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1, 13, 121, 1093, 9841, 88573, 797161, 7174453, 64570081, 581130733, 5230176601, 47071589413, 423644304721, 3812798742493, 34315188682441, 308836698141973, 2779530283277761, 25015772549499853, 225141952945498681
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (10,-9).
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FORMULA
| 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. - 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)
a(n) = 9*a(n-1)+4. - Vincenzo Librandi, Nov 01 2011
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PROG
| (MAGMA) [(3*9^n-1)/2: n in [0..20]]; // Vincenzo Librandi, Nov 01 2011
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CROSSREFS
| Cf. A083420, A096045, A096046, A096047, A096045, A096054.
Cf. A107903.
Sequence in context: A081033 A091111 A196921 * A033470 A016230 A201382
Adjacent sequences: A096050 A096051 A096052 * A096054 A096055 A096056
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KEYWORD
| nonn,easy
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 18 2004
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EXTENSIONS
| Edited by N. J. A. Sloane, at the suggestion of Andrew Plewe, Jun 15 2007
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