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%I #42 Sep 08 2022 08:45:14
%S 1,13,121,1093,9841,88573,797161,7174453,64570081,581130733,
%T 5230176601,47071589413,423644304721,3812798742493,34315188682441,
%U 308836698141973,2779530283277761,25015772549499853,225141952945498681
%N a(n) = (3*9^n - 1)/2.
%C Generalized NSW numbers. - _Paul Barry_, May 27 2005
%C 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
%C 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.
%C Sum of divisors of 9^n. - _Altug Alkan_, Nov 10 2015
%H Vincenzo Librandi, <a href="/A096053/b096053.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-9).
%F From _Paul Barry_, May 27 2005: (Start)
%F G.f.: (1+3*x)/(1-10*x+9*x^2);
%F a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*4^k;
%F a(n) = ((1+sqrt(4))*(5+2*sqrt(4))^n+(1-sqrt(4))*(5-2*sqrt(4))^n)/2. (End)
%F 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.
%F a(n) = 10*a(n-1) - 9*a(n-2).
%F a(n) = 9*a(n-1) + 4. - _Vincenzo Librandi_, Nov 01 2011
%F a(n) = A000203(A001019(n)). - _Altug Alkan_, Nov 10 2015
%F a(n) = A320030(3^n-1). - _Nathan M Epstein_, Jan 02 2019
%t Table[(3*9^n - 1)/2, {n, 0, 18}] (* _L. Edson Jeffery_, Feb 13 2015 *)
%o (Magma) [(3*9^n-1)/2: n in [0..20]]; // _Vincenzo Librandi_, Nov 01 2011
%o (PARI) a(n)=(3*9^n-1)/2 \\ _Charles R Greathouse IV_, Sep 28 2015
%o (PARI) vector(30, n, n--; sigma(9^n)) \\ _Altug Alkan_, Nov 10 2015
%Y Cf. A083420, A096045, A096046, A096047, A096054.
%Y Cf. A107903, A138894 ((5*9^n-1)/4).
%K nonn,easy
%O 0,2
%A _Benoit Cloitre_, Jun 18 2004
%E Edited by _N. J. A. Sloane_, at the suggestion of _Andrew S. Plewe_, Jun 15 2007
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