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Search: id:A059100
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| 2, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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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.
Binomial transformation yields A081908, with A081908(0)=1 dropped. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 05 2008]
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]
a(n) = A000196(A156798(n) - A000290(n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 16 2009]
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,1000
Eric Weisstein's World of Mathematics, Near-Square Prime
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FORMULA
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G.f.: (2-3x+3x^2)/(1-x)^3. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 05 2008]
a(n) = ((n-2)^2 + 2*(n+1)^2)/3. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 13 2009]
a(n)=2*n+a(n-1)-3 (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]
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EXAMPLE
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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]
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MAPLE
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with(combinat, fibonacci):seq(fibonacci(3, i)+1, i=0..49); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008
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MATHEMATICA
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a[n_]:=n^2+2; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 2008]
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PROGRAM
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(Other) sage: [lucas_number1(3, n, -2) for n in xrange(0, 50)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
(PARI) { for (n = 0, 1000, write("b059100.txt", n, " ", n^2+2); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 24 2009]
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CROSSREFS
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Cf. A000290, A002522, A056899. Apart from initial terms, same as A010000.
Cf. A000027, A069987 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]
Sequence in context: A049794 A121617 A157656 this_sequence A131512 A147388 A003479
Adjacent sequences: A059097 A059098 A059099 this_sequence A059101 A059102 A059103
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KEYWORD
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easy,nonn
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Feb 13 2001
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