A306462 -id:A306462 - cp's OEIS frontend

A306460 Number of ways to write n as x(2x-1) + y(y+1)/2 + z(z+1)(z+2)/6, where x,y,z are nonnegative integers with x > 0.

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

Offset: 1

Zhi-Wei Sun, Feb 17 2019

nonn

Conjecture 1: a(n) > 0 for all n > 0. In other words, any positive integer n can be written as the sum of a positive hexagonal number, a triangular number and a tetrahedral number.
We have verified a(n) > 0 for all n = 1..10^7.
Conjecture 2: Let c be 1 or 3. Then each n = 0,1,... can be written as c*x(x+1) + y*(y+1)/2 + z*(z+1)*(z+2)/6 with x,y,z nonnegative integers.
Conjecture 3: Let t(x) = x*(x+1)*(x+2)/6. Then each n = 0,1,... can be written as 2*t(w) + t(x) + t(y) + t(z) with w,x,y,z nonnegative integers.
We have verified Conjecture 3 for all n = 0..2*10^5. Clearly, Conjecture 3 implies Pollock's conjecture which states that any natural number is the sum of five tetrahedral numbers.

			a(3) = 1 with 3 = 1*(2*1-1) + 1*2/2 + 1*2*3/6.
a(14) = 1 with 14 = 1*(2*1-1) + 2*3/2 + 3*4*5/6.
a(75) = 1 with 75 = 5*(2*5-1) + 4*5/2 + 4*5*6/2.
a(349) = 1 with 349 = 5*(2*5-1) + 24*25/2 + 2*3*4/6.
a(369) = 1 with 369 = 4*(2*4-1) + 10*11/2 + 11*12*13/6.
a(495) = 1 with 495 = 8*(2*8-1) + 20*21/2 + 9*10*11/6.
a(642) = 1 with 642 = 16*(2*16-1) + 16*17/2 + 3*4*5/6.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000217, A000292, A000384, A000797, A104246, A262813, A306462, A306471.

f[n_]:=f[n]=n(n+1)(n+2)/6;
TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[f[z]>=n, Goto[aa]]; Do[If[TQ[n-f[z]-x(2x-1)],r=r+1],{x,1,(Sqrt[8(n-f[z])+1]+1)/4}];Label[aa],{z,0,n}];tab=Append[tab,r],{n,1,100}];Print[tab]

A306477 Number of ways to write n as C(w+2,2) + C(x+3,4) + C(y+5,6) + C(z+7,8) with w,x,y,z nonnegative integers, where C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).

Original entry on oeis.org

1, 3, 4, 4, 3, 3, 5, 6, 5, 5, 8, 8, 6, 4, 6, 10, 10, 8, 6, 6, 6, 10, 9, 6, 6, 7, 7, 6, 8, 10, 10, 7, 4, 7, 7, 9, 13, 12, 9, 6, 5, 6, 11, 12, 12, 13, 10, 9, 8, 9, 11, 15, 12, 8, 8, 10, 14, 11, 7, 8, 12, 9, 8, 9, 10, 11, 13, 8, 5, 9, 10, 13, 14, 12, 8, 7, 6, 12, 14, 14

Offset: 1

Zhi-Wei Sun, Feb 18 2019

nonn

Conjecture: a(n) > 0 for all n > 0. In other words, any positive integer n can be written as C(w,2) + C(x,4) + C(y,6) + C(z,8), where w,x,y,z are integers greater than one.
I'd like to call this conjecture "the 2-4-6-8 conjecture". I have verified it for all n = 1..3*10^7.
On Feb. 20, 2019, Yaakov Baruch reported on Mathoverflow that he had verified the 2-4-6-8 conjecture for n up to 5*10^8. - Zhi-Wei Sun, Feb 20 2019
On Feb. 24, 2019, Max A. Alekseyev reported on Mathoverflow that he had verified the 2-4-6-8 conjecture for n up to 2*10^11.
I'd like to offer 2468 US dollars as the prize for the first correct proof of my 2-4-6-8 conjecture, or 2468 RMB as the prize for the first explicit counterexample. - Zhi-Wei Sun, Feb 24 2019
Yaakov Baruch reported on March 12, 2019 that he had checked the 2-4-6-8 conjecture for all n = 1..2*10^12 with no counterexample found. - Zhi-Wei Sun, Mar 12 2019

			a(1) = 1 with 1 = C(2,2) + C(3,4) + C(5,6) + C(7,8).
a(4655) = 2 with 4655 = C(85,2) + C(14,4) + C(9,6) + C(7,8) = C(94,2) + C(7,4) + C(9,6) + C(11,8).
a(9590) = 2 with 9590 = C(35,2) + C(21,4) + C(7,6) + C(14,8) = C(136,2) + C(7,4) + C(10,6) + C(11,8).
a(24935) = 2 with 24935 = C(49,2) + C(29,4) + C(7,6) + C(8,8) = C(140,2) + C(26,4) + C(10,6) + C(10,8).
a(33845) = 2 with 33845 = C(104,2) + C(8,4) + C(19,6) + C(13,8) = C(148,2) + C(26,4) + C(16,6) + C(9,8).
a(192080) = 2 with 192080 = C(7,2) + C(26,4) + C(25,6) + C(9,8) = C(414,2) + C(39,4) + C(8,6) + C(17,8).
a(23343989) = 1 with 23343989 = C(365,2) + C(76,4) + C(40,6) + C(34,8).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Positive integers written as C(w,2) + C(x,4) + C(y,6) + C(z,8) with w,x,y,z in {2,3,...}, Question 323541 on Mathoverflow, Feb. 19, 2019.

Cf. A000217, A000332, A000579, A000581, A306459, A306460, A306462, A306471.

f[m_,n_]:=f[m,n]=Binomial[m+n-1,m]; TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[f[8,z]>=n,Goto[cc]];Do[If[f[6,y]>=n-f[8,z],Goto[bb]];Do[If[f[4,x]>=n-f[8,z]-f[6,y],Goto[aa]];If[TQ[n-f[8,z]-f[6,y]-f[4,x]],r=r+1],{x,0,n-1-f[8,z]-f[6,y]}];Label[aa],{y,0,n-1-f[8,z]}];Label[bb],{z,0,n-1}];Label[cc];tab=Append[tab,r],{n,1,80}];Print[tab]

A306471 Number of ways to write n as C(2w+1,2) + C(x+2,3) + C(y+3,4) + C(z+4,5) with w,x,y,z nonnegative integers, where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!).

Original entry on oeis.org

1, 3, 3, 2, 4, 6, 5, 4, 4, 5, 7, 8, 6, 4, 5, 8, 8, 5, 4, 6, 7, 10, 10, 6, 6, 12, 13, 8, 7, 7, 6, 11, 9, 4, 3, 8, 16, 12, 8, 9, 9, 13, 14, 10, 7, 9, 18, 12, 6, 5, 4, 11, 10, 4, 2, 5, 19, 21, 11, 9, 13, 20, 16, 9, 6, 8, 17, 17, 4, 2, 9, 20, 17, 6, 9, 9, 15, 23, 14, 9, 15

Offset: 0

Zhi-Wei Sun, Feb 17 2019

nonn

Conjecture 1: a(n) > 1 for all n > 0.
We have verified a(n) > 0 for all n = 0..5*10^6.
Conjecture 2: For each r = 0, 1, any positive integer can be written as w^2 + C(x,3) + C(y,4) + C(z,5), where w,x,y,z are nonnegative integers with w - r even.
See also A306462 and A306477 for similar conjectures.

			a(0) = 1 with 0 = C(1,2) + C(2,3) + C(3,4) + C(4,5).
a(3) = 2 with 3 = C(3,2) + C(2,3) + C(3,4) + C(4,5) = C(1,2) + C(3,3) + C(4,4) + C(5,5).
a(54) = 2 with 54 = C(3,2) + C(7,3) + C(6,4) + C(5,5) = C(3,2) + C(5,3) + C(7,4) + C(6,5).
a(69) = 1 with 69 = C(3,2) + C(5,3) + C(7,4) + C(7,5) = C(3,2) + C(5,3) + C(3,4) + C(8,5).

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

Cf. A000217, A000290, A000292, A000332, A000389, A000797, A014105, A262813, A306460, A306462, A306477.

f[m_,n_]:=f[m,n]=Binomial[m+n-1,m];
HQ[n_]:=HQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==1;
tab={};Do[r=0;Do[If[f[5,z]>n,Goto[cc]];Do[If[f[4,y]>n-f[5,z],Goto[bb]];Do[If[f[3,x]>n-f[5,z]-f[4,y],Goto[aa]];If[HQ[n-f[5,z]-f[4,y]-f[3,x]],r=r+1],{x,0,n-f[5,z]-f[4,y]}];Label[aa],{y,0,n-f[5,z]}];Label[bb],{z,0,n}];Label[cc];tab=Append[tab,r],{n,0,80}];Print[tab]

A306459 Number of ways to write n as w^3 + C(x+2,3) + C(y+2,3) + C(z+2,3), where w,x,y,z are nonnegative integers with x <= y <= z, and C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 20 2019

nonn

Conjecture: a(n) > 0 for all n >= 0. In other words, each nonnegative integer can be written as the sum of a nonnegative cube and three tetrahedral numbers.
It seems that a(n) = 1 only for n = 0, 7, 17, 27, 34, 53, 54, 110, 118, 163, 207, 263, 270, 309, 362, 443, 1174, 1284.
We have verified a(n) > 0 for all n = 0..2*10^6.

			a(0) = 1 with 0 = 0^3 + C(2,3) + C(2,3) + C(2,3).
a(17) = 1 with 17 = 2^3 + C(3,3) + C(4,3) + C(4,3).
a(27) = 1 with 27 = 3^3 + C(2,3) + C(2,3) + C(2,3).
a(362) = 1 with 362 = 0^3 + C(6,3) + C(8,3) + C(13,3).
a(443) = 1 with 443 = 3^3 + C(5,3) + C(10,3) + C(13,3).
a(1174) = 1 with 1174 = 1^3 + C(9,3) + C(10,3) + C(19,3).
a(1284) = 1 with 1284 = 10^3 + C(7,3) + C(9,3) + C(11,3).

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

Cf. A000292, A000578, A000797, A262813, A306460, A306462, A306471, A306477.

f[n_]:=f[n]=Binomial[n+2,3];
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={};Do[r=0;Do[If[f[x]>n/3,Goto[cc]];Do[If[f[y]>(n-f[x])/2,Goto[bb]];Do[If[f[z]>n-f[x]-f[y],Goto[aa]];If[CQ[n-f[x]-f[y]-f[z]],r=r+1],{z,y,n-f[x]-f[y]}];Label[aa],{y,x,(n-f[x])/2}];Label[bb],{x,0,n/3}];Label[cc];tab=Append[tab,r],{n,0,100}];Print[tab]

A306571 Number of ways to write n as w(w+1) + C(x+3,4) + C(y+5,6) + C(z+7,8) with w,x,y,z nonnegative integers, where C(m,k) denotes the binomial coefficient m!/(k!(m-k)!).

Original entry on oeis.org

1, 3, 4, 4, 3, 2, 3, 6, 7, 5, 4, 4, 6, 7, 7, 6, 6, 6, 6, 5, 4, 8, 9, 6, 3, 4, 4, 6, 7, 7, 6, 8, 5, 5, 4, 8, 10, 9, 5, 5, 5, 5, 8, 10, 10, 9, 8, 7, 9, 9, 7, 8, 8, 7, 7, 6, 7, 12, 12, 8, 2, 3, 6, 11, 9, 8, 9, 7, 2, 4, 5, 8, 13, 14, 8, 6, 6, 8, 9, 9, 11, 8, 7, 7, 10, 9, 10, 11, 8, 7, 9, 11, 13, 11, 8, 5, 6, 7, 10, 10, 13

Offset: 0

Zhi-Wei Sun, Feb 24 2019

nonn

Conjecture: a(n) > 0 for all n >= 0. In other words, each n = 0,1,2,... can be written as 2*C(w,2) + C(x,4) + C(y,6) + C(z,8) with w,x,y,z positive integers.
We have verified a(n) > 0 for all n = 0..2*10^7.
Note that 10413917 is the least positive integer not representable as w^2 + C(x,4) + C(y,6) + C(z,8) with w,x,y,z nonnegative integers.
See also A306477 for a similar conjecture.

			a(0) = 1 with 0 = 0*1 + C(3,4) + C(5,6) + C(7,8).
a(60) = 2 with 60 = 0*1 + C(6,4) + C(5,6) + C(10,8) = 5*6 + C(4,4) + C(8,6) + C(8,8).
a(220544) = 1 with 220544 = 151*152 + C(48,4) + C(14,6) + C(9,8).
a(809165) = 1 with 809165 = 295*296 + C(63,4) + C(10,6) + C(20,8).
a(16451641) = 1 with 16451641 = 2256*2257 + C(130,4) + C(12,6) + C(10,8).

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

Cf. A000332, A000579, A000581, A002378, A306459, A306460, A306462, A306471, A306477.

f[m_,n_]:=f[m,n]=Binomial[m+n-1,m];
TQ[n_]:=TQ[n]=IntegerQ[Sqrt[4n+1]];
tab={};Do[r=0;Do[If[f[8,z]>n,Goto[cc]];Do[If[f[6,y]>n-f[8,z],Goto[bb]];Do[If[f[4,x]>n-f[8,z]-f[6,y],Goto[aa]];If[TQ[n-f[8,z]-f[6,y]-f[4,x]],r=r+1],{x,0,n-f[8,z]-f[6,y]}];Label[aa],{y,0,n-f[8,z]}];Label[bb],{z,0,n}];Label[cc];tab=Append[tab,r],{n,0,100}];Print[tab]

cp's OEIS Frontend

A306460 Number of ways to write n as x*(2x-1) + y*(y+1)/2 + z*(z+1)*(z+2)/6, where x,y,z are nonnegative integers with x > 0.

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A306477 Number of ways to write n as C(w+2,2) + C(x+3,4) + C(y+5,6) + C(z+7,8) with w,x,y,z nonnegative integers, where C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).

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A306471 Number of ways to write n as C(2w+1,2) + C(x+2,3) + C(y+3,4) + C(z+4,5) with w,x,y,z nonnegative integers, where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!).

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A306459 Number of ways to write n as w^3 + C(x+2,3) + C(y+2,3) + C(z+2,3), where w,x,y,z are nonnegative integers with x <= y <= z, and C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).

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A306571 Number of ways to write n as w*(w+1) + C(x+3,4) + C(y+5,6) + C(z+7,8) with w,x,y,z nonnegative integers, where C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).

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A306460 Number of ways to write n as x(2x-1) + y(y+1)/2 + z(z+1)(z+2)/6, where x,y,z are nonnegative integers with x > 0.

A306571 Number of ways to write n as w(w+1) + C(x+3,4) + C(y+5,6) + C(z+7,8) with w,x,y,z nonnegative integers, where C(m,k) denotes the binomial coefficient m!/(k!(m-k)!).