A274108 -id:A274108 - cp's OEIS frontend

A216665 Triangular array read by rows: T(n,k) is the number of partitions of n into k parts of 2 different sizes; n>=3, 2<=k<=n-1.

1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 3, 3, 2, 2, 1, 3, 3, 2, 2, 2, 1, 4, 3, 2, 3, 2, 2, 1, 4, 4, 5, 1, 3, 2, 2, 1, 5, 5, 3, 4, 2, 3, 2, 2, 1, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1, 6, 6, 4, 5, 2, 4, 2, 3, 2, 2, 1, 6, 6, 7, 5, 5, 1, 4, 2, 3, 2, 2, 1, 7, 6, 4, 3, 4, 4, 2, 4, 2, 3, 2, 2, 1

Offset: 3

Geoffrey Critzer, Sep 13 2012

Row sums = A002133.

First column (corresponding to k=2) = floor( (n-1)/2 ).

			T(8,3) = 3 because we have: 6+1+1, 4+2+2, 3+3+2.
Triangle indexed from n=3 and k=2:
1;
1, 1;
2, 2, 1;
2, 1, 2, 1;
3, 3, 2, 2, 1;
3, 3, 2, 2, 2, 1;
4, 3, 2, 3, 2, 2, 1;
4, 4, 5, 1, 3, 2, 2, 1;

Alois P. Heinz, Rows n = 3..143, flattened
N. B. Tani and S. Bouroubi, Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes, J. Integer Seqs., Vol. 14 (2011), #11.3.6. (This sequence is Table 1 on p. 10).

Cf. A117955, A117956, A274108, A274109.

nn=15;ss=Sum[Sum[y^2 x^(i+j)/(1-y x^i)/(1-y x^j),{j,1,i-1}],{i,1,nn}]; f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[ss,{x,0,nn}], {x,y}]]//Flatten

G.f.: Sum_{i>=1} Sum_{j=1..n-1} y^2*x^(i+j)/((1-y*x^j)*(1-y*x^i)).

A274109 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts with exactly two different sizes, the sizes being relatively prime (n >= 3, 2 <= k <= n-1).

Original entry on oeis.org

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

Offset: 3

N. J. A. Sloane, Jun 17 2016

			Triangle T(n,k) (with columns n >= 3 and k >= 2) begins as follows:
  1;
  1, 1;
  2, 2, 1;
  1, 1, 2, 1;
  3, 3, 2, 2, 1;
  2, 2, 2, 2, 2, 1;
  3, 3, 2, 3, 2, 2, 1;
  2, 2, 4, 1, 3, 2, 2, 1;
  5, 5, 3, 4, 2, 3, 2, 2, 1;
  2, 2, 2, 2, 3, 2, 3, 2, 2, 1;
  6, 6, 4, 5, 2, 4, 2, 3, 2, 2, 1;
  3, 3, 5, 3, 4, 1, 4, 2, 3, 2, 2, 1;
  4, 4, 3, 3, 4, 4, 2, 4, 2, 3, 2, 2, 1;
  ...

N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire 03 (2015), 18-27.
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Elemente der Mathematik 72(2) (2017), 66-74.

Row sums give A274108.

Cf. A002133, A117955, A117956, A216665.

A377812 Number of quadruples of positive integers (x,y,a,b) such that a < b, gcd(a,b) = gcd(x,y) = 1 and ax + by = n.

Original entry on oeis.org

0, 0, 1, 2, 5, 4, 11, 9, 15, 12, 27, 14, 37, 22, 32, 31, 59, 26, 71, 38, 58, 48, 97, 42, 99, 62, 93, 68, 141, 48, 157, 91, 120, 94, 150, 78, 207, 112, 154, 108, 241, 84, 259, 138, 170, 150, 295, 116, 289, 144, 232, 178, 353, 136, 304, 188, 274, 210, 413, 132

Offset: 1

Anshveer Bindra, Nov 08 2024

nonn

Number of partitions of n into parts with exactly two different sizes, the sizes being relatively prime and also the multiplicities of the two part sizes being relatively prime. - Andrew Howroyd, Nov 10 2024

Cf. A002133, A274108.

a(n)={sum(b=2, n-1, sum(y=1, (n-1)\b, my(s=n-b*y); sumdiv(s, a, aAndrew Howroyd, Nov 10 2024

PARI

seq(n)={my(v=Vec(sum(k=1, n-1, numdiv(k)*x^k, O(x^n))^2, -n), u=vector(n, n, moebius(n))); dirmul(dirmul(u,u), vector(#v, n, v[n]+numdiv(n)-sigma(n))/2)} \\ Andrew Howroyd, Nov 10 2024

Python

def a(n): count = 0 for a in range(1, n+1): for b in range(a + 1, n+1): if gcd(a, b) == 1: for x in range(1, n+1): for y in range(1, n+1): if gcd(x, y) == 1 and a * x + b * y == n: count += 1 return count print([a(n) for n in range(1,21)])

Python

from math import gcd from sympy import divisors def A377812(n): return sum(1 for ax in range(1,n-1) for a in divisors(ax,generator=True) for b in divisors(n-ax,generator=True) if aChai Wah Wu, Dec 11 2024

Formula

Moebius transform of A274108. - Andrew Howroyd, Nov 10 2024

Extensions

a(21) onwards from Andrew Howroyd, Nov 10 2024

cp's OEIS Frontend

A216665 Triangular array read by rows: T(n,k) is the number of partitions of n into k parts of 2 different sizes; n>=3, 2<=k<=n-1.

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A274109 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts with exactly two different sizes, the sizes being relatively prime (n >= 3, 2 <= k <= n-1).

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A377812 Number of quadruples of positive integers (x,y,a,b) such that a < b, gcd(a,b) = gcd(x,y) = 1 and ax + by = n.

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PARI

PARI

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Python

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cp's OEIS Frontend

A216665 Triangular array read by rows: T(n,k) is the number of partitions of n into k parts of 2 different sizes; n>=3, 2<=k<=n-1.

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Author

Keywords

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Examples

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Crossrefs

Programs

Mathematica

Formula

A274109 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts with exactly two different sizes, the sizes being relatively prime (n >= 3, 2 <= k <= n-1).

Views

Author

Keywords

Examples

Links

Crossrefs

A377812 Number of quadruples of positive integers (x,y,a,b) such that a < b, gcd(a,b) = gcd(x,y) = 1 and a*x + b*y = n.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

PARI

Python

Python

Formula

Extensions

A377812 Number of quadruples of positive integers (x,y,a,b) such that a < b, gcd(a,b) = gcd(x,y) = 1 and ax + by = n.