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 1-3 of 3 results.

A306240 Number of ways to write n as x^9 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2 and z <= w.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Jan 31 2019

Keywords

Comments

Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 11, 17, 18, 47, 108, 109, 234, 359. Also, any nonnegative integer can be written as x^6 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2.
We have verified a(n) > 0 for all n = 0..2*10^7.

Examples

			a(11) = 1 with 11 = 1^9 + 2^3 + 0*1 + 1*2.
a(18) = 1 with 18 = 0^9 + 0^3 + 2*3 + 3*4.
a(109) = 1 with 109 = 1^9 + 4^3 + 1*2 + 6*7.
a(234) = 1 with 234 = 0^9 + 6^3 + 2*3 + 3*4.
a(359) = 1 with 359 = 1^9 + 2^3 + 10*11 + 15*16.
a(1978) = 3 with 1978 = 2^9 + 2^3 + 26*27 + 27*28 = 2^9 + 6^3 + 19*20 + 29*30 = 2^9 + 6^3 + 24*25 + 25*26.

Links

Crossrefs

Cf. A000578, A001014, A001017, A002378, A262813, A262815, A270488, A306225, A306227, A306239.

Programs

  • Mathematica
    TQ[n_]:=TQ[n]=IntegerQ[Sqrt[4n+1]];
tab={};Do[r=0;Do[If[TQ[n-x^9-y^3-z(z+1)],r=r+1],{x,0,Min[2,n^(1/9)]},{y,0,(n-x^9)^(1/3)},{z,0,(Sqrt[2(n-x^9-y^3)+1]-1)/2}];tab=Append[tab,r],{n,0,100}];Print[tab]

A306249 Number of ways to write n as x*(2x-1) + y*(3y-1) + z*(4z-1) + w*(5w-1), where x,y,z are nonnegative integers and w is 0 or 1.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Jan 31 2019

Keywords

Comments

Conjecture: a(n) > 0 for any nonnegative integer n.
This has been verified for n up to 10^6. By Theorem 1.2 of the linked 2017 paper of the author, any nonnegative integer can be written as x*(2x-1) + y*(3y-1) + z*(4z-1) with x,y,z integers.
We have some other similar conjectures. For example, we conjecture that each n = 0,1,2,... can be written as x*(3x-1)/2 + y*(5y-1)/2 + z*(7z-1)/2 + w*(9w-1)/2) (or x*(x-1) + y*(2y-1) + z*(3z-1) + w*(4w-1)) with x,y,z,w nonnegative integers.

Examples

			a(1) = 1 with 1 = 1*(2*1-1) + 0*(3*0-1) + 0*(4*0-1) + 0*(5*0-1).
a(2) = 1 with 2 = 0*(2*0-1) + 1*(3*1-1) + 0*(4*0-1) + 0*(5*0-1).
a(12) = 1 with 12 = 2*(2*2-1) + 1*(3*1-1) + 0*(4*0-1) + 1*(5*1-1).
a(26) = 1 with 26 = 2*(2*2-1) + 1*(3*1-1) + 2*(4*2-1) + 1*(5*1-1).
a(220) = 1 with 220 = 6*(2*6-1) + 7*(3*7-1) + 2*(4*2-1) + 0*(5*0-1).
a(561) = 1 with 561 = 17*(2*17-1) + 0*(3*0-1) + 0*(4*0-1) + 0*(5*0-1).
a(1356) = 1 with 1356 = 23*(2*23-1) + 1*(3*1-1) + 9*(4*9-1) + 1*(5*1-1).

Links

Crossrefs

Cf. A000326, A000384, A002378, A033991, A049450, A255350, A306225, A306227, A306239, A306242.

Programs

  • Mathematica
    HexQ[n_]:=HexQ[n]=IntegerQ[Sqrt[8n+1]]&&(n==0||Mod[Sqrt[8n+1]+1,4]==0);
tab={};Do[r=0;Do[If[HexQ[n-x(5x-1)-y(4y-1)-z(3z-1)],r=r+1],{x,0,Min[1,(Sqrt[20n+1]+1)/10]},{y,0,(Sqrt[16(n-x(5x-1))+1]+1)/8},{z,0,(Sqrt[12(n-x(5x-1)-y(4y-1))+1]+1)/6}];tab=Append[tab,r],{n,0,100}];Print[tab]

A306260 Number of ways to write n as w*(4w+1) + x*(4x-1) + y*(4y-2) + z*(4z-3) with w,x,y,z nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 2, 3, 3, 2, 4, 4, 3, 1, 2, 1, 2, 3, 1, 2, 5, 5, 4, 5, 5, 4, 3, 1, 2, 4, 4, 4, 4, 5, 5, 7, 2, 2, 5, 3, 4, 5, 5, 3, 7, 4, 2, 5, 2, 4, 7, 6, 6, 6, 5, 6, 5, 3, 5, 6, 5, 8, 9, 8, 4, 7, 2, 4, 9, 2, 6, 5, 8, 6, 7, 7, 2, 6, 4, 4, 12, 6, 5, 5, 7, 9, 8, 5, 6, 9, 8
Offset: 0

Views

Author

Zhi-Wei Sun, Feb 01 2019

Keywords

Comments

Conjecture 1: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 1, 2, 4, 7, 9, 11, 14, 23, 25, 28, 37.
Conjecture 2: Each n = 0,1,2,... can be written as w*(4w+2) + x*(4x-1) + y*(4y-2) + z*(4z-3) with w,x,y,z nonnegative integers.
Conjecture 3: Each n = 0,1,2,... can be written as 4*w^2 + x*(4x+1) + y*(4y-2) + z*(4z-3) with w,x,y,z nonnegative integers.
We have verified that a(n) > 0 for all n = 0..2*10^6. By Theorem 1.3 in the linked 2017 paper of the author, any nonnegative integer can be written as x*(4x-1) + y*(4y-2) + z*(4z-3) with x,y,z integers.

Examples

			a(11) = 1 with 11 = 1*(4*1+1) + 1*(4*1-1) + 1*(4*1-2) + 1*(4*1-3).
a(23) = 1 with 23 = 2*(4*2+1) + 1*(4*1-1) + 1*(4*1-2) + 0*(4*0-3).
a(25) = 1 with 25 = 0*(4*0+1) + 1*(4*1-1) + 2*(4*2-2) + 2*(4*2-3).
a(28) = 1 with 28 = 2*(4*2+1) + 0*(4*0-1) + 0*(4*0-2) + 2*(4*2-3).
a(37) = 1 with 37 = 1*(4*1+1) + 1*(4*1-1) + 1*(4*1-2) + 3*(4*3-3).

Links

Crossrefs

Cf. A001107, A002939, A007742, A033991, A255350, A306225, A306227, A306239, A306240, A306249, A306250.

Programs

  • Mathematica
    QQ[n_]:=QQ[n]=IntegerQ[Sqrt[16n+1]]&&Mod[Sqrt[16n+1],8]==1;
tab={};Do[r=0;Do[If[QQ[n-x(4x-1)-y(4y-2)-z(4z-3)],r=r+1],{x,0,(Sqrt[16n+1]+1)/8},{y,0,(Sqrt[4(n-x(4x-1))+1]+1)/4},{z,0,(Sqrt[16(n-x(4x-1)-y(4y-2))+9]+3)/8}];tab=Append[tab,r],{n,0,100}];Print[tab]
Showing 1-3 of 3 results.

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