A002665 -id:A002665 - cp's OEIS frontend

A002065 a(n+1) = a(n)^2 + a(n) + 1.

0, 1, 3, 13, 183, 33673, 1133904603, 1285739649838492213, 1653126447166808570252515315100129583, 2732827050322355127169206170438813672515557678636778921646668538491883473

Offset: 0

N. J. A. Sloane

a(n) is the 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..., gives the number h(n) of these trees whose 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) gives sequence 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

This is a divisibility sequence, that is, if n divides m, then a(n) divides a(m). This is a particular case of the result: if p(x) is an integral polynomial then the sequence of n-th iterates p^n(x) (:= p(p^(n-1)(x)) with p^1(x) := p(x)), n = 1,2,..., of p(x) evaluated at x = 0 is a divisibility sequence. In this case p(x) = 1 + x + x^2. - Peter Bala, Mar 28 2018

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..12
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Steven R. Finch, Lehmer's Constant [Broken link]
Steven R. Finch, Lehmer's Constant [From the Wayback machine]
Stan C. Kalman and Barry L. Kwasny, Tail-recursive distributed representations and simple recurrent networks, Connection Science, 7 (1995), 61-80.
D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]
H. P. Robinson, Letter to N. J. A. Sloane, Jul 12 1971
Eric Weisstein's World of Mathematics, Lehmer's Constant
Eric Weisstein's World of Mathematics, Lehmer Cotangent Expansion
Wikipedia, Herbrand Structure
J. W. Wrench, Jr., Letters to N. J. A. Sloane, Feb 1974
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A002665, A002794, A002795, A030125, A063573, A232806.

[n le 1 select 0 else Self(n-1)^2 + Self(n-1) + 1: n in [1..15]]; // Vincenzo Librandi, Oct 05 2015

f[x_] := 1 + x + x^2 ; NestList[f, 1, 7] (* Geoffrey Critzer, May 04 2010 *)

a(n) := if n = 0 then 1 else a(n-1)^2+a(n-1)+1 $
makelist(a(n),n,0,8); /* Emanuele Munarini, Mar 23 2017 */

PARI
```
a(n)=if(n<1,0,a(n-1)^2+a(n-1)+1)
```

a(n) = floor(c^(2^n)) for n > 0, where c = 1.385089248334672909882206535871311526236739234374149506334120193387331772... - Benoit Cloitre, Nov 29 2002

a(n) = (A232806(n) - 1)/2 = (A232806(n-1)^2 + 3)/4. - Peter Bala, Mar 28 2018

A030125 Decimal expansion of Lehmer's constant.

Original entry on oeis.org

5, 9, 2, 6, 3, 2, 7, 1, 8, 2, 0, 1, 6, 3, 6, 1, 9, 7, 1, 0, 4, 0, 7, 8, 6, 0, 4, 9, 9, 5, 7, 0, 1, 4, 6, 9, 0, 8, 4, 2, 7, 5, 4, 0, 7, 1, 9, 7, 1, 6, 1, 0, 7, 1, 0, 9, 9, 5, 6, 2, 6, 0, 8, 1, 5, 8, 2, 4, 7, 3, 5, 2, 3, 6, 4, 1, 6, 0, 0, 0, 8, 5, 1, 0, 6, 6, 4, 7, 8, 4, 2, 9, 7, 1, 0, 1, 2, 5, 7, 0, 5, 1, 1, 8

Offset: 0

Eric W. Weisstein

Digits 999 and 1000 should be "48" not "65" as given in the Plouffe links. - Sean A. Irvine, Aug 24 2014

Named after the American mathematician Derrick Henry Lehmer (1905-1991). - Amiram Eldar, Jun 22 2021

			0.592632718201636197104078604995701469084275407197161...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.

Simon Plouffe and Sean A. Irvine, Table of n, a(n) for n = 0..2000 [Replaces terms 0..998 from Harry J. Smith.]
Steven R. Finch, Lehmer's Constant. [Broken link]
Steven R. Finch, Lehmer's Constant. [From the Wayback machine]
D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., Vol. 4, No. 2 (1938), pp. 323-340. [Annotated scanned copy]
Simon Plouffe, The Lehmer Constant to 1000 digits.
Simon Plouffe, The Lehmer constant to 1000 digits.
Eric Weisstein's World of Mathematics, Lehmer's Constant.

Cf. A002065, A002794, A002795, A030125.

Cf. A002665 (continued fraction). - Harry J. Smith, May 14 2009

RealDigits[With[{nn=15},Cot[Total[Last[#]ArcCot[First[#]]&/@Thread[ {NestList[ #^2+#+1&,0,nn],PadRight[{},nn+1,{1,-1}]}]]]],10,120][[1]] (* Harvey P. Dale, Jan 29 2012 *)

b=0.;1/tan(suminf(k=1,b=b^2+b+1;(-1)^k*atan(1/b))+Pi/2) \\ Charles R Greathouse IV, Jan 21 2016

Equals cot(Sum_{k>=0} (-1)^k * arccot(A002065(k))). - Amiram Eldar, Aug 18 2020

More terms from David W. Wilson

A002794 Numerators of convergents to Lehmer's constant.

Original entry on oeis.org

1, 1, 3, 16, 547, 538811, 620245817465, 692770666469127829226736, 1025344764595988314871439243086711931108916434521

Offset: 0

N. J. A. Sloane

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. W. Wrench, Jr., personal communication.

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]
J. W. Wrench, Jr., Letters to N. J. A. Sloane, Feb 1974

Cf. A002795, A002665, A030125, A002065.

Block[{$MaxExtraPrecision=1000},Numerator[Convergents[With[{nn=15},Cot[ Total[Last[#] ArcCot[First[#]]&/@Thread[{NestList[#^2+#+1&,0,nn], PadRight[{},nn+1,{1,-1}]}]]]],10]]] (* Harvey P. Dale, Jan 29 2012 *)

A002795 Denominators of convergents to Lehmer's constant.

Original entry on oeis.org

1, 2, 5, 27, 923, 909182, 1046593950039, 1168971346319460027570137, 1730152138254248421873938035305987364739567671241

Offset: 0

N. J. A. Sloane

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. W. Wrench, Jr., personal communication.

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]
J. W. Wrench, Jr., Letters to N. J. A. Sloane, Feb 1974.

Cf. A002794, A002665, A030125, A002065.

cp's OEIS Frontend

A096407 a(n) = f(n)/f(n-2) where f(k) = A002665(k).

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A002065 a(n+1) = a(n)^2 + a(n) + 1.

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A030125 Decimal expansion of Lehmer's constant.

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A002794 Numerators of convergents to Lehmer's constant.

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A002795 Denominators of convergents to Lehmer's constant.

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A174143 Partials sums of A002794.

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