A356770 -id:A356770 - cp's OEIS frontend

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.

1, 2, 20, 280, 4620, 84084, 1633632, 33256080, 701149020, 15191562100, 336424047960, 7584833081280, 173575987821600, 4022766574898400, 94247674040476800, 2228957491057276320, 53150802525726081660, 1276661433215969608500, 30863850087221160009000

Offset: 0

Paul D. Hanna, Oct 10 2023

Equals the central terms of triangle A365770.
Conjectures: given A033042 is the sums of distinct powers of 5, then
(1) a(5*A033042(n)) == 4 (mod 5) for n > 0,
(2) a(5*A033042(n) + 1) == 2 (mod 5) for n > 0,
(3) a(n) == 0 (mod 5) for n > 0 except when n or n-1 equals 5*A033042(k) for some k >= 0.

Paolo Xausa, Table of n, a(n) for n = 0..500

Cf. A007004, A000108, A165817, A033042, A319578, A356770.

A365771[n_] := Binomial[2*n + 1, n]/(2*n + 1)*Binomial[3*n - 1, n];
Array[A365771, 20, 0] (* Paolo Xausa, Oct 12 2024 *)

{a(n) = binomial(2*n+1, n)/(2*n+1) * binomial(3*n-1, n)}
for(n=0,30,print1(a(n),", "))

from math import comb
def A365771(n): return comb(m:=(n<<1)+1,n)*comb(m+n-2,n)//m if n else 1 # Chai Wah Wu, Oct 11 2023

a(n) = A365770(2*n,n) for n >= 0.
a(n) = A000108(n) * A165817(n) for n >= 0.
a(n) = 2*A319578(n) = (2/3) * A007004(n) for n >= 1. - Peter Bala, Aug 25 2025

A357714 a(n) is the number of equations in the set E_{n,b} := {x+2^by=n^b, 2^bx+3^by=n^b, ..., k^bx+(k+1)^by=n^b, ..., n^bx+(n+1)^b*y=n^b} which admit at least one nonnegative integer solution when b is sufficiently large.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 4, 6, 5, 6, 4, 8, 5, 7, 7, 8, 5, 9, 5, 9, 8, 8, 6, 12, 7, 8, 8, 10, 6, 12, 7, 11, 9, 9, 9, 14, 7, 9, 9, 13, 7, 13, 8, 12, 12, 10, 8, 16, 9, 12, 10, 12, 8, 14, 10, 14, 11, 11, 9, 19, 9, 11, 13, 14, 11, 15, 9, 13, 11, 15, 9, 19, 10, 12, 14, 14, 12, 16, 10, 18, 13

Offset: 1

Luca Onnis, Oct 10 2022

nonn

Defining a(n,b) as the number of equations of the set E_{n,b} which admit at least one nonnegative integer solution, it's possible to prove the existence of b_0 such that for all b > b_0, a(n,b) = a(n) whose value does not depend on b anymore.

a(n) is the number of positive integers k such that k(k+1) <= n or k divides n or k+1 divides n.

			a(11) = 4 since for all b >= 29 the number of equations of the set E_{11,b} which admit at least one nonnegative integer solution is exactly equal to 4.
a(4) = 4 since for all b >= 1 the number of equations of the set E_{11,b} which admit at least one nonnegative integer solution is exactly equal to 4.

Cf. A000005, A356770.

Table[Ceiling[Sqrt[n] - 3/2] + Length[Divisors[n]], {n, 1, 100}]

a(n) = ceiling(sqrt(n) - 3/2) + A000005(n).
a(n) ~ A356770(n)/2 as n->infinity.
a(n) <= A356770(n) for all n >= 1.

A361197 a(n) is the number of equations in the set {x^2 + 2y^2 = n, 2x^2 + 3y^2 = n, ..., kx^2 + (k+1)y^2 = n, ..., nx^2 + (n+1)y^2 = n} which admit at least one nonnegative integer solution.

Original entry on oeis.org

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

Offset: 1

Luca Onnis, Mar 04 2023

nonn

Compared to the "linear" case given by A356770, the "quadratic" case given by this sequence has a more chaotic behavior.
a(n) >= 2 for all n > 1 since (n-1)*x^2 + n*y^2 = n and n*x^2 + (n+1)*y^2 = n always admit one integer solution (respectively (0,1) and (1,0)).
Conjecture: a(n) = 2 for infinitely many n.

			a(5) = 3. Consider the equations: x^2 + 2y^2 = 5, 2x^2 + 3y^2 = 5, 3x^2 + 4y^2 = 5, 4x^2 + 5y^2 = 5, 5x^2 + 6y^2 = 5. Only three of them admit at least one nonnegative integer solution, since 3x^2 + 4y^2 = 5 and x^2 + 2y^2 = 5 have no nonnegative integer solutions.

Cf. A356770.

b[m_] := m;
f[n_] := Table[Dimensions[Solve[b[k]*x^2 + b[k + 1]*y^2 == n, {x, y}, NonNegativeIntegers]][[1]], {k, 1, n}];

cp's OEIS Frontend

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.

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A357714 a(n) is the number of equations in the set E_{n,b} := {x+2^by=n^b, 2^bx+3^by=n^b, ..., k^bx+(k+1)^by=n^b, ..., n^bx+(n+1)^b*y=n^b} which admit at least one nonnegative integer solution when b is sufficiently large.

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A361197 a(n) is the number of equations in the set {x^2 + 2y^2 = n, 2x^2 + 3y^2 = n, ..., kx^2 + (k+1)y^2 = n, ..., nx^2 + (n+1)y^2 = n} which admit at least one nonnegative integer solution.

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cp's OEIS Frontend

A365771 a(n) = binomial(2*n+1, n)/(2*n+1) * binomial(3*n-1, n) for n >= 0.

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Programs

Mathematica

PARI

Python

Formula

A357714 a(n) is the number of equations in the set E_{n,b} := {x+2^b*y=n^b, 2^b*x+3^b*y=n^b, ..., k^b*x+(k+1)^b*y=n^b, ..., n^b*x+(n+1)^b*y=n^b} which admit at least one nonnegative integer solution when b is sufficiently large.

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Formula

A361197 a(n) is the number of equations in the set {x^2 + 2y^2 = n, 2x^2 + 3y^2 = n, ..., k*x^2 + (k+1)*y^2 = n, ..., n*x^2 + (n+1)*y^2 = n} which admit at least one nonnegative integer solution.

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Programs

Mathematica

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.

A357714 a(n) is the number of equations in the set E_{n,b} := {x+2^by=n^b, 2^bx+3^by=n^b, ..., k^bx+(k+1)^by=n^b, ..., n^bx+(n+1)^b*y=n^b} which admit at least one nonnegative integer solution when b is sufficiently large.

A361197 a(n) is the number of equations in the set {x^2 + 2y^2 = n, 2x^2 + 3y^2 = n, ..., kx^2 + (k+1)y^2 = n, ..., nx^2 + (n+1)y^2 = n} which admit at least one nonnegative integer solution.