A144788 -id:A144788

A144811

Decimal expansion of the constant c = 1.429887738657309204890861721408999... arising in A144788.

+20
9

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..105.

FORMULA

c = 1.429887738657309204890861721408999...; A144788(n) = round(c^(3^n)).

CROSSREFS

Cf. A076393, A144788, A144802, A144803, A144804, A144805, A144806, A144807, A144808, A144809, A144810, A144812.

KEYWORD

cons,nonn

AUTHOR

Artur Jasinski, Sep 22 2008

STATUS

approved

A082732

a(1) = 1, a(2) = 3, a(n) = LCM of all the previous terms + 1.

+10
24

1, 3, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The LCM is in fact the product of all previous terms. From a(5) onwards the terms alternately end in 57 and 93.

LINKS

Table of n, a(n) for n=1..10.

FORMULA

For n>=3, a(n+1) = a(n)^2 - a(n) + 1.

For n>=3, a(n) = A004168(n-3) + 1. - Max Alekseyev, Aug 09 2019

1/3 = Sum_{n=3..oo} 1/a(n) = 1/4 + 1/13 + 1/157 + 1/24493 + ... or 1 = Sum_{n=3..oo} 3/a(n) = 3/4 + 3/13 + 3/157 + 3/24493 + .... If we take segment of length 1 and cut off in each step fragment of maximal length such that numerator of fraction is 3, denominators of such fractions will be successive numbers of this sequence. - Artur Jasinski, Sep 22 2008

a(n+2)=1.8806785436830780944921917650127503562630617563236301969047995953391\

4798717695395204087358090874194124503892563356447954254847544689332763...^(2^n). - Artur Jasinski, Sep 22 2008

MATHEMATICA

a[1] = 1; a[2] = 3; a[n_] := Apply[LCM, Table[a[i], {i, 1, n - 1}]] + 1; Table[ a[n], {n, 1, 10}]

c=1.8806785436830780944921917650127503562630617563236301969047995953391479871\

7695395204087358090874194124503892563356447954254847544689332763; Table[c^(2^n), {n, 1, 6}] or a = {}; k = 4; Do[AppendTo[a, k]; k = k^2 - k + 1, {n, 1, 10}]; a (* Artur Jasinski, Sep 22 2008 *)

CROSSREFS

Cf. A000058, A004168, A144743, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Apr 14 2003

EXTENSIONS

More terms from Robert G. Wilson v, Apr 15 2003

STATUS

approved

A144780

Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 6.

+10
17

6, 31, 931, 865831, 749662454731, 561993796032558961827631, 315837026779085485103718410756049100028793244531

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..11

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.

FORMULA

a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 6.

a(n) ~ c^(2^n) where is c is 2.350117384... (A144804).

MATHEMATICA

a = {}; k = 6; Do[AppendTo[a, k]; k = k^2 - k + 1, {n, 1, 10}]; a

CROSSREFS

Cf. A000058, A082732, A144779, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788, A144804.

KEYWORD

nonn,easy

AUTHOR

Artur Jasinski, Sep 21 2008

EXTENSIONS

a(8) moved to b-file by Hugo Pfoertner, Aug 30 2020

STATUS

approved

A144784

Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 11.

+10
17

11, 111, 12211, 149096311, 22229709804712411, 494159998001727075769152612720511, 244194103625066907517263589918036880566782292998362610615987380611

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For the "exact" formula, compare the Aho-Sloane reference in A000058. - N. J. A. Sloane, Apr 07 2014

LINKS

Table of n, a(n) for n=1..7.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.

FORMULA

a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 11.

a(n) ~ c^(2^n) where c = 3.242214... (see A144808).

MATHEMATICA

a = {}; r = 11; Do[AppendTo[a, r]; r = r^2 - r + 1, {n, 1, 10}]; a

CROSSREFS

Cf. A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788, A144808.

See A239900 for another version.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Sep 21 2008

STATUS

approved

A144779

Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 5.

+10
16

5, 21, 421, 176821, 31265489221, 977530816197201697621, 955566496615167328821993756200407115362021, 913107329453384594090655605142589591944556891901674138343716072975722193082773842421

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..8.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution, College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.

FORMULA

a(n) = round(2.127995907464107054577351...)^(2^n) = round(A144803^(2^n)). [corrected by Joerg Arndt, Jan 15 2021]

a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 5.

EXAMPLE

a(0) = 4, a(1) = 4+1 = 5, a(2) = 4*5+1 = 21, a(3) = 4*5*21+1 = 421, a(4) = 4*5*21*421+1 = 176821, ... - Philippe Deléham, Apr 19 2013

MATHEMATICA

a = {}; k = 5; Do[AppendTo[a, k]; k = k^2 - k + 1, {n, 1, 10}]; a (* Artur Jasinski, Sep 21 2008 *)

NestList[#^2-#+1&, 5, 8] (* Harvey P. Dale, Jan 17 2012 *)

CROSSREFS

Cf. A000058, A082732, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Sep 21 2008

STATUS

approved

A144782

Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 9.

+10
15

9, 73, 5257, 27630793, 763460694178057, 582872231554839914154126117193, 339740038317718918529575265905277902175236102890836244082057

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..7.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.

FORMULA

a(n) ~ c^(2^n) with c = 2.918012...

a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 9.

MATHEMATICA

a = {}; r = 9; Do[AppendTo[a, r]; r = r^2 - r + 1, {n, 1, 10}]; a

NestList[#^2-#+1&, 9, 10] (* Harvey P. Dale, Aug 31 2014 *)

CROSSREFS

Cf. A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Sep 21 2008

STATUS

approved

A144783

Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 10.

+10
15

10, 91, 8191, 67084291, 4500302031888391, 20252718378218776104731448680491, 410172601707440572557971589875869064610540321970215293555320591, 168241563191450680898537024308131628447885486994777537422995633998657738457104605412468520116391629012196009150161991233268691

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

REFERENCES

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342

LINKS

Table of n, a(n) for n=1..8.

FORMULA

a(n) = round((3.08435104906918990233569320020272148875011089837398848476442237096569...)^(2^n)) = round(A144807^(2^n)). [corrected by Joerg Arndt, Jan 15 2021]

a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 10.

MATHEMATICA

a = {}; r = 10; Do[AppendTo[a, r]; r = r^2 - r + 1, {n, 1, 10}]; a (* or *)

Table[Round[3.08435104906918990233569320020272148875011089837398848476442237096569188195734783139337492942278549518507672786196650938869338548385641623^(2^n)], {n, 1, 8}] (* Artur Jasinski *)

NestList[#^2-#+1&, 10, 8] (* Harvey P. Dale, May 07 2017 *)

CROSSREFS

Cf. A000058, A082732, A144779, A144780, A144781, A144782, A144783, A144784, A144785, A144786, A144787, A144788.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Sep 21 2008

STATUS

approved

A144785

Variant of Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 12

+10
15

12, 133, 17557, 308230693, 95006159799029557, 9026170399758739819525199160586693, 81471752085480849000657595909467634426991447160798281416700808089557

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

REFERENCES

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342

LINKS

Table of n, a(n) for n=1..7.

FORMULA

a(n) =3.39277252592669675143137065018187376847206615308598784654603692312172475924599026837940758013759324881455503678006543568111163817496672898^(2^n) a(n+1) = a(n)^2 - a(n) + 1, with a(1) = 11

MATHEMATICA

a = {}; r = 12; Do[AppendTo[a, r]; r = r^2 - r + 1, {n, 1, 10}]; a or Table[Round[3.39277252592669675143137065018187376847206615308598784654603692312172475924599026837940758013759324881455503678006543568111163817496672898^(2^n)], {n, 1, 8}] (*Artur Jasinski*)

NestList[#^2-#+1&, 12, 6] (* Harvey P. Dale, Jan 01 2016 *)

CROSSREFS