A254573 -id:A254573 - cp's OEIS frontend

A254574 Number of ways to write n = x(x+1)/2 + y(3y+1)/2 + z(3*z-1)/2 with x,y,z nonnegative integers.

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

Offset: 0

Zhi-Wei Sun, Feb 01 2015

nonn

Conjecture: (i) a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 49.
(ii) Any nonnegative integer not equal to 18 can be written as x*(x+1)/2 + y*(3*y+1) + z*(3*z-1) with x,y,z nonnegative integers.
See also the comments of A254573 for a similar conjecture. We have proved that any nonnegative integer can be written as x*(x+1)/2 + y*(3*y+1)/2 + z*(3*z-1)/2 with x,y,z integers.
Note that Zhi-Wei Sun conjectured in 2009 (cf. Conjecture 1.10 of arXiv:0905.0635) that every n = 0,1,... can be expressed as the sum of a triangular number and two pentagonal numbers.

			a(14) = 2 since 14 = 0*1/2 + 1*(3*1+1)/2 + 3*(3*3-1)/2 = 3*4/2 + 2*(3*2+1)/2 + 1*(3*1-1)/2.
a(49) = 1 since 49 = 1*2/2 + 4*(3*4+1)/2 + 4*(3*4-1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000217, A000326, A005449, A254573.

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[r=0;Do[If[TQ[n-y(3y+1)/2-z(3z-1)/2],r=r+1],{y,0,(Sqrt[24n+1]-1)/6},{z,0,(Sqrt[24(n-y(3y+1)/2)+1]+1)/6}];
Print[n," ",r];Continue,{n,0,70}]

A254595 Least positive integer m such that m can be written as x(x+1) + y(3y+1)/2 + z(3*z-1)/2 in exactly n ways, where x,y,z are nonnegative integers.

Original entry on oeis.org

1, 2, 9, 7, 14, 37, 64, 68, 57, 119, 112, 168, 194, 147, 267, 259, 222, 477, 427, 404, 519, 652, 567, 497, 512, 749, 722, 719, 952, 1209, 904, 1139, 1267, 1184, 1069, 1737, 1594, 1667, 1734, 2077, 1799, 1659, 1729, 1814, 1762, 1862, 2577, 2444, 2997, 2072, 2457, 2842, 3029, 3249, 3094, 3589, 3999, 4208, 3479, 3232

Offset: 1

Zhi-Wei Sun, Feb 02 2015

nonn

Conjecture: (i) a(n) exists for any n > 0. Moreover, n^2 is the main term of a(n) as n tends to the infinity.

(ii) a(n) is never divisible by 5. Also, for any n > 1 the term a(n) is not congruent to 1 modulo 5.

			a(3) = 9 since 9 is the first positive integer m with A254573(m)=3. Note that 9 = 2*3 +1*(3*1+1)/2 + 1*(3*1-1)/2 = 1*2 + 1*(3*1+1)/2 + 2*(3*2-1)/2 = 1*2 + 2*(3*2+1)/2 + 0*(3*0-1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000326, A002378, A005449, A254573, A254617.

sQ[n_]:=IntegerQ[Sqrt[4n+1]]
Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[sQ[m-y(3y+1)/2-z(3z-1)/2],r=r+1;If[r>n,Goto[aa]]],{y,0,(Sqrt[24m+1]-1)/6}, {z,0,(Sqrt[24(m-y(3y+1)/2)+1]+1)/6}]; If[r==n,Print[n," ",m];Goto[bb],Goto[aa]]]; Label[bb];Continue,{n,1,60}]

A254617 Least positive integer m such that A254574(m) = n.

Original entry on oeis.org

49, 1, 3, 20, 8, 15, 30, 22, 58, 48, 93, 78, 92, 148, 113, 127, 155, 198, 197, 323, 268, 272, 288, 345, 358, 338, 555, 568, 443, 498, 612, 547, 653, 730, 708, 687, 722, 778, 1002, 897, 1107, 1030, 1112, 1205, 1535, 1343, 1458, 1093, 1203, 1588, 1548, 1822, 1623, 2162, 2208, 1577, 1497, 1948, 2228, 2473

Offset: 1

Zhi-Wei Sun, Feb 03 2015

nonn

Conjecture: (i) a(n) exists for any n > 0. Also, the main term of a(n) is n^2/2 as n tends to the infinity.
(ii) No term a(n) with n>2 is congruent to 1 or -1 modulo 5.
See also the comments in A254595 for a similar conjecture.

			a(4) = 20 since 20 is the first positive integer m with A254574(m) = 4. Note that 20 = 0*1/2 + 3*(3*3+1)/2 + 2*(3*2-1)/2 = 1*2/2 + 2*(3*2+1)/2 + 3*(3*3-1)/2 = 3*4/2 + 1*(3*1+1)/2 + 3*(3*3-1)/2 = 5*6/2 + 0*(3*0+1)/2 + 2*(3*2-1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000217, A000326, A005449, A254573, A254574, A254595.

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[TQ[m-y(3y+1)/2-z(3z-1)/2],r=r+1;If[r>n,Goto[aa]]],{y,0,(Sqrt[24m+1]-1)/6}, {z,0,(Sqrt[24(m-y(3y+1)/2)+1]+1)/6}];
If[r==n,Print[n," ",m];Goto[bb],Goto[aa]]]; Label[bb];Continue,{n,1,60}]

A254631 Number of ways to write n as x(x+1)/2 + y(3y+2) + z(3*z-2) with x,y,z nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 03 2015

nonn

Conjecture: a(n) > 0 for all n, and a(n) > 1 for all n > 35.

We have proved that any nonnegative integer n can be written as x*(x+1)/2 + y*(3*y+2) + z*(3*z-2) with x,y,z integers.

			a(13) = 1 since 13 = 0*1/2 + 1*(3*1+2) + 2*(3*2-2).
a(30) = 1 since 30 = 3*4/2 + 2*(3*2+2) + 2*(3*2-2).
a(35) = 1 since 35 = 1*2/2 + 3*(3*3+2) + 1*(3*1-2).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000217, A000567, A045944, A254573, A254574.

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[r=0;Do[If[TQ[n-y(3y+2)-z(3z-2)],r=r+1],{y,0,(Sqrt[3n+1]-1)/3},{z,0,(Sqrt[3(n-y(3y+2))+1]+1)/3}];
Print[n," ",r];Continue,{n,0,100}]

A254629 Number of ways to write n as x^2 + y(y+1) + z(4*z+1) with x,y,z nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 03 2015

nonn

Conjecture: (i) a(n) > 0 for all n, and a(n) > 1 for all n > 338.
(ii) If f(x) is one of the polynomials 3x^2+x, (7x^2+3x)/2, (9x^2-x)/2, (11x^2-7x)/2, (15x^2-7x)/2, (15x^2-11x)/2, then any nonnegative integer n can be written as x^2 + y*(y+1) + f(z) with x,y,z nonnegative integers.
We have proved that for each n = 0,1,... there are integers x,y,z such that n = x^2 + y*(y+1) + z*(4z+1).
It is known that {x^2+y*(y+1): x,y=0,1,...} = {x*(x+1)/2+y*(y+1)/2: x,y=0,1,...}.

			a(103) = 1 since 103 = 8^2 + 0*1 + 3*(4*3+1).
a(122) = 1 since 122 = 9^2 + 1*2 +3*(4*3+1).
a(143) = 1 since 143 = 6^2 + 1*2 + 5*(4*5+1).
a(167) = 1 since 167 = 3^2 + 9*10 + 4*(4*4+1).
a(248) = 1 since 248 = 5^2 + 4*5 + 7*(4*7+1).
a(338) = 1 since 338 = 5^2 + 10*11 + 7*(4*7+1).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000217, A000290, A007742, A254573, A254574, A254623.

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-y(y+1)-z*(4z+1)],r=r+1],{y,0,(Sqrt[4n+1]-1)/2},{z,0,(Sqrt[16(n-y(y+1))+1]-1)/8}];
Print[n," ",r];Continue,{n,0,100}]

A254639 Least positive integer m such that A254631(m) = n.

Original entry on oeis.org

2, 1, 6, 16, 27, 62, 71, 92, 122, 161, 176, 216, 286, 386, 351, 491, 577, 492, 781, 866, 1023, 617, 736, 1002, 1504, 1441, 1402, 1297, 1451, 1562, 1842, 2166, 1682, 1331, 2626, 2311, 2332, 2969, 3177, 2761, 2876, 3641, 3261, 3697, 3586, 4894, 3576, 3921, 4482, 4542

Offset: 1

Zhi-Wei Sun, Feb 04 2015

nonn

Conjecture: a(n) exists for any n > 0. Moreover, no term a(n) is divisible by 5.

It seems that no term a(n) is congruent to 8 modulo 10.

			a(3) = 6 since 6 is the least positive integer m with A254631(m) = 3. Note that 6 = 0*1/2 + 1*(3*1+2) + 1*(3*1-2) = 1*2/2 + 1*(3*1+2) + 0*(3*0-2) = 3*4/2 + 0*(3*0+2) + 0*(3*0-2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..160
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000217, A000567, A045944, A254573, A254574, A254595, A254617, A254631.

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[Do[m=0;Label[aa];m=m+1;r=0;Do[If[TQ[m-y(3y+2)-z(3z-2)],r=r+1;If[r>n, Goto[aa]]],{y,0,(Sqrt[3m+1]-1)/3},{z,0,(Sqrt[3(m-y(3y+2))+1]+1)/3}];
If[r==n,Print[n, " ", m];Goto[bb],Goto[aa]]];Label[bb];Continue,{n,1,50}]

A256855 Number of ordered ways to write n as x(3x-1)/2 + y(3y+1)/2 + z(3z+1), where x and y are nonnegative integers and z is an integer.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Apr 11 2015

nonn

Conjecture: a(n) > 0 for all n. Also, any nonnegative integer can be written as x*(3*x-1)/2 + y*(3*y-1) + z*(3*z+1)/2 with x,y nonnegative integers and z an integer, and each n = 0,1,2,... can be expressed as x*(3*x+1)/2 + 3*y*(3*y+1)/2 + z*(3*z+1)/2 with x,y nonnegative integers and z an integer.

			 a(8) = 1 since 8 = 1*(3*1-1)/2 + 2*(3*2+1)/2 + 0*(3*0+1).
a(23) = 1 since 23 = 3*(3*3-1)/2 + 2*(3*2+1)/2 + 1*(3*1+1).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000326, A001318, A005449, A253187, A254573, A254574.

PQ[n_]:=IntegerQ[Sqrt[12n+1]]
Do[r=0;Do[If[PQ[n-x(3x-1)/2-y(3y+1)/2],r=r+1],{x,0,(Sqrt[24n+1]+1)/6},{y,0,(Sqrt[24(n-x(3x-1)/2)+1]-1)/6}];
Print[n," ",r];Continue,{n,0,100}]

cp's OEIS Frontend

A254574 Number of ways to write n = x*(x+1)/2 + y*(3*y+1)/2 + z*(3*z-1)/2 with x,y,z nonnegative integers.

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A254595 Least positive integer m such that m can be written as x*(x+1) + y*(3*y+1)/2 + z*(3*z-1)/2 in exactly n ways, where x,y,z are nonnegative integers.

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A254617 Least positive integer m such that A254574(m) = n.

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A254631 Number of ways to write n as x*(x+1)/2 + y*(3*y+2) + z*(3*z-2) with x,y,z nonnegative integers.

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A254629 Number of ways to write n as x^2 + y*(y+1) + z*(4*z+1) with x,y,z nonnegative integers.

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A254639 Least positive integer m such that A254631(m) = n.

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A256855 Number of ordered ways to write n as x*(3*x-1)/2 + y*(3*y+1)/2 + z*(3*z+1), where x and y are nonnegative integers and z is an integer.

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A254574 Number of ways to write n = x(x+1)/2 + y(3y+1)/2 + z(3*z-1)/2 with x,y,z nonnegative integers.

A254595 Least positive integer m such that m can be written as x(x+1) + y(3y+1)/2 + z(3*z-1)/2 in exactly n ways, where x,y,z are nonnegative integers.

A254631 Number of ways to write n as x(x+1)/2 + y(3y+2) + z(3*z-2) with x,y,z nonnegative integers.

A254629 Number of ways to write n as x^2 + y(y+1) + z(4*z+1) with x,y,z nonnegative integers.

A256855 Number of ordered ways to write n as x(3x-1)/2 + y(3y+1)/2 + z(3z+1), where x and y are nonnegative integers and z is an integer.