

A270516


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


18



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

0,2


COMMENTS

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and the only values of n > 1428 with a(n) = 1 are 2205, 2259, 3556, 4107, 4337, 5387, 9331, 16561, 22237, 27569, 63947, 78610.
(ii) Any natural number can be written as x*(x^3+2) + y*(y+1)/2 + z*(3z+1), where x and y are nonnegative integers, and z is an integer.
(iii) Every n = 0,1,2,... can be written as x*(x^3+x^2+6) + y*(y+1)/2 + z*(3z+2) (or x*(x^3+x^2+4x+1) + y*(y+1)/2 + z*(3z+1)), where x and y are nonnegative integers, and z is an integer.
See also A270533 for a similar conjecture.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..10000
ZhiWei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103113.
ZhiWei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190211.


EXAMPLE

a(72) = 1 since 72 = 2^3*3 + 5*6/2 + 3*(3*3+2).
a(75) = 1 since 75 = 0^3*1 + 4*5/2 + (5)*(3*(5)+2).
a(5387) = 1 since 5387 = 7^3*8 + 2*3/2 + (30)*(3*(30)+2).
a(9331) = 1 since 9331 = 8^3*9 + 2*3/2 + (40)*(3*(40)+2).
a(16561) = 1 since 16561 = 1^3*2 + 101*102/2 + (62)*(3*(62)+2).
a(22237) = 1 since 22237 = 6^3*7 + 104*105/2 + 71*(3*71+2).
a(27569) = 1 since 27569 = 2^3*3 + 49*50/2 + (94)*(3*(94)+2).
a(63947) = 1 since 63947 = 0^3*1 + 173*174/2 + (128)*(3*(128)+2).
a(78610) = 1 since 78610 = 16^3*17 + 52*53/2 + 50*(3*50+2).


MATHEMATICA

OQ[x_]:=OQ[x]=IntegerQ[Sqrt[3x+1]]
Do[r=0; Do[If[OQ[ny(y+1)/2x^3*(x+1)], r=r+1], {y, 0, (Sqrt[8n+1]1)/2}, {x, 0, (ny(y+1)/2)^(1/4)}]; Print[n, " ", r]; Continue, {n, 0, 80}]


CROSSREFS

Cf. A000217, A000578, A001082, A001318, A262813, A262815, A262816, A262941, A262944, A262945, A262954, A262955, A262956, A270469, A270488, A270533.
Sequence in context: A122443 A262945 A309674 * A099318 A187128 A255911
Adjacent sequences: A270513 A270514 A270515 * A270517 A270518 A270519


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 18 2016


STATUS

approved



