A061979 -id:A061979 - cp's OEIS frontend

A061980 Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.

1, 0, 3, 0, 2, 9, 0, 1, 8, 27, 0, 0, 6, 26, 81, 0, 0, 4, 23, 80, 243, 0, 0, 3, 20, 76, 242, 729, 0, 0, 3, 17, 72, 237, 728, 2187, 0, 0, 1, 17, 66, 232, 722, 2186, 6561, 0, 0, 1, 11, 66, 222, 716, 2179, 6560, 19683, 0, 0, 1, 11, 54, 222, 701, 2172, 6552, 19682, 59049

Offset: 0

Henry Bottomley, May 24 2001

			Array begins as:
    1,   0,   0,   0,   0,   0,   0, ...;
    3,   2,   1,   0,   0,   0,   0, ...;
    9,   8,   6,   4,   3,   3,   1, ...;
   27,  26,  23,  20,  17,  17,  11, ...;
   81,  80,  76,  72,  66,  66,  54, ...;
  243, 242, 237, 232, 222, 222, 202, ...;
  729, 728, 722, 716, 701, 701, 671, ...;
Antidiagonal rows begin as:
  1;
  0, 3;
  0, 2, 9;
  0, 1, 8, 27;
  0, 0, 6, 26, 81;
  0, 0, 4, 23, 80, 243;
  0, 0, 3, 20, 76, 242, 729;
  0, 0, 3, 17, 72, 237, 728, 2187;
  0, 0, 1, 17, 66, 232, 722, 2186, 6561;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Row sums are 6^n: A000400.
Columns are A000244, A024023, A060188, A061981, A061982 twice, A061983 twice, etc.
Cf. A000244, A061290, A061930, A061979, A061984, A061987.

A[n_, k_]:= A[n, k]= If[n==0, Boole[k==0], A[n-1,k] +A[n-1,Floor[k/2]] +A[n-1, Floor[k/3]]];
T[n_, k_]:= A[k, n-k];
Table[A[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 18 2022 *)

@CachedFunction
def A(n,k):
    if (n==0): return 0^k
    else: return A(n-1, k) + A(n-1, (k//2)) + A(n-1, (k//3))
def T(n, k): return A(k, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 18 2022

A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1.
T(n, k) = A(k, n-k).
Sum_{k=0..n} A(n, k) = A000400(n).
T(n, n) = A(n, 0) = A000244(n). - G. C. Greubel, Jun 18 2022

A076145 Floor of arithmetic-geometric mean of n and n^2.

Original entry on oeis.org

1, 2, 5, 8, 13, 17, 23, 28, 35, 42, 50, 58, 67, 76, 86, 96, 107, 118, 130, 143, 156, 169, 183, 198, 213, 228, 244, 260, 277, 295, 313, 331, 350, 369, 389, 409, 430, 451, 473, 495, 517, 540, 564, 588, 612, 637, 662, 688, 714, 741, 768, 795, 823, 852, 880, 910

Offset: 1

Zak Seidov, Nov 02 2002

nonn

Floor of arithmetic-geometric mean of n and prime(n) = A061979(n).

Cf. A061979.

Array[Floor[ArithmeticGeometricMean[#,#^2]]&,60] (* Harvey P. Dale, Sep 30 2012 *)

A234363 a(n) = floor(agm(n, triangular(n))), where agm denotes the arithmetic-geometric mean.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 15, 19, 23, 27, 32, 37, 42, 48, 54, 60, 66, 73, 80, 88, 95, 103, 111, 120, 129, 138, 147, 156, 166, 176, 187, 197, 208, 219, 231, 242, 254, 267, 279, 292, 305, 318, 331, 345, 359, 373, 388, 402, 417, 432, 448, 464, 479, 496, 512, 529, 546

Offset: 0

Alex Ratushnyak, Dec 24 2013

nonn

Cf. A000217, A061979, A076145, A234362.

Table[Floor[ArithmeticGeometricMean[n,(n(n+1))/2]],{n,0,60}] (* Harvey P. Dale, Aug 17 2014 *)

a(n) ~ Pi*n^2/(4*log(2*n)). - Vaclav Kotesovec, May 09 2016

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

M. F. Hasler, Sep 18 2020

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.)

			1.62885808884493884077629027798870804765763752833626903647603...

User Mathlover, To find the limit of three terms mean iteration, math.StackExchange.com, Jul 12 2013.
Wikipedia, Arithmetic-geometric mean, created Jan 2, 2002.
Wikipedia, Elementary symmetric polynomial, created Jan 28, 2005.

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

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

M. F. Hasler, Sep 18 2020

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]

			1.90992623354081532372267510978753355913562440802728405833885556866...

Brad Klee, Iterated averaging of triples, math-fun list (available for subscribers), Sep 18 2020.
User Mathlover, To find the limit of three terms mean iteration, math.StackExchange.com, Jul 12 2013.
Vladimir Reshetnikov, Arithmetic-geometric mean of 3 numbers, math.StackExchange.com, May 22 2016.
Wikipedia, Arithmetic-geometric mean, created Jan 2, 2002.
Wikipedia, Elementary symmetric polynomial, created Jan 28, 2005.

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.

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)

A234362 Floor(AGM(n^2, n^3)), where AGM denotes the arithmetic-geometric mean.

Original entry on oeis.org

0, 1, 5, 16, 35, 65, 106, 161, 231, 318, 425, 551, 700, 872, 1069, 1293, 1545, 1827, 2140, 2486, 2866, 3281, 3734, 4225, 4755, 5327, 5942, 6601, 7306, 8057, 8856, 9706, 10606, 11558, 12565, 13626, 14744, 15919, 17154, 18449, 19805, 21225, 22709, 24259, 25876, 27561

Offset: 0

Alex Ratushnyak, Dec 24 2013

nonn

Cf. A000290, A000578, A061979, A076145, A234363.

import math
for n in range(67):
  x = n*n
  y = x*n
  a = (x+y)*0.5
  g = math.sqrt(x*y)
  while abs(a-g)>0.00000000001:
    x = a
    y = g
    a = (x+y)*0.5
    g = math.sqrt(x*y)
  print(int(a), end=', ')

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

M. F. Hasler, Sep 18 2020

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).

			1.294575108116612643448643498210035367403797272156424586808664172...

Vladimir Reshetnikov, Arithmetic-geometric mean of 3 numbers, math.StackExchange.com, May 22 2016.
User Mathlover, To find the limit of three terms mean iteration, math.StackExchange.com, Jul 12 2013.
Wikipedia, Arithmetic-geometric mean, created Jan 2, 2002.

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.

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)

cp's OEIS Frontend

A061980 Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.

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A076145 Floor of arithmetic-geometric mean of n and n^2.

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A234363 a(n) = floor(agm(n, triangular(n))), where agm denotes the arithmetic-geometric mean.

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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)).

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PARI

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)).

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PARI

A234362 Floor(AGM(n^2, n^3)), where AGM denotes the arithmetic-geometric mean.

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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)).

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PARI