A303234 -id:A303234 - cp's OEIS frontend

A303233 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

0, 1, 2, 3, 4, 5, 4, 6, 7, 7, 7, 9, 7, 8, 9, 9, 8, 12, 11, 11, 11, 11, 11, 14, 11, 13, 12, 11, 10, 14, 11, 12, 17, 15, 12, 16, 14, 15, 17, 19, 15, 16, 13, 15, 17, 17, 16, 20, 16, 14, 17, 17, 14, 22, 17, 14, 14, 17, 15, 19

Offset: 1

Zhi-Wei Sun, Apr 20 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two triangular numbers and two powers of 2.
a(n) > 0 for all n = 2..10^9. See A303234 for numbers of the form x*(x+1)/2 + 2^y with x and y nonnegative integers. See also A303363 for a stronger conjecture.
In contrast, Crocker proved in 2008 that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of 2.

			a(2) = 1 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^0.
a(3) = 2 with 3 = 0*(0+1)/2 + 1*(1+1)/2 + 2^0 + 2^0 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^1.
a(4) = 3 with 4 = 1*(1+1)/2 + 1*(1+1)/2 + 2^0 + 2^0 = 0*(0+1)/2 + 1*(1+1)/2 + 2^0 + 2^1 = 0*(0+1)/2 + 0*(0+1)/2 + 2^1 + 2^1.

R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000217, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303234, A303338, A303363.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-2^k-2^j)+1],Do[If[SQ[8(n-2^k-2^j-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-2^k-2^j)+1]-1)/2}]],{k,0,Log[2,n]-1},{j,k,Log[2,n-2^k]}];tab=Append[tab,r],{n,1,60}];Print[tab]

A303363 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 2, 4, 6, 3, 5, 6, 4, 6, 7, 4, 4, 9, 6, 6, 8, 4, 9, 9, 5, 7, 7, 5, 7, 9, 4, 8, 13, 7, 6, 11, 7, 10, 13, 8, 9, 10, 7, 9, 11, 7, 9, 15, 8, 8, 14, 6, 9, 16, 6, 8, 11, 11, 10, 12, 8, 7, 15, 10, 8, 11, 9, 14, 15, 9

Offset: 1

Zhi-Wei Sun, Apr 22 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This is stronger than the author's conjecture in A303233. I have verified a(n) > 0 for all n = 2..10^9.
In contrast, Corcker proved in 2008 that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of 2.

			a(2) = 1 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^0.
a(3) = 2 with 3 = 0*(0+1)/2 + 1*(1+1)/2 + 2^0 + 2^0 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000217, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303389.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-4^j-2^k)+1],Do[If[SQ[8(n-4^j-2^k-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-4^j-2^k)+1]-1)/2}]],{j,0,Log[4,n/2]},{k,2j,Log[2,n-4^j]}];tab=Append[tab,r],{n,1,70}];Print[tab]

A303389 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c + 5^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 2, 2, 2, 4, 3, 2, 2, 3, 3, 3, 2, 2, 2, 4, 3, 2, 1, 5, 4, 3, 2, 5, 5, 5, 5, 3, 3, 5, 5, 4, 4, 4, 5, 5, 2, 5, 3, 5, 4, 7, 2, 4, 6, 6, 5, 4, 4, 5, 8, 4, 4, 4, 7, 6, 4, 3, 4, 8, 4, 7, 3, 3, 6, 8, 2, 5, 6, 5, 4, 6, 4, 3

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integers n > 1 can be written as the sum of two triangular numbers and two powers of 5.
This has been verified for all n = 2..10^10.
See A303393 for the numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.
See also A303401, A303432 and A303540 for similar conjectures.

			a(4) = 1 with 4 = 1*(1+1)/2 + 1*(1+1)/2 + 5^0 + 5^0.
a(5) = 1 with 5 = 0*(0+1)/2 + 2*(2+1)/2 + 5^0 + 5^0.
a(7) = 1 with 7 = 0*(0+1)/2 + 1*(1+1)/2 + 5^0 + 5^1.
a(25) = 1 with 25 = 0*(0+1)/2 + 5*(5+1)/2 + 5^1 + 5^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000217, A000351, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303363, A303393, A303401, A303432, A303540.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-5^j-5^k)+1],Do[If[SQ[8(n-5^j-5^k-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-5^j-5^k)+1]-1)/2}]],{j,0,Log[5,n/2]},{k,j,Log[5,n-5^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A303393 Numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 11, 15, 16, 20, 22, 25, 26, 28, 29, 31, 33, 35, 37, 40, 41, 46, 50, 53, 56, 60, 61, 67, 70, 71, 79, 80, 83, 91, 92, 96, 103, 106, 110, 116, 121, 125, 126, 128, 130, 131, 135, 137, 140, 141, 145, 146, 153, 154, 158, 161, 170, 172, 176

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

The author's conjecture in A303389 has the following equivalent version: Each integer n > 1 can be expressed as the sum of two terms of the current sequence.

This has been verified for all n = 2..2*10^8.

			a(1) = 1 with 1 = 0*(0+1)/2 + 5^0.
a(2) = 2 with 2 = 1*(1+1)/2 + 5^0.
a(3) = 4 with 4 = 2*(2+1)/2 + 5^0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000217, A000351, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303363, A303389.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[Do[If[TQ[m-5^k],tab=Append[tab,m];Goto[aa]],{k,0,Log[5,m]}];Label[aa],{m,1,176}];Print[tab]

A303399 Number of ordered pairs (a, b) with 0 <= a <= b such that n - 5^a - 5^b can be written as the sum of two triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 1, 4, 3, 3, 2, 5, 4, 4, 4, 3, 3, 4, 4, 3, 4, 4, 4, 3, 2, 4, 3, 3, 3, 5, 2, 4, 5, 4, 4, 4, 4, 3, 5, 3, 4, 4, 4, 4, 4, 3, 3, 5, 4, 5, 3, 3, 5, 5, 2, 4, 6, 3, 3, 4, 4, 3

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This is equivalent to the author's conjecture in A303389. It has been verified that a(n) > 0 for all n = 2..6*10^9.
Note that a nonnegative integer m is the sum of two triangular numbers if and only if 4*m + 1 can be written as the sum of two squares.

			a(6) = 2 with 6 - 5^0 - 5^0 = 1*(1+1)/2 + 2*(2+1)/2 and 6 - 5^0 - 5^1 = 0*(0+1)/2 + 0*(0+1)/2.
a(7) = 1 with 7 - 5^0 - 5^1 = 0*(0+1)/2 + 1*(1+1)/2.
a(25) = 1 with 25 - 5^1 - 5^1 = 0*(0+1)/2 + 5*(5+1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000217, A000351, A271518, A273812, A281976, A299924, A300219, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303363, A303389, A303393.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-5^j-5^k)+1],r=r+1],{j,0,Log[5,n/2]},{k,j,Log[5,n-5^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A303540 Number of ways to write n as a^2 + b^2 + binomial(2c,c) + binomial(2d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

Original entry on oeis.org

0, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 6, 4, 2, 2, 4, 4, 2, 2, 5, 5, 5, 4, 4, 4, 4, 5, 6, 5, 5, 4, 5, 4, 4, 3, 4, 5, 5, 6, 5, 5, 5, 4, 7, 3, 4, 5, 6, 4, 2, 4, 6, 7, 4, 4, 5, 7, 6, 2, 5, 4, 6, 3, 2, 5, 5, 5, 4, 4, 3, 7, 9, 6, 5, 6, 11, 7, 3, 4, 8

Offset: 1

Zhi-Wei Sun, Apr 25 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two squares and two central binomial coefficients.
It has been verified that a(n) > 0 for all n = 2..10^10.
See also A303539 and A303541 for related information.
Jiao-Min Lin (a student at Nanjing University) has verified a(n) > 0 for all 1 < n <= 10^11. - Zhi-Wei Sun, Jul 30 2022

			a(2) = 1 since 2 = 0^2 + 0^2 + binomial(2*0,0) + binomial(2*0,0).
a(10) = 2 with 10 = 2^2 + 2^2 + binomial(2*0,0) + binomial(2*0,0) = 1^2 + 1^2 + binomial(2*1,1) + binomial(2*2,2).
a(2435) = 1 with 2435 = 32^2 + 33^2 + binomial(2*4,4) + binomial(2*5,5).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303541, A303543, A303601, A303639.

N:= 100: # for a(1)..a(N)
A:= Vector(N):
for b from 0 to floor(sqrt(N)) do
  for a from 0 to min(b, floor(sqrt(N-b^2))) do
    t:= a^2+b^2;
    for d from 0 do
      s:= t + binomial(2*d,d);
      if s > N then break fi;
      for c from 0 to d do
        u:= s + binomial(2*c,c);
        if u > N then break fi;
        A[u]:= A[u]+1;
od od od od:
convert(A,list); # Robert Israel, May 30 2018

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
c[n_]:=c[n]=Binomial[2n,n];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;k=0;Label[bb];If[c[k]>n,Goto[aa]];Do[If[QQ[n-c[k]-c[j]],Do[If[SQ[n-c[k]-c[j]-x^2],r=r+1],{x,0,Sqrt[(n-c[k]-c[j])/2]}]],{j,0,k}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,80}];Print[tab]

A303434 Numbers of the form x(3x-1)/2 + 3^y with x and y nonnegative integers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 13, 14, 15, 21, 23, 25, 27, 28, 31, 32, 36, 38, 39, 44, 49, 52, 54, 60, 62, 71, 73, 78, 79, 81, 82, 86, 93, 95, 97, 101, 103, 116, 118, 119, 120, 126, 132, 144, 146, 148, 151, 154, 172, 173, 177, 179, 185

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

The author's conjecture in A303401 has the following equivalent version: Each integer n > 1 can be written as the sum of two terms of the current sequence.

This has been verified for all n = 2..7*10^6.

			a(1) = 1 with 1 = 0*(3*0-1)/2 + 3^0.
a(2) = 2 with 2 = 1*(3*1-1)/2 + 3^0.
a(5) = 6 with 6 = 2*(3*2-1)/2 + 3^0.
a(6) = 8 with 8 = 2*(3*2-1)/2 + 3^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000244, A000326, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432.

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0);
tab={};Do[Do[If[PenQ[m-3^k],n=n+1;tab=Append[tab,m];Goto[aa]],{k,0,Log[3,m]}];Label[aa],{m,1,185}];Print[tab]

A303656 Number of ways to write n as a^2 + b^2 + 3^c + 5^d, where a,b,c,d are nonnegative integers with a <= b.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 2, 4, 4, 3, 2, 4, 4, 3, 2, 4, 3, 4, 1, 4, 5, 6, 4, 6, 5, 5, 6, 6, 5, 8, 4, 6, 6, 5, 4, 7, 5, 7, 5, 6, 4, 5, 3, 4, 7, 6, 7, 8, 5, 4, 7, 5, 5, 9, 3, 6, 5, 6, 4, 6, 5, 7, 7, 4, 5, 5, 5, 4, 6, 5, 6, 10, 5, 4, 5, 7, 4, 9, 2, 9, 8, 5, 6, 6

Offset: 1

Zhi-Wei Sun, Apr 27 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two squares, a power of 3 and a power of 5.
It has been verified that a(n) > 0 for all n = 2..2*10^10.
It seems that any integer n > 1 also can be written as the sum of two squares, a power of 2 and a power of 3.
The author would like to offer 3500 US dollars as the prize for the first proof of his conjecture that a(n) > 0 for all n > 1. - Zhi-Wei Sun, Jun 05 2018
Jiao-Min Lin (a student at Nanjing University) has verified a(n) > 0 for all 1 < n <= 2.4*10^11. - Zhi-Wei Sun, Jul 30 2022

			a(2) = 1 with 2 = 0^2 + 0^2 + 3^0 + 5^0.
a(5) = 1 with 5 = 0^2 + 1^2 + 3^1 + 5^0.
a(25) = 1 with 25 = 1^2 + 4^2 + 3^1 + 5^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000244, A000290, A000351, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303429, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303702, A303821.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[n-3^k-5^m],Do[If[SQ[n-3^k-5^m-x^2],r=r+1],{x,0,Sqrt[(n-3^k-5^m)/2]}]],{k,0,Log[3,n]},{m,0,If[n==3^k,-1,Log[5,n-3^k]]}];tab=Append[tab,r],{n,1,90}];Print[tab]

A303539 Number of ordered pairs (k, m) with 0 <= k <= m such that n - binomial(2k,k) - binomial(2m,m) can be written as the sum of two squares.

Original entry on oeis.org

0, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 6, 4, 2, 2, 4, 4, 2, 2, 5, 5, 5, 4, 4, 4, 4, 4, 5, 4, 5, 4, 4, 3, 4, 3, 4, 4, 5, 6, 5, 5, 5, 4, 7, 3, 3, 4, 6, 4, 2, 3, 5, 6, 3, 4, 5, 6, 5, 2, 5, 4, 5, 3, 2, 4, 5, 4, 3, 3, 3, 6, 7, 5, 5, 6, 10, 6, 3, 4, 8

Offset: 1

Zhi-Wei Sun, Apr 25 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
a(n) > 0 for all n = 2..10^10.
See also A303540 and A303541 for related sequences.

			a(2) = 1 with 2 - binomial(2*0,0) - binomial(2*0,0) = 0^2 + 0^2.
a(3) = 2 with 3 - binomial(2*0,0) - binomial(2*0,0) = 0^2 + 1^2 and 3 - binomial(2*0,0) - binomial(2*1,1) = 0^2 + 0^2.
a(5) = 2 with 5 - binomial(2*0,0) - binomial(2*1,1) = 1^2 + 1^2 and 5 - binomial(2*1,1) - binomial(2*1,1) = 0^2 + 1^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303540, A303541, A303543.

c[n_]:=c[n]=Binomial[2n,n];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;k=0;Label[bb];If[c[k]>n,Goto[aa]];Do[If[QQ[n-c[k]-c[j]],r=r+1],{j,0,k}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,80}];Print[tab]

A303541 Numbers of the form k^2 + binomial(2*m,m) with k and m nonnegative integers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 15, 17, 18, 20, 21, 22, 24, 26, 27, 29, 31, 36, 37, 38, 42, 45, 50, 51, 55, 56, 65, 66, 69, 70, 71, 74, 79, 82, 83, 84, 86, 87, 95, 101, 102, 106, 119, 120, 122, 123, 127, 134

Offset: 1

Zhi-Wei Sun, Apr 25 2018

nonn

The conjecture in A303540 has the following equivalent version: Each integer n > 1 can be written as the sum of two terms of the current sequence.

This has been verified for all n = 2..10^10.

			a(1) = 1 with 0^2 + binomial(2*0,0) = 1.
a(7) = 10 with 2^2 + binomial(2*2,2) = 10.
a(8) = 11 with 3^2 + binomial(2*1,1) = 11.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303543.

c[n_]:=c[n]=Binomial[2n,n];
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};n=0;Do[k=0;Label[bb];If[c[k]>m,Goto[aa]];If[SQ[m-c[k]],n=n+1;tab=Append[tab,m];Goto[aa],k=k+1;Goto[bb]];Label[aa],{m,1,134}];Print[tab]

cp's OEIS Frontend

A303233 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A303363 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A303389 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 5^c + 5^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A303393 Numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A303399 Number of ordered pairs (a, b) with 0 <= a <= b such that n - 5^a - 5^b can be written as the sum of two triangular numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A303540 Number of ways to write n as a^2 + b^2 + binomial(2*c,c) + binomial(2*d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A303434 Numbers of the form x*(3*x-1)/2 + 3^y with x and y nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A303656 Number of ways to write n as a^2 + b^2 + 3^c + 5^d, where a,b,c,d are nonnegative integers with a <= b.

Views

Author

Keywords

Comments

Examples

Links

A303233 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

A303363 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.

A303389 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c + 5^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

A303540 Number of ways to write n as a^2 + b^2 + binomial(2c,c) + binomial(2d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

A303434 Numbers of the form x(3x-1)/2 + 3^y with x and y nonnegative integers.

A303539 Number of ordered pairs (k, m) with 0 <= k <= m such that n - binomial(2k,k) - binomial(2m,m) can be written as the sum of two squares.