0,2

Generalized NSW numbers. - Paul Barry (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), 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.

G.f.: (1+3x)/(1-10x+9x^2); a(n)=sum{k=0..n, binomial(2n+1, 2k)4^k}; a(n)=((1+sqrt(4))(5+2sqrt(4))^n+(1-sqrt(4))(5-2sqrt(4))^n)/2. - Paul Barry (pbarry(AT)wit.ie), May 27 2005

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)

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

Cf. A107903.

Sequence in context: A014918 A081033 A091111 this_sequence A033470 A016230 A101186

Adjacent sequences: A096050 A096051 A096052 this_sequence A096054 A096055 A096056

nonn

Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 18 2004

Edited by njas at the suggestion of Andrew Plewe, Jun 15 2007