A002065 -id:A002065 - cp's OEIS frontend

A002665 Continued fraction expansion of Lehmer's constant.

0, 1, 1, 2, 5, 34, 985, 1151138, 1116929202845, 1480063770341062927127746, 1846425204836010506550936273411258268076151412465

Offset: 0

N. J. A. Sloane

			0.592632718201636... = 0 + 1/(1 + 1/(1 + 1/(2 + 1/(5 + ...)))). - _Harry J. Smith_, May 14 2009

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).

Harry J. Smith, Table of n, a(n) for n = 0..12
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]
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A030125 (decimal expansion).
Cf. A002794, A002795, A002665, A002065.
Starting with n=2, a(n)/a(n-2) are in A096407.

digits = 1200; c[0] = 0; c[n_] := c[n] = c[n-1]^2 + c[n-1] + 1; LC[m_] := LC[m] = Cot[Sum[(-1)^k*ArcCot[c[k]], {k, 0, m}]] // N[#, digits+10]&; LC[10]; LC[m = 20]; While[Abs[LC[m] - LC[m-10]] > 10^-digits, m = m+10]; ContinuedFraction[LC[m]] (* Jean-François Alcover, Oct 08 2013 *)

default(realprecision, 2000);b=0.;
Lehmers=1/tan(suminf(k=1,b=b^2+b+1;(-1)^k*atan(1/b))+Pi/2);
x=contfrac(Lehmers);
for (n=1, 13, write("b002665.txt", n-1, " ", x[n])) \\ Harry J. Smith, May 14 2009; edited by Charles R Greathouse IV, Jan 21 2016

With a different offset: a(0)=1, a(1)=1, a(n+1)=(b(n)+b(n-1)+1)*a(n-1), n >= 1, b()=A002065, with b(0)=0, b(1)=1, b(2)=3, ...

More terms from Jeffrey Shallit

First two terms inserted by Harry J. Smith, May 14 2009

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.

A166105 Quadratic recurrence from Sylvester's sequence, but starting with a(0)=1 and a(1)=2.

Original entry on oeis.org

1, 2, 4, 14, 184, 33674, 1133904604, 1285739649838492214, 1653126447166808570252515315100129584, 2732827050322355127169206170438813672515557678636778921646668538491883474

Offset: 0

Jaume Oliver Lafont, Oct 06 2009

nonn

a(n) is the size of the set S(n) constructed recursively as follows: Let S(1) = {a,b} and let P(S) be the set of pairs (s,t) where s,t are members of S and s not equal to t. We define S(n+1) as the union of S(n) and P(S(n)). - David M. Cerna, Feb 07 2018

David M. Cerna, Table of n, a(n) for n = 0..12

Cf. A000058.

a:= [1, 2];; for n in [3..13] do a[n]:= a[n-1]^2 - a[n-2]^2 + a[n-2]; od; a; # Muniru A Asiru, Feb 07 2018

a := proc(n) option remember: if n=0 then 1 elif n=1 then 2 elif n>=2 then procname(n-1)^2 - procname(n-2)^2 + procname(n-2) fi; end:
seq(a(n), n = 0..10); # Muniru A Asiru, Feb 07 2018
a:=1:A:=a : to 10 do a:=a*(a-1)+2 : A:=A,a od:
print(A); # Robert FERREOL, May 05 2020

RecurrenceTable[{a[n]==a[n-1]^2-a[n-2]^2+a[n-2],a[0]==1,a[1]==2}, a, {n,0,10}] (* Vaclav Kotesovec, Jan 19 2015 *)

a(n)=if(n<2,[1,2][n+1],a(n-1)^2-a(n-2)^2+a(n-2));

Sum_{n>=0} 1/a(n) = 1.82689305142092757947757234878575... (compare with Sum_{n>=0} 1/A000058(n) = 1).
a(n) ~ c^(2^n), where c = 1.385089248334672909882206535871311526236739234374149506334120193387331772... . - Vaclav Kotesovec, Jan 19 2015
Sum_{n>=1} arctan(1/a(n)) = Pi/4. - Carmine Suriano, Apr 07 2015
a(0)=1, a(n+1) = a(n)*(a(n)-1) + 2. - Robert FERREOL, May 05 2020
a(n) = A002065(n) + 1 = (A232806(n) + 1)/2. - Robert FERREOL, May 31 2020

A379506 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the semiperimeter of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

Original entry on oeis.org

3, 4, 5, 13, 84, 85, 183, 16744, 16745, 33673, 566935464, 566935465, 1133904603, 642869824352293804, 642869824352293805, 1285739649838492213, 826563223583404284483387832630818684, 826563223583404284483387832630818685

Offset: 1

Miguel-Ángel Pérez García-Ortega, Dec 23 2024

The only Pythagorean triple whose inradius is equal to r and such that its long leg and its hypotenuse are consecutive is (2r+1,2r^2+2r,2r^2+2r+1).

			Triples begin:
  3, 4, 5;
  13, 84, 85;
  183, 16744, 16745;
  33673, 566935464, 566935465;

Miguel-Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz, and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint, 2024.

Cf. A002065 (short leg), A378395, A365577, A378963

{a0,b0,c0}={3,4,5};f[n_]:= Module[{fn0=a0+b0+c0+1,fn1=((a0+b0+c0+1)^2+1)/2},Do[{fn0,fn1}={2fn1+fn0,((2fn1+fn0)^2+1)/2},{n}];fn0];t[n_]:={f[n-1],(f[n-1]^2-1)/2,(f[n 1]^2+1)/2};ternas={a0,b0,c0};For[i=1,i<=6,i++,ternas=Join[ternas,t[i]]];ternas

A060136 Square array read by antidiagonals with T(n,k)=T(n,k-1)^2+n*T(n,k-1)+1 and T(n,0)=0.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 5, 3, 1, 0, 26, 13, 4, 1, 0, 677, 183, 25, 5, 1, 0, 458330, 33673, 676, 41, 6, 1, 0, 210066388901, 1133904603, 458329, 1805, 61, 7, 1, 0, 44127887745906175987802, 1285739649838492213, 210066388900, 3263441, 3966

Offset: 0

Henry Bottomley, Mar 05 2001

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Rows include A003095, A002065, A004019. Columns include A000004, A000012, A000027 (offset), A001844. Cf. A060137.

A232806 a(n) = a(n-1)^2/2 + 5/2, a(0) = 1.

Original entry on oeis.org

1, 3, 7, 27, 367, 67347, 2267809207, 2571479299676984427, 3306252894333617140505030630200259167

Offset: 0

Richard R. Forberg, Nov 30 2013

nonn

First differences are 2*A063573.

Cf. A063573, A002065, A166105.

a(n) = 2*A002065(n) + 1.
a(n) = 2*A166105(n) - 1. - Robert FERREOL, May 31 2020
a(n) = A002065(n) + A166105(n). - Alois P. Heinz, Jun 07 2020

A328701 Period in residues modulo n in iteration of x^2 + x + 1 starting at 0.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 4, 2, 1, 2, 2, 3, 3, 2, 8, 1, 2, 2, 2, 6, 2, 4, 4, 4, 3, 2, 6, 7, 2, 7, 16, 2, 1, 3, 2, 3, 2, 6, 4, 7, 6, 5, 2, 2, 4, 5, 8, 3, 4, 2, 6, 1, 2, 2, 12, 2, 7, 11, 2, 4, 7, 6, 32, 3, 2, 2, 2, 4, 3, 10, 4, 18, 3, 4, 2, 6, 6, 3, 8, 2, 7, 2, 6, 1, 5, 14, 4, 1, 2, 3

Offset: 1

Jianing Song, Oct 26 2019

nonn

a(n) is the period of {A002065 mod n}.
Let f(0) = 0, f(k+1) = (f(k)^2+f(k)+1) mod n, then a(n) is the smallest t such that f(i) = f(i+t) for all sufficiently large i.
Obviously a(n) <= A290731(n): f(1), f(2), ..., f(A290731(n)+1) are all of the form (s^2+s+1) mod n, so there must exists 1 <= i < j <= A290731(n)+1 such that f(i) = f(j), and a(n) <= j - i <= A290731(n). The equality seems to hold only for n = 3, 6 or n is a power of 2.

			In the following example, () denotes the cycles.
A002065(n) mod 4: 0, (1, 3), so a(4) = 2.
A002065(n) mod 7: 0, (1, 3, 6), so a(7) = 3.
A002065(n) mod 29: 0, (1, 3, 13, 9, 4, 21, 28), so a(29) = 7.
A002065(n) mod 61: (0, 1, 3, 13). {A002065(n) mod 61} enters into the cycle (0, 1, 3, 13) from the very beginning, so a(61) = 0.
A002065(n) mod 64: 0, (1, 3, 13, 55, 9, 27, 53, 47, 17, 51, 29, 39, 25, 11, 5, 31, 33, 35, 45, 23, 41, 59, 21, 15, 49, 19, 61, 7, 57, 43, 37, 63), so a(64) = 32.

Cf. A002065, A328702 (indices to enter the cycles), A290731.

a(n) = my(v=[0],k); for(i=2, n+1, k=(v[#v]^2+v[#v]+1)%n; v=concat(v, k); for(j=1, i-1, if(v[j]==k, return(i-j))))

a(n1*n2) = lcm(a(n1),a(n2)) if gcd(n1,n2) = 1.
It seems that for e > 0, a(3^e) = 2; a(5^e) = 1 if e = 1, 4*5^(e-2) otherwise; a(7^e) = 3; a(11^e) = 2 if e = 1, 10*11^(e-2) otherwise; a(13^e) = 3 if e = 1, 12*13^(e-2) otherwise ...
Proof that a(2^e) = 2^(e-1) by induction: we will show that {f(1), f(2), ..., f(2^(e-1))} is a reduced system modulo 2^e, where f is defined in the comment section. It is easy to see that this is true for e = 1, 2.
Suppose that {f(1), f(2), ..., f(2^(e-1))} is a reduced system modulo 2^e, e = 1, 2. For each 1 <= i <= 2^(e-1), f(2^(e-1)+i) - f(i) = Sum_{j=i..2^(e-1)+i-1} (f(j+1)-f(j)) = Sum_{j=i..2^(e-1)+i-1} (f(j)^2+1) = 2^(e-1) + Sum_{j=i..2^(e-1)+i-1} f(j)^2. Of course, {f(i), f(i+1), ..., f(2^(e-1)+i-1)} is also a reduced system modulo 2^e.
Note that if x == y (mod 2^e), then x^2 == y^2 (mod 2^(e+1)). So f(2^(e-1)+i) - f(i) == 2^(e-1) + (1^2+3^2+5^2+...+(2^e-1)^2) == 2^e (mod 2^(e+1)), 1 <= i <= 2^(e-1). This shows that {f(1), f(2), ..., f(2^(e-1)), f(2^(e-1)+1), f(2^(e-1)+2), ..., f(2^e)} is a reduced system modulo 2^(e+1). QED.

A328702 Start with 0, a(n) is the smallest number of iterations: x -> (x^2+x+1) mod n needed to run into a cycle.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 1, 1, 2, 2, 3, 1, 0, 1, 2, 1, 3, 2, 4, 2, 1, 3, 1, 1, 2, 1, 4, 1, 1, 2, 2, 1, 3, 3, 2, 2, 2, 4, 0, 2, 4, 1, 5, 3, 2, 1, 3, 1, 3, 2, 3, 1, 9, 4, 3, 1, 4, 1, 6, 2, 0, 2, 2, 1, 2, 3, 7, 3, 1, 2, 2, 2, 3, 2, 2, 4, 3, 1, 4, 2, 4, 4, 14, 1, 3, 5, 1, 3, 8, 2, 1

Offset: 1

Jianing Song, Oct 26 2019

nonn

Let f(0) = 0, f(k+1) = (f(k)^2+f(k)+1) mod n, then a(n) is the smallest i such that f(i) = f(j) for some j > i.
Obviously a(n) <= A290731(n): f(1), f(2), ..., f(A290731(n)+1) are all of the form (s^2+s+1) mod n, so there must exists 0 <= i < j <= A290731(n)+1 such that f(i) = f(j), and a(n) <= i <= A290731(n). The equality seems to hold only for n = 2.
k divides A002065(m) for some m > 0 if and only if a(k) = 0, in which case all the indices m such that k divides A002065(m) are m = t*A328701(k), t = 0, 1, 2, 3, ...

			A002065(n) mod 4: 0, (1, 3). {A002065(n) mod 4} enters into the cycle (1, 3) from the 1st term on, so a(4) = 1.
A002065(n) mod 7: 0, (1, 3, 6). {A002065(n) mod 7} enters into the cycle (1, 3, 6) from the 1st term on, so a(7) = 1.
A002065(n) mod 11: 0, 1, 3, (2, 7). {A002065(n) mod 11} enters into the cycle (2, 7) from the 3rd term on, so a(11) = 3.
A002065(n) mod 19: 0, 1, 3, 13, (12, 5). {A002065(n) mod 19} enters into the cycle (12, 5) from the 4th term on, so a(19) = 4.
A002065(n) mod 61: (0, 1, 3, 13). {A002065(n) mod 61} enters into the cycle (0, 1, 3, 13) from the very beginning, so a(61) = 0.

Cf. A002065, A328701 (cycle length), A328703, A290731.

a(n) = my(v=[0],k); for(i=2, n+1, k=(v[#v]^2+v[#v]+1)%n; v=concat(v, k); for(j=1, i-1, if(v[j]==k, return(j-1))))

cp's OEIS Frontend

A002665 Continued fraction expansion of Lehmer's constant.

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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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A166105 Quadratic recurrence from Sylvester's sequence, but starting with a(0)=1 and a(1)=2.

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A379506 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the semiperimeter of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

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A060136 Square array read by antidiagonals with T(n,k)=T(n,k-1)^2+n*T(n,k-1)+1 and T(n,0)=0.

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A232806 a(n) = a(n-1)^2/2 + 5/2, a(0) = 1.

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