This site is supported by donations to The OEIS Foundation. Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A275301 Number of ordered ways to write n as x^2 + y^2 + z^2 + 2*w^2 with x + 2*y a cube, where x,y,z,w are nonnegative integers. 3
 1, 2, 2, 2, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 3, 4, 4, 5, 3, 2, 5, 2, 4, 5, 2, 3, 4, 3, 1, 3, 4, 4, 5, 3, 3, 5, 6, 3, 4, 3, 2, 4, 3, 4, 4, 3, 3, 7, 3, 4, 5, 3, 6, 4, 4, 4, 3, 3, 2, 3, 2, 2, 8, 3, 4, 8, 4, 3, 8, 3, 4, 9, 3, 4, 3, 4, 1, 3, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 5, 6, 7, 8, 11, 14, 15, 30, 78, 90, 93, 106, 111, 117, 125, 223, 335. (ii) Any natural number can be written as x^2 + y^2 + z^2 + 2*w^3 with x,y,z,w nonnegative integers such that x + 2*y is a square. See also A275344 for a similar conjecture. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016. EXAMPLE a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 2*0^2 with 0 + 2*0 = 0^3. a(2) = 2 since 2 = 1^2 + 0^2 + 1^2 + 2*0^2 with 1 + 2*0 =1^3, and 2 = 0^2 + 0^2 + 0^2 + 2*1^2 with 0 + 2*0 = 0^3. a(5) = 1 since 5 = 1^2 + 0^2 + 2^2 + 2*0^2 with 1 + 2*0 = 1^3. a(6) = 1 since 6 = 0^2 + 0^2 + 2^2 + 2*1^2 with 0 + 2*0 = 0^3. a(7) = 1 since 7 = 1^2 + 0^2 + 2^2 + 2*1^2 with 1 + 2*0 = 1^3. a(8) = 1 since 8 = 0^2 + 0^2 + 0^2 + 2*2^2 with 0 + 2*0 = 0^3. a(11) = 1 since 11 = 0^2 + 0^2 + 3^2 + 2*1^2 with 0 + 2*0 = 0^3. a(14) = 1 since 14 = 2^2 + 3^2 + 1^2 + 2*0^2 with 2 + 2*3 = 2^3. a(15) = 1 since 15 = 2^2 + 3^2 + 0^2 + 2*1^2 with 2 + 2*3 = 2^3. a(30) = 1 since 30 = 2^2 + 3^2 + 3^2 + 2*2^2 with 2 + 2*3 = 2^3. a(78) = 1 since 78 = 6^2 + 1^2 + 3^2 + 2*4^2 with 6 + 2*1 = 2^3. a(90) = 1 since 90 = 1^2 + 0^2 + 9^2 + 2*2^2 with 1 + 2*0 = 1^3. a(93) = 1 since 93 = 4^2 + 2^2 + 1^2 + 2*6^2 with 4 + 2*2 = 2^3. a(106) = 1 since 106 = 4^2 + 2^2 + 6^2 + 2*5^2 with 4 + 2*2 = 2^3. a(111) = 1 since 111 = 2^2 + 3^2 + 0^2 + 2*7^2 with 2 + 2*3 = 2^3. a(117) = 1 since 117 = 4^2 + 2^2 + 5^2 + 2*6^2 with 4 + 2*2 = 2^3. a(125) = 1 since 125 = 6^2 + 1^2 + 4^2 + 2*6^2 with 6 + 2*1 = 2^3. a(223) = 1 since 223 = 11^2 + 8^2 + 6^2 + 2*1^2 with 11 + 2*8 = 3^3. a(335) = 1 since 335 = 11^2 + 8^2 + 10^2 + 2*5^2 with 11 + 2*8 = 3^3. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]] CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)] Do[r=0; Do[If[SQ[n-2w^2-x^2-y^2]&&CQ[x+2*y], r=r+1], {w, 0, (n/2)^(1/2)}, {x, 0, Sqrt[n-2w^2]}, {y, 0, Sqrt[n-2w^2-x^2]}]; Print[n, " ", r]; Continue, {n, 0, 80}] CROSSREFS Cf. A000290, A000578, A271518, A275297, A275300, A275344. Sequence in context: A037805 A327144 A327051 * A282542 A271518 A106825 Adjacent sequences:  A275298 A275299 A275300 * A275302 A275303 A275304 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jul 22 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 10 00:54 EST 2019. Contains 329885 sequences. (Running on oeis4.)