A308547 -id:A308547 - cp's OEIS frontend

A308566 Number of ways to write n as w^2 + x(x+1) + 4^y5^z with w,x,y,z nonnegative integers.

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

Offset: 1

Zhi-Wei Sun, Jun 07 2019

nonn

Recall an observation of Euler: {w^2 + x*(x+1): w,x = 0,1,2,...} = {a*(a+1)/2 + b*(b+1)/2: a,b = 0,1,...}.
Conjecture: a(n) > 0 for all n > 0. Equivalently, each n = 1,2,3,... can be written as a*(a+1)/2 + b*(b+1)/2 + 4^c*5^d with a,b,c,d nonnegative integers.
See also A308584 for a similar conjecture.
We have verified a(n) > 0 for all n = 1..5*10^8.
a(n) > 0 for 0 < n < 10^10. - Giovanni Resta, Jun 08 2019

			a(1) = 1 with 1 = 0^2 + 0*1 + 4^0*5^0.
a(2) = 1 with 2 = 1^2 + 0*1 + 4^0*5^0.
a(3) = 1 with 3 = 0^2 + 1*2 + 4^0*5^0.
a(9) = 1 with 9 = 2^2 + 0*1 + 4^0*5^1.
a(303) = 1 with 303 = 16^2 + 6*7 + 4^0*5^1.
a(585) = 1 with 585 = 5^2 + 15*16 + 4^3*5^1.
a(37863) = 2 with 37863 = 166^2 + 101*102 + 4^0*5^1 = 179^2 + 26*27 + 4^5*5^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A000217, A000290, A000302, A000351, A002378, A303656, A303637, A308411, A308547, A308584.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-4^k*5^m-x(x+1)],r=r+1],{k,0,Log[4,n]},{m,0,Log[5,n/4^k]},{x,0,(Sqrt[4(n-4^k*5^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

A308584 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jun 08 2019

nonn

Conjecture: a(n) > 0 for all n > 0. Equivalently, each n = 1,2,3,... can be written as w^2 + x*(x+1) + 5^y*8^z with w,x,y,z nonnegative integers.
We have verified a(n) > 0 for all n = 1..4*10^8.
See also A308566 for a similar conjecture.
a(n) > 0 for all 0 < n < 10^10. - Giovanni Resta, Jun 10 2019

			a(13) = 1 with 13 = 3*4/2 + 3*4/2 + 5^0*8^0.
a(48) = 1 with 48 = 5*6/2 + 7*8/2 + 5^1*8^0.
a(87) = 1 with 87 = 1*2/2 + 12*13/2 + 5^0*8^1.
a(90) = 1 with 90 = 4*5/2 + 10*11/2 + 5^2*8^0.
a(423) = 1 with 423 = 9*10/2 + 22*23/2 + 5^3*8^0.
a(517) = 1 with 517 = 17*18/2 + 24*25/2 + 5^0*8^2.
a(985) = 1 with 985 = 19*20/2 + 34*35/2 + 5^2*8^1.
a(2694) = 1 with 2694 = 7*8/2 + 68*69/2 + 5^1*8^2.
a(42507) = 1 with 42507 = 178*179/2 + 223*224/2 + 5^2*8^2.
a(544729) = 1 with 544729 = 551*552/2 + 857*858/2 + 5^5*8^1.
a(913870) = 1 with 913870 = 559*560/2 + 700*701/2 + 5^3*8^4.
a(1843782) = 1 with 1843782 = 808*809/2 + 1668*1669/2 + 5^6*8^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A000217, A000351, A001018, A303656, A303637, A308411, A308547, A308566.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[TQ[n-5^k*8^m-x(x+1)/2],r=r+1],{k,0,Log[5,n]},{m,0,Log[8,n/5^k]},{x,0,(Sqrt[4(n-5^k*8^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

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A308566 Number of ways to write n as w^2 + x(x+1) + 4^y5^z with w,x,y,z nonnegative integers.

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A308584 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.

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cp's OEIS Frontend

A308566 Number of ways to write n as w^2 + x*(x+1) + 4^y*5^z with w,x,y,z nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A308584 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A308566 Number of ways to write n as w^2 + x(x+1) + 4^y5^z with w,x,y,z nonnegative integers.

A308584 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.