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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 (list; graph; refs; listen; history; text; internal format)
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

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.

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[n-x^4-y(y+1)/2], r=r+1], {x, 0, n^(1/4)}, {y, 1, (Sqrt[8(n-x^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

Zhi-Wei Sun, Oct 05 2015

STATUS

approved

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Last modified October 16 06:10 EDT 2019. Contains 328046 sequences. (Running on oeis4.)