cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A090854 a(n) is the least positive integer such that the integer part of the arithmetic-geometric mean of a(n) and 1 is equal to Fibonacci(n).

Original entry on oeis.org

1, 1, 4, 7, 13, 24, 43, 77, 137, 241, 421, 732, 1266, 2178, 3733, 6376, 10858, 18439, 31237, 52804, 89082, 150014, 252206, 423367, 709697, 1188136, 1986730, 3318386, 5536857, 9229483, 15370775, 25576584, 42524547, 70649205, 117290710
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2003

Keywords

Crossrefs

Cf. A090852, A090853, A090855.

Formula

floor( agm(a(n), 1) ) = Fibonacci(n), for n>=0.

A090855 a(n) is the least positive integer such that the integer part of the arithmetic-geometric mean of a(n) and 1 is equal to 2^n.

Original entry on oeis.org

1, 4, 10, 24, 55, 127, 288, 640, 1408, 3069, 6642, 14281, 30544, 65028, 137896, 291399, 613885, 1289715, 2702909, 5652038, 11795170, 24570079, 51095155, 106092067, 219972452, 455493427, 942031726, 1946056082, 4015916211
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2003

Keywords

Crossrefs

Cf. A090852, A090854, A090856.

Programs

  • Mathematica
    Flatten[{1, Table[Ceiling[y /. FindRoot[Log[Pi/(2*EllipticK[1 - y^2])] == n*Log[2], {y, n*2^n}, MaxIterations -> 1000]], {n, 1, 50}]}] (* Vaclav Kotesovec, Sep 28 2019 *)

Formula

floor( agm(a(n), 1) ) = 2^n, for n>=0.

A090853 a(n) is the least positive integer such that the arithmetic-geometric mean satisfies: floor( agm(a(n),a(n-2)) ) = a(n-1) for n>2, with a(1)=1, a(2)=2.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 28, 40, 55, 73, 94, 118, 145, 176, 211, 250, 293, 340, 391, 446, 505, 568, 635, 706, 781, 860, 943, 1030, 1121, 1216, 1315, 1418, 1525, 1637, 1754, 1876, 2003, 2135, 2272, 2414, 2561, 2713, 2870, 3032, 3199, 3371, 3548, 3730, 3917, 4109
Offset: 1

Views

Author

Paul D. Hanna, Dec 10 2003

Keywords

Crossrefs

Cf. A090852, A090854, A090855.

A090856 a(n) is the least positive integer such that the integer part of the arithmetic-geometric mean of a(n) and 1 is equal to 3^n.

Original entry on oeis.org

1, 7, 27, 104, 378, 1327, 4553, 15351, 51072, 168147, 548915, 1779377, 5734022, 18384612, 58688163, 186632570, 591509670, 1869118923, 5890466415, 18518945789, 58094637801, 181884111404, 568416743474, 1773443888599
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2003

Keywords

Crossrefs

Cf. A090852, A090854, A090857.

Formula

floor( agm(a(n), 1) ) = 3^n, for n>=0.

A332092 Decimal expansion of Arithmetic-geometric mean AGM(1, 2, 2) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 2, 2) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Sep 18 2020

Keywords

Comments

The Arithmetic-geometric mean of two values, AGM(x,y), is the limit of the sequence defined by iterations of (x,y) -> ((x+y)/2, sqrt(xy)). This can be generalized to any number of m variables by taking the vector of the k-th roots of the normalized k-th elementary symmetric polynomials in these variables, i.e., the average of all products of k among these m variables, with k = 1 .. m. After each iteration these m components are in strictly decreasing order unless they are all equal. Once they are in this order, the first one is strictly decreasing, the last one is strictly increasing, therefore they must all have the same limit.
Has this multi-variable AGM already been studied somewhere? Any contributions in that sense are welcome. (Other generalizations have also been proposed, cf. comments on StackExchange.)

Examples

			1.62885808884493884077629027798870804765763752833626903647603...

Links

Crossrefs

Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.

Programs

  • PARI
    f(k,x,S)={forvec(i=vector(k,i,[1,#x]), S+=vecprod(vecextract(x,i)),2); S/binomial(#x,k)} \\ normalized k-th elementary symmetric polynomial in x
AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
default(realprecision,100);digits(AGM([1,2,2])\.1^100)

A332093 Decimal expansion of Arithmetic-geometric mean AGM(1, 2, 3) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 2, 3) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Sep 18 2020

Keywords

Comments

The Arithmetic-geometric mean of two values, AGM(x,y), is the limit of the sequence defined by iterations of (x,y) -> ((x+y)/2, sqrt(xy)). This can be generalized to any number of m variables by taking the vector of the k-th roots of the normalized k-th elementary symmetric polynomials in these variables, i.e., the average of all products of k among these m variables, with k = 1 .. m. After each iteration these m components are in strictly decreasing order unless they are all equal. Once they are in this order, the first one is strictly decreasing, the last one is strictly increasing, therefore they both converge, and their limits (thus that of all components) must be the same.
Has this multi-variable AGM already been studied somewhere? Any references in that sense or formulas are welcome.
Other 3-argument generalizations of the AGM have been proposed, which all give different values whenever the three arguments are not all equal: replacing P(a,b,c) by (agm(a,b), agm(b,c), agm(a,c)) or (agm(a,agm(b,c)), cyclic...) one gets 1.9091574... resp. 1.9091504..., but these are less straightforwardly generalized to a symmetric function in more than 3 arguments. Using the average of the k-th roots rather than the root of the average (normalized elementary symmetric polynomial) yields 1.89321.... See the two StackExchange links and discussion on the math-fun list. [Edited by M. F. Hasler, Sep 23 2020]

Examples

			1.90992623354081532372267510978753355913562440802728405833885556866...

Links

Crossrefs

Cf. A332091 = AGM(1,1,2), A332092 = AGM(1,2,2).
Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.

Programs

  • PARI
    f(k,x,S)={forvec(i=vector(k,i,[1,#x]), S+=vecprod(vecextract(x,i)),2); S/binomial(#x,k)} \\ normalized k-th elementary symmetric polynomial in x
AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
default(realprecision,100);digits(AGM([1,2,3])\.1^100)

A090857 a(n) is the least positive integer such that the integer part of the arithmetic-geometric mean of a(n) and 2^n is equal to 3^n.

Original entry on oeis.org

1, 5, 17, 59, 203, 690, 2308, 7621, 24913, 80794, 260303, 834057, 2660049, 8449715, 26747224, 84407894, 265647824, 834016199, 2612728134, 8168761695, 25494031748, 79434416090, 247130166428, 767788267178, 2382328079245
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2003

Keywords

Examples

			a(6)=2308 since floor(agm(2308,2^6))=729=3^6, but floor(agm(2307,2^6))=728.

Crossrefs

Cf. A090852, A090855, A090856.

Formula

floor( agm(a(n), 2^n) ) = 3^n, for n>=0.

A231603 Floor of the arithmetic-geometric mean of n+n and n*n.

Original entry on oeis.org

0, 1, 4, 7, 11, 16, 22, 28, 35, 43, 52, 61, 70, 81, 92, 103, 115, 128, 141, 155, 170, 185, 200, 216, 233, 250, 268, 286, 305, 325, 344, 365, 386, 408, 430, 452, 475, 499, 523, 548, 573, 598, 625, 651, 678, 706, 734, 763, 792, 822, 852, 883, 914, 945, 978, 1010, 1043, 1077, 1111, 1145, 1180, 1216, 1252, 1288, 1325
Offset: 0

Views

Author

John R Phelan, Nov 11 2013

Keywords

Comments

Arithmetic-geometric mean of n+n and n*n.
AGM of the sum and product of n and n.
a(n) = agm(A005843(n), A000290(n)).

Examples

			a(2) = floor(agm(2+2, 2*2)) = floor(agm(4, 4)) = 4.
a(5) = floor(agm(10.0, 25.0)) = floor(agm(17.5, 15.811388)) = floor(agm(16.655695, 16.634281)) = floor(agm(16.644987, 16.644983)) = floor(16.644987) = 16.

Links

Crossrefs

Cf. A005843, A005843, A090852.

Programs

  • Java
    public class Agmnxn {
    private static final double TOLERANCE = Math.pow(10, -4);
    private static final long LENGTH = 250;
    public static void main(String[] args) {
       String series="";
       long n=0;
       while (series.length()
  • Mathematica
    Table[Floor[ArithmeticGeometricMean[2n,n^2]],{n,0,70}] (* Harvey P. Dale, Aug 03 2014 *)

Formula

a(n) = floor(agm(n+n,n*n)).

A332091 Decimal expansion of the arithmetic-geometric mean AGM(1, 1, 2) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 1, 2) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Sep 18 2020

Keywords

Comments

See the main entry A332093 for more information on the multi-argument AGM(...) used here. One main motivation for these entries is to find exact formulas for this function which seems not yet well studied in the literature, or at least for particular values like this one, A332092 = AGM(1,2,2) and A332093 = AGM(1,2,3). Any references to possibly existing works using this definition would be welcome.
Other 3-argument generalizations of the AGM have been proposed (cf. A332093) which will give different values for AGM(1,1,2).

Examples

			1.294575108116612643448643498210035367403797272156424586808664172...

Links

Crossrefs

Cf. A332092 (AGM(1,2,2)), A332093 (AGM(1,2,3)).
Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.

Programs

  • PARI
    f(k,x,S)={forvec(i=vector(k,i,[1,#x]), S+=vecprod(vecextract(x,i)),2); S/binomial(#x,k)} \\ normalized k-th elementary symmetric polynomial in x
AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
default(realprecision,100);digits(AGM([1,1,2])\.1^100)
Showing 1-9 of 9 results.

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