A262857 - OEIS

A262857

Number of ordered ways to write n as w^3 + 2*x^3 + y^2 + 2*z^2, where w, x, y and z are nonnegative integers.

1, 2, 3, 4, 4, 3, 3, 2, 3, 5, 5, 6, 6, 3, 4, 1, 4, 6, 7, 10, 7, 5, 4, 2, 5, 8, 8, 9, 9, 6, 6, 2, 6, 10, 8, 13, 9, 6, 7, 5, 5, 8, 6, 9, 10, 6, 9, 4, 5, 9, 6, 13, 10, 7, 11, 6, 8, 10, 8, 10, 12, 9, 9, 7, 8, 13, 10, 16, 12, 6, 12, 8, 10, 13, 12, 13, 12, 8, 11, 7, 10, 16, 15, 17, 16, 6, 11, 7, 12, 16, 11, 16, 9, 10, 5, 6, 10, 15, 17, 18, 16 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	COMMENTS	`Conjecture: We have {aw^3+bx^3+cy^2+dz^2: w,x,y,z = 0,1,2,...} = {0,1,2,...} if (a,b,c,d) is among the following 63 quadruples:` (1,1,1,2),(1,1,2,4),(1,2,1,1),(1,2,1,2),(1,2,1,3),(1,2,1,4),(1,2,1,6),(1,2,1,13),(1,2,2,3),(1,2,2,4),(1,2,2,5),(1,3,1,1),(1,3,1,2),(1,3,1,3),(1,3,1,5),(1,3,1,6),(1,3,2,3),(1,3,2,4),(1,3,2,5),(1,4,1,1),(1,4,1,2),(1,4,1,3),(1,4,2,2),(1,4,2,3),(1,4,2,5),(1,5,1,1),(1,5,1,2),(1,6,1,1),(1,6,1,3),(1,7,1,2),(1,8,1,2),(1,9,1,2),(1,9,2,4),(1,10,1,2),(1,11,1,2),(1,11,2,4),(1,12,1,2),(1,14,1,2),(1,15,1,2),(2,3,1,1),(2,3,1,2),(2,3,1,3),(2,3,1,4),(2,4,1,1),(2,4,1,2),(2,4,1,6),(2,4,1,8),(2,4,1,10),(2,5,1,3),(2,6,1,1),(2,7,1,3),(2,8,1,1),(2,8,1,4),(2,10,1,1),(2,13,1,1),(3,4,1,2),(3,5,1,2),(3,7,1,2),(3,9,1,2),(4,5,1,2),(4,6,1,2),(4,8,1,2),(4,11,1,2). `Conjecture verified up to 10^11 for all quadruples. - Mauro Fiorentini, Jul 18 2023`
	LINKS	`Zhi-Wei Sun, Table of n, a(n) for n = 0..10000`
	EXAMPLE	`a(7) = 2 since 7 = 1^3 + 20^3 + 2^2 + 21^2 = 1^3 + 21^3 + 2^2 + 20^2.` `a(15) = 1 since 15 = 1^3 + 21^3 + 2^2 + 22^2.`
	MATHEMATICA	`SQ[n_]:=IntegerQ[Sqrt[n]]` `Do[r=0; Do[If[SQ[n-x^3-2y^3-2z^2], r=r+1], {x, 0, n^(1/3)}, {y, 0, ((n-x^3)/2)^(1/3)}, {z, 0, Sqrt[(n-x^3-2y^3)/2]}]; Print[n, " ", r]; Continue, {n, 0, 100}]`
	CROSSREFS	`Cf. A000290, A000578, A262813, A262815, A262816, A262824, A262827.` `Sequence in context: A316823 A372982 A051951 * A107898 A128863 A117391` `Adjacent sequences: A262854 A262855 A262856 * A262858 A262859 A262860`
	KEYWORD	`nonn`
	AUTHOR	`Zhi-Wei Sun, Oct 03 2015`
	STATUS	`approved`