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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.)