

A262956


Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n  x^4  y*(y+1)/2 is a square or a square minus 1.


14



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

1,2


COMMENTS

Conjecture: a(n) > 0 for all n > 0. In other words, for any positive integer n, either n or n + 1 can be written as the sum of a fourth power, a square and a positive triangular number.
We also guess that a(n) = 1 only for n = 1, 89, 244, 464, 5243, 14343.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103113.


EXAMPLE

a(1) = 1 since 1 = 0^4 + 1*2/2 + 0^2.
a(89) = 1 since 89 = 2^4 + 4*5/2 + 8^2  1.
a(244) = 1 since 244 = 2^4 + 2*3/2 + 15^2.
a(464) = 1 since 464 = 2^4 + 22*23/2 + 14^2  1.
a(5243) = 1 since 5243 = 0^4 + 50*51/2 + 63^2  1.
a(14343) = 1 since 14343 = 2^4 + 163*164/2 + 31^2.


MATHEMATICA

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


CROSSREFS

Cf. A000217, A000290, A000583, A262941, A262944, A262945, A262954, A262955, A262959.
Sequence in context: A130971 A051776 A270920 * A073734 A231335 A271237
Adjacent sequences: A262953 A262954 A262955 * A262957 A262958 A262959


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 05 2015


STATUS

approved



