A030450 -id:A030450 - cp's OEIS frontend

A258621 Decimal expansion of sum(1/A030450).

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

Offset: 1

Jani Melik, Jun 06 2015

			2.2569474585265979554627367244423422105592365088936959533....

Cf. A030450.

#include 
#include 
#define dBIT   512
#define dLIST  n, t, u
int main (void) {
mpfr_t dLIST;
mpfr_rnd_t trnd;
unsigned int i;
trnd = MPFR_RNDU;
mpfr_inits2 (dBIT, dLIST, (mpfr_ptr) 0);
mpfr_set_d (n, 1.0, trnd);
mpfr_set_d (t, 1.0, trnd);
for (i = 1; i <= 9; i++) {
    mpfr_mul_ui (t, t, i, trnd);
    mpfr_mul (t, t, t, trnd);
    mpfr_set_d (u, 1.0, trnd);
    mpfr_div (u, u, t, trnd);
    mpfr_add (n, n, u, trnd);
}
mpfr_printf ("%.106Rg", n);
mpfr_clears (dLIST, (mpfr_ptr) 0);
return 0;
}

def A030450(n) :
   return prod((n-i+1)^(2^i) for i in (1..n))
N(sum(1/A030450(n) for n in (0..9)), digits=106)

A052129 a(0) = 1; thereafter a(n) = n*a(n-1)^2.

Original entry on oeis.org

1, 1, 2, 12, 576, 1658880, 16511297126400, 1908360529573854283038720000, 29134719286683212541013468732221146917416153907200000000

Offset: 0

Reinhard Zumkeller, Feb 12 2002

Somos's quadratic recurrence sequence.
Iff n is prime (n>2), the n-adic valuation of a(2n) is 3*A001045(n) (three times the values at the prime indices of Jacobsthal numbers), which is 2^n+1. For example: the 11-adic valuation at a(22) = 2049 = 3*A001045(11)= 683.  3*683 = 2^11+1 = 2049.  True because: When n is prime, n-adic valuation is 1 at A052129(n), then doubles as n-increases to 2n, at which point 1 is added; thus A052129(2n) = 2^n+1. Since 3*A001045(n) = 2^n+1, n-adic valuation of A052129(2n) = 3*A001045(n) when n is prime. - Bob Selcoe, Mar 06 2014
Unreduced denominators of: f(1) = 1, f(n) = f(n-1) + f(n-1)/(n-1). - Daniel Suteu, Jul 29 2016

			a(3) = 3*a(2)^2 = 3*(2*a(1)^2)^2 = 3*(2*(1*a(0)^2)^2)^2 = 3*(2*(1*1^2)^2)^2 = 3*(2*1)^2 = 3*4 = 12.
G.f. = 1 + x + 2*x^2 + 12*x^3 + 576*x^4 + 1658880*x^5 + 16511297126400*x^6 + ...

S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

Vincenzo Librandi, Table of n, a(n) for n = 0..12
Sung-Hyuk Cha, On the k-ary Tree Combinatorics.
Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 172 (2017), 145-159.
Chao-Ping Chen and X. F. Han, On Somos' quadratic recurrence constant, Journal of Number Theory, Volume 166, September 2016, Pages 31-40.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 16.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008), 247-270.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article 6.
Gergo Nemes, On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant, Applicable Analysis and Discrete Mathematics, 5(1) (2011), 60-66.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.

Cf. A000142, A001045, A030450, A088679, A112302, A116603, A123851, A123852, A123853, A123854, A238462 (2-adic valuation).

Join[{1},RecurrenceTable[{a[1]==1,a[n]==n a[n-1]^2},a,{n,10}]]  (* Harvey P. Dale, Apr 26 2011 *)
a[ n_] := If[ n < 1, Boole[n == 0], Product[ (n - k)^2^k, {k, 0, n - 1}]]; (* Michael Somos, May 24 2013 *)
a[n_] := Product[ k^(2^(n - k)), {k,1,n}] (* Jonathan Sondow, Mar 17 2014 *)
NestList[{#[[1]]+1,#[[1]]*#[[2]]^2}&,{1,1},10][[All,2]] (* Harvey P. Dale, Jul 30 2018 *)

{a(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^2^k))}; /* Michael Somos, May 24 2013 */

a(n) ~ s^(2^n) / (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...) where s = 1.661687949633... (see A112302) and A116603. - Michael Somos, Apr 02 2006
a(n) = n * A030450(n - 1) if n>0. - Michael Somos, Oct 22 2006
a(n) = (a(n-1) + a(n-2)^2) * (a(n-1) / a(n-2))^2. - Michael Somos, Mar 20 2012
a(n) = product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 17 2014
A088679(n+1)/a(n) = n+1. -Daniel Suteu, Jul 29 2016

A005345 Number of elements of a free idempotent monoid on n letters.

Original entry on oeis.org

1, 2, 7, 160, 332381, 2751884514766, 272622932796281408879065987, 3641839910835401567626683593436003894250931310990279692, 848831867913830760986671126293000918118297635181600248839480614255059539078136221019132415247551725144817958905

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

An idempotent monoid satisfies the equation xx=x for any element x.

A squarefree word may be equivalent to a smaller or larger word as a consequence of the idempotent equation.

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 32.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Morgan Rogers, From free idempotent monoids to free multiplicatively idempotent rigs, arXiv:2408.17440 [math.RA], 2024. See pp. 21, 23.
Eric Weisstein's World of Mathematics, Monoid.
Eric Weisstein's World of Mathematics, Free Idempotent Monoid
Index entries for sequences related to monoids

A030449(n) = a(n) - 1.

Array[Sum[Binomial[#, k]* Product[(k - i + 1)^(2^i), {i, k}], {k, 0, #}] &, 10, 0] (* Michael De Vlieger, Sep 05 2024 *)

{a(n)=sum(k=0, n, binomial(n, k)*prod(i=1, k, (k-i+1)^2^i))} /* Michael Somos, Oct 22 2006 */

a(n) = Sum_{k=0..n} (C(n, k) Prod_{i=1..k} (k-i+1)^(2^i)).

Binomial transform of A030450. - Michael Somos, Oct 22 2006

One more term from Gabriel Cunningham (gcasey(AT)mit.edu), Nov 14 2004

A030449 Number of elements in the free band (idempotent semigroup) on n generators.

Original entry on oeis.org

1, 6, 159, 332380, 2751884514765, 272622932796281408879065986, 3641839910835401567626683593436003894250931310990279691

Offset: 1

marcel_j(AT)hilbert.maths.utas.edu.au (Marcel Jackson)

nonn

An idempotent semigroup satisfies the equation xx=x for any element x.

J. Howie, Fundamentals of Semigroup Theory, Oxford University Press 1995, p. 123.

Index entries for sequences related to semigroups

Cf. A030450. A005345(n)=a(n)+1.

a(n) = Sum_{k=1..n} C(n, k) A030450(k).

A296042 Decimal expansion of sqrt(1 + 2sqrt(2 + 3sqrt(3 + 4sqrt(4 + 5sqrt(5 + 6*sqrt(6 + ...)))))).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 03 2017

Decimal expansion of sqrt(1 + sqrt(8 + sqrt(432 + sqrt(1327104 + ... + sqrt(k*A030450(k) + ...))))).

			3.0833551418306944580511425800881719306014784933...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A030450, A072449, A099876, A099879, A105546, A105817, A239349, A254689, A257574, A257575, A257576, A257577, A257578.

cp's OEIS Frontend

A258621 Decimal expansion of sum(1/A030450).

Views

Author

Keywords

Examples

Crossrefs

Programs

C

Sage

A052129 a(0) = 1; thereafter a(n) = n*a(n-1)^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A005345 Number of elements of a free idempotent monoid on n letters.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A030449 Number of elements in the free band (idempotent semigroup) on n generators.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A296042 Decimal expansion of sqrt(1 + 2*sqrt(2 + 3*sqrt(3 + 4*sqrt(4 + 5*sqrt(5 + 6*sqrt(6 + ...)))))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A296042 Decimal expansion of sqrt(1 + 2sqrt(2 + 3sqrt(3 + 4sqrt(4 + 5sqrt(5 + 6*sqrt(6 + ...)))))).