Search: id:A059100
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%I A059100
%S A059100 2,3,6,11,18,27,38,51,66,83,102,123,146,171,198,227,258,291,326,363,
%T A059100 402,443,486,531,578,627,678,731,786,843,902,963,1026,1091,1158,1227,
%U A059100 1298,1371,1446,1523,1602,1683,1766,1851,1938,2027,2118,2211,2306,2403
%N A059100 n^2+2.
%C A059100 Let s(n)=sum(k>=1,1/n^(2^k)). Then I conjecture that the maximum element 
               in the continued fraction for s(n) is n^2+2. - Benoit Cloitre, Aug 
               15, 2002.
%C A059100 Binomial transformation yields A081908, with A081908(0)=1 dropped. [From 
               R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 05 2008]
%C A059100 Except for the first term of [A059100], if X=[A069987], Y=[A000027], 
               A= [A059100], we have, for all other terms, Pell's equation: [A069987]^2 
               - [A059100]*[A000027]^2=1; (X^2-A*Y^2=1); example: 2^2-3*1^2=1; 5^2-6*2^2=1; 
               101^2-102*10^2=1; and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%C A059100 a(n) = A000196(A156798(n) - A000290(n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 16 2009]
%H A059100 Harry J. Smith, <a href="b059100.txt">Table of n, a(n) for n=0,...,1000</
               a>
%H A059100 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Near-SquarePrime.html">Near-Square Prime</a>
%F A059100 G.f.: (2-3x+3x^2)/(1-x)^3. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 05 2008]
%F A059100 a(n) = ((n-2)^2 + 2*(n+1)^2)/3. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 13 2009]
%F A059100 a(n)=2*n+a(n-1)-3 (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 07 2009]
%e A059100 For n=2, a(2)=2*2+2-3=3; n=3, a(3)=2*3+3-3=6; n=4, a(4)=2*4+6-3=11 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]
%p A059100 with(combinat, fibonacci):seq(fibonacci(3, i)+1, i=0..49); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008
%t A059100 a[n_]:=n^2+2; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 
               2008]
%o A059100 (Other) sage: [lucas_number1(3,n,-2) for n in xrange(0, 50)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
%o A059100 (PARI) { for (n = 0, 1000, write("b059100.txt", n, " ", n^2+2); ) } [From 
               Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 24 2009]
%Y A059100 Cf. A000290, A002522, A056899. Apart from initial terms, same as A010000.
%Y A059100 Cf. A000027, A069987 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%Y A059100 Sequence in context: A049794 A121617 A157656 this_sequence A131512 A147388 
               A003479
%Y A059100 Adjacent sequences: A059097 A059098 A059099 this_sequence A059101 A059102 
               A059103
%K A059100 easy,nonn,new
%O A059100 0,1
%A A059100 Henry Bottomley (se16(AT)btinternet.com), Feb 13 2001

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