A306225 -id:A306225 - cp's OEIS frontend

A306227 Number of ways to write n as w + x^4 + y(y+1)/2 + z(z+1)/2, where w is 0 or 1, and x, y, z are nonnegative integers with x >= w and y < z.

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

Offset: 1

Zhi-Wei Sun, Jan 30 2019

nonn

Conjecture: a(n) > 0 for all n > 0. In other words, any positive integer can be written as the sum of two fourth powers one of which is 0 or 1, and two distinct triangular numbers.
We have verified a(n) > 0 for all n = 1..10^6. The conjecture implies that the set S = {x^4 + y*(y+1)/2: x,y = 0,1,2,...} is an additive basis of order two (i.e., the sumset S + S coincides with {0,1,2,...}).
See also A306225 for a similar conjecture.

			a(1) = 1 with 1 = 0 + 0^4 + 0*1/2 + 1*2/2.
a(2) = 1 with 2 = 0 + 1^4 + 0*1/2 + 1*2/2.
a(14) = 1 with 14 = 0 + 1^4 + 2*3/2 + 4*5/2.
a(3774) = 1 with 3774 = 1 + 5^4 + 52*53/2 + 59*60/2.
a(7035) = 1 with 7035 = 0 + 3^4 + 48*49/2 + 107*108/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000217, A000583, A262813, A262941, A262954, A262956, A262959, A270566, A306225.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[TQ[n-x-y^4-z(z+1)/2],r=r+1],{x,0,Min[1,(n-1)/2]},{y,x,(n-1-x)^(1/4)},{z,0,(Sqrt[4(n-1-x-y^4)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

A306239 Number of ways to write n as x^3 + y^3 + pen(z) + pen(w), where x, y, z, w are nonnegative integers with x <= y and z <= w, and pen(k) denotes the pentagonal number k(3k-1)/2.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Jan 31 2019

nonn

Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 4, 5, 16, 20, 46. Also, any nonnegative integer not equal to 16 can be written as x^6 + y^3 + pen(z) + pen(w) with x, y, z, w nonnegative integers.

We have verified a(n) > 0 for all n = 0..5*10^6.

			a(4) = 1 with 4 = 1^3 + 1^3 + pen(1) + pen(1).
a(5) = 1 with 5 = 0^3 + 0^3 + pen(0) + pen(2).
a(16) = 1 with 16 = 2^3 + 2^3 + pen(0) + pen(0).
a(20) = 1 with 20 = 0^3 + 2^3 + pen(0) + pen(3).
a(46) = 1 with 46 = 1^3 + 1^3 + pen(4) + pen(4).

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

Cf. A000326, A000578, A306225, A306227.

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0);
tab={};Do[r=0;Do[If[PenQ[n-x^3-y^3-z(3z-1)/2],r=r+1],{x,0,(n/2)^(1/3)},{y,x,(n-x^3)^(1/3)},{z,0,(Sqrt[12(n-x^3-y^3)+1]+1)/6}];tab=Append[tab,r],{n,0,100}];Print[tab]

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

Zhi-Wei Sun, Jan 31 2019

nonn

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.

			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.

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

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

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

Zhi-Wei Sun, Jan 31 2019

nonn

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.

			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).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.

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

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

Zhi-Wei Sun, Feb 01 2019

nonn

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.

			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).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.

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

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]

cp's OEIS Frontend

A306227 Number of ways to write n as w + x^4 + y*(y+1)/2 + z*(z+1)/2, where w is 0 or 1, and x, y, z are nonnegative integers with x >= w and y < z.

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A306239 Number of ways to write n as x^3 + y^3 + pen(z) + pen(w), where x, y, z, w are nonnegative integers with x <= y and z <= w, and pen(k) denotes the pentagonal number k*(3*k-1)/2.

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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.

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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.

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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.

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A306227 Number of ways to write n as w + x^4 + y(y+1)/2 + z(z+1)/2, where w is 0 or 1, and x, y, z are nonnegative integers with x >= w and y < z.

A306239 Number of ways to write n as x^3 + y^3 + pen(z) + pen(w), where x, y, z, w are nonnegative integers with x <= y and z <= w, and pen(k) denotes the pentagonal number k(3k-1)/2.

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.

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.

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.