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A306240 Number of ways to write n as x^9 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2 and z <= w. 2
1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 1, 3, 5, 4, 3, 2, 1, 1, 2, 4, 4, 3, 3, 2, 2, 3, 4, 3, 2, 3, 4, 5, 6, 4, 2, 2, 2, 2, 3, 5, 5, 4, 4, 4, 4, 2, 1, 3, 4, 5, 5, 3, 2, 2, 2, 3, 4, 4, 5, 4, 2, 4, 6, 5, 2, 2, 3, 4, 6, 6, 4, 4, 5, 3, 3, 6, 6, 4, 3, 3, 3, 3, 3, 5, 7, 6, 5, 3, 3, 4, 3, 5, 6, 4, 3, 4, 4, 3, 5, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 11, 17, 18, 47, 108, 109, 234, 359. Also, any nonnegative integer can be written as x^6 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

EXAMPLE

a(11) = 1 with 11 = 1^9 + 2^3 + 0*1 + 1*2.

a(18) = 1 with 18 = 0^9 + 0^3 + 2*3 + 3*4.

a(109) = 1 with 109 = 1^9 + 4^3 + 1*2 + 6*7.

a(234) = 1 with 234 = 0^9 + 6^3 + 2*3 + 3*5.

a(359) = 1 with 359 = 1^9 + 2^3 + 10*11 + 15*16.

a(1978) = 3 with 1978 = 2^9 + 2^3 + 26*27 + 27*28 = 2^9 + 6^3 + 19*20 + 29*30 = 2^9 + 6^3 + 24*25 + 25*26.

MATHEMATICA

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[4n+1]];

tab={}; Do[r=0; Do[If[TQ[n-x^9-y^3-z(z+1)], r=r+1], {x, 0, Min[2, n^(1/9)]}, {y, 0, (n-x^9)^(1/3)}, {z, 0, (Sqrt[2(n-x^9-y^3)+1]-1)/2}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

CROSSREFS

Cf. A000578, A001014, A001017, A002378, A262813, A262815, A270488, A306225, A306227, A306239.

Sequence in context: A066088 A139326 A029243 * A109829 A054125 A177227

Adjacent sequences:  A306237 A306238 A306239 * A306241 A306242 A306243

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 31 2019

STATUS

approved

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Last modified November 14 10:04 EST 2019. Contains 329111 sequences. (Running on oeis4.)