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Search: id:A002065
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| A002065 |
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a(n+1) = a(n)^2 + a(n) + 1.
(Formerly M2961 N1197)
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+0
10
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| 0, 1, 3, 13, 183, 33673, 1133904603, 1285739649838492213, 1653126447166808570252515315100129583, 2732827050322355127169206170438813672515557678636778921646668538491883473
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n) = number of trees of height <= n, generated by unary and binary composition: S = x + (S) + (S,S) = x + (x) + (x,x) + (x,(x)) + ((x),x) + ((x)) + ((x),(x)) + (x,(x,x)) + ((x,x),x) + ((x),(x,x)) + ((x,x),(x)) + ((x,x)) + ((x,x),(x,x)) + ... (x is of height 1); the first difference sequence (beginning with 1), 1 2 10 170 33490 1133870930..., give the number h(n) of these trees whose the height is n, h(n + 1) = h(n) + h(n)*h(n) + 2h(n)*a(n-1), h(1) = 1; As h(n + 1)/h(n) = 1 + a(n) + a(n-1), 1 2 10 = 2*5 170 = 2*5*17 33490 = 2*5*17*197 1133870930 = 2*5*17*197*33877... - Claude Lenormand (claude.lenormand(AT)free.fr), Sep 05 2001
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.
D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
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LINKS
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A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.
S. R. Finch, Lehmer's Constant
Index entries for sequences of form a(n+1)=a(n)^2 + ...
Eric Weisstein's World of Mathematics, Lehmer's Constant
Eric Weisstein's World of Mathematics, Lehmer Cotangent Expansion
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FORMULA
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a(n)=floor(c^(2^n)) for n>0, where c=1.385089248334672909882206535871311526236739234374149506334120193387331772... - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 29 2002
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PROGRAM
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(PARI) a(n)=if(n<1, 0, a(n-1)^2+a(n-1)+1)
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CROSSREFS
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Cf. A002794, A002795, A002665, A030125, A002065, A063573.
Adjacent sequences: A002062 A002063 A002064 this_sequence A002066 A002067 A002068
Sequence in context: A114317 A081299 A117808 this_sequence A087601 A112093 A085010
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KEYWORD
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easy,nice,nonn
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AUTHOR
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njas
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