 1 mod p. - Altug Alkan, Mar 17 2016

Index entries for linear recurrences with constant coefficients, signature (8,-4).

Cf. A001834, A099156, A102591.

Table[Sum[Binomial[2 n + 1, 2 k] 3^k, {k, 0, n}], {n, 0, 20}] (* or *) CoefficientList[Series[(1 + 2 x)/(1 - 8 x + 4 x^2), {x, 0, 20}], x] (* Michael De Vlieger, Mar 17 2016 *)

Vec((1+2*x)/(1-8*x+4*x^2) + O(x^40)) \\ Michel Marcus, Mar 17 2016

G.f.: (1+2*x)/(1-8*x+4*x^2). [corrected by Ralf Stephan, Nov 30 2010]
a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*3^k.
a(n) = ((1+sqrt(3))*(4+2*sqrt(3))^n+(1-sqrt(3))*(4-2*sqrt(3))^n)/2 = A099156(n+1)+2*A099156(n).
a(n) = 8*a(n-1) - 4*a(n-2); a(0) = 1, a(1) = 10. - Lekraj Beedassy, Apr 19 2020
a(n) = 2^n*A001834(n). - Philippe Deléham, Mar 18 2023

Typo corrected and link added by Johannes W. Meijer, Aug 07 2010

cp's OEIS Frontend

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

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A107903 Generalized NSW numbers.

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