A306614 -id:A306614 - cp's OEIS frontend

A306606 Number of ways to write n as w^2 + T(x)^2 + Pen(y)^2 + 2Pen(z)^2, where w,x,y,z are integers with w > 0 and x >= 0, and T(m) = m(m+1)/2 and Pen(m) = m*(3m-1)/2.

1, 2, 2, 3, 4, 3, 3, 3, 3, 6, 5, 4, 7, 5, 2, 5, 5, 6, 6, 3, 4, 5, 1, 2, 5, 6, 5, 5, 6, 3, 3, 2, 2, 6, 5, 3, 9, 7, 3, 6, 4, 5, 8, 3, 6, 7, 2, 4, 7, 6, 8, 9, 9, 6, 6, 5, 3, 9, 9, 6, 11, 7, 4, 10, 5, 6, 13, 5, 5, 9, 2, 5, 7, 7, 7, 10, 7, 3, 5, 3, 3, 6, 6, 5, 11, 6, 6, 9, 4, 6, 11, 3, 5, 9, 2, 5, 4, 4, 7, 11

Offset: 1

Zhi-Wei Sun, Feb 28 2019

nonn

Conjecture: a(n) > 0 for all n > 0. In other words, any positive integer n can be written as s^2 + t^2 + u^2 + 2*v^2, where s is a positive integer, t is a triangular number, u and v are generalized pentagonal numbers.
I have verified a(n) > 0 for all n = 1..5*10^6.
It is well known that any positive odd integer can be written as x^2 + y^2 + 2*z^2 with x,y,z integers.
See also A306614 for similar conjectures.

			a(1) = 1 with 1 = 1^2 + T(0)^2 + Pen(0)^2 + 2*Pen(0)^2.
a(23) = 1 with 23 = 4^2 + T(1)^2 + Pen(-1)^2 + 2*Pen(1)^2.
a(335) = 1 with 335 = 18^2 + T(2)^2 + Pen(0)^2 + 2*Pen(1)^2.
a(3695) = 1 with 3695 = 53^2 + T(7)^2 + Pen(-1)^2 + 2*Pen(-2)^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000217, A000290, A001318, A275344, A306614.

t[x_]:=t[x]=x(x+1)/2; p[x_]:=p[x]=x(3x-1)/2;
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-t[x]^2-p[y]^2-2*p[z]^2],r=r+1],{x,0,(Sqrt[8*Sqrt[n-1]+1]-1)/2},{y,-Floor[(Sqrt[24*Sqrt[n-1-t[x]^2]+1]-1)/6],(Sqrt[24*Sqrt[n-1-t[x]^2]+1]+1)/6},
{z,-Floor[(Sqrt[24*Sqrt[(n-1-t[x]^2-p[y]^2)/2]+1]-1)/6],(Sqrt[24*Sqrt[(n-1-t[x]^2-p[y]^2)/2]+1]+1)/6}];tab=Append[tab,r],{n,1,100}];Print[tab]

A306690 Number of ways to write n as u^4 + (v(v+1)/2)^2 + (x(3x+1)/2)^2 + (y(5y+1)/2)^2 + (z(9z+1)/2)^2, where u and v are nonnegative integers and x,y,z are integers.

Original entry on oeis.org

1, 3, 3, 1, 2, 5, 4, 1, 1, 4, 6, 4, 1, 3, 5, 2, 2, 6, 6, 3, 5, 8, 6, 2, 2, 9, 14, 9, 2, 9, 14, 7, 2, 5, 10, 12, 9, 6, 8, 7, 5, 9, 10, 6, 4, 10, 10, 4, 1, 4, 12, 11, 5, 4, 10, 6, 5, 5, 5, 8, 8, 7, 8, 5, 1, 7, 11, 5, 3, 5, 8, 5, 3, 1, 6, 10, 4, 4, 6, 4, 1, 8, 8, 8, 6, 7, 11, 6, 1, 2, 10, 8, 3, 2, 7, 6, 1, 4, 8, 9, 4

Offset: 0

Zhi-Wei Sun, Mar 05 2019

nonn

Conjecture 1: a(n) > 0 for any nonnegative integer n.
Conjecture 2: Each n = 0,1,2,... can be written as f(u,v,x,y,z) with u,v,x,y,z integers, where f is any of the following polynomials: u^4 + (v*(v+1)/2)^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(5z+3)/2)^2, u^4 + (v*(v+1)/2)^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(3z+2))^2, (u*(u+1)/2)^2 + (v*(3v+1)/2)^2 + (x*(5x+1)/2)^2 + (y*(5y+3)/2)^2 + (z*(3z+2))^2, (u*(u+1)/2)^2 + (v*(3v+1)/2)^2 + (x*(5x+1)/2)^2 + (y*(5y+3)/2)^2 + (z*(4z+3))^2, (u*(u+1)/2)^2 + (v*(3v+1)/2)^2 + (x*(5x+1)/2)^2 + (y*(5y+3)/2)^2 + (z*(9z+7)/2)^2.
We have verified Conjectures 1 and 2 for n up to 2*10^6 and 10^6 respectively.

			a(8) = 1 with 8 = 0^4 + (0*(0+1)/2)^2 + (1*(3*1+1)/2)^2 + ((-1)*(5*(-1)+1)/2)^2 + (0*(9*0+1)/2)^2.
a(2953) = 1 with 2953 = 6^4 + (8*(8+1)/2)^2 + (0*(3*0+1)/2)^2 + (0*(5*0+1)/2)^2 + (2*(9*2+1)/2)^2.
a(8953) = 1 with 8953 = 2^4 + (7*(7+1)/2)^2 + (6*(3*6+1)/2)^2 + ((-1)*(5*(-1)+1)/2)^2 + ((-4)*(9*(-4)+1)/2)^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000118, A000217, A000290, A000583, A001318, A057569, A306606, A306614.

t[x_]:=t[x]=(x(x+1)/2)^2; f[x_]:=f[x]=(x(5x+1)/2)^2; g[x_]:=g[x]=(x(9x+1)/2)^2; SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; PQ[n_]:=PQ[n]=SQ[n]&&SQ[24*Sqrt[n]+1];
tab={};Do[r=0;Do[If[PQ[n-k^4-t[x]-f[y]-g[z]],r=r+1],{k,0,n^(1/4)},{x,0,(Sqrt[8*Sqrt[n-k^4]+1]-1)/2},{y,-Floor[(Sqrt[40*Sqrt[n-k^4-t[x]]+1]+1)/10],(Sqrt[40*Sqrt[n-k^4-t[x]]+1]-1)/10},{z,-Floor[(Sqrt[72*Sqrt[n-k^4-t[x]-f[y]]+1]+1)/18],(Sqrt[72*Sqrt[n-k^4-t[x]-f[y]]+1]-1)/18}];tab=Append[tab,r],{n,0,100}];Print[tab]

cp's OEIS Frontend

A306606 Number of ways to write n as w^2 + T(x)^2 + Pen(y)^2 + 2Pen(z)^2, where w,x,y,z are integers with w > 0 and x >= 0, and T(m) = m(m+1)/2 and Pen(m) = m*(3m-1)/2.

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A306690 Number of ways to write n as u^4 + (v(v+1)/2)^2 + (x(3x+1)/2)^2 + (y(5y+1)/2)^2 + (z(9z+1)/2)^2, where u and v are nonnegative integers and x,y,z are integers.

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cp's OEIS Frontend

A306606 Number of ways to write n as w^2 + T(x)^2 + Pen(y)^2 + 2*Pen(z)^2, where w,x,y,z are integers with w > 0 and x >= 0, and T(m) = m*(m+1)/2 and Pen(m) = m*(3m-1)/2.

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A306690 Number of ways to write n as u^4 + (v*(v+1)/2)^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(9z+1)/2)^2, where u and v are nonnegative integers and x,y,z are integers.

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A306606 Number of ways to write n as w^2 + T(x)^2 + Pen(y)^2 + 2Pen(z)^2, where w,x,y,z are integers with w > 0 and x >= 0, and T(m) = m(m+1)/2 and Pen(m) = m*(3m-1)/2.

A306690 Number of ways to write n as u^4 + (v(v+1)/2)^2 + (x(3x+1)/2)^2 + (y(5y+1)/2)^2 + (z(9z+1)/2)^2, where u and v are nonnegative integers and x,y,z are integers.