# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a270566 Showing 1-1 of 1 %I A270566 #21 Jul 24 2023 10:28:35 %S A270566 1,2,2,2,3,5,4,2,2,2,2,2,2,2,2,5,7,4,4,4,5,5,3,3,1,3,5,4,3,3,5,8,4,3, %T A270566 4,6,6,2,6,4,4,5,4,3,3,4,5,1,3,3,2,6,2,4,5,8,8,4,3,5,6,6,2,1,4,3,5,3, %U A270566 2,3,7,8,3,5,5,4,3,4,1,1,4 %N A270566 Number of ordered ways to write n as x^4 + y*(3y+1)/2 + z*(7z+1)/2, where x, y and z are integers with x nonnegative. %C A270566 Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 24, 47, 63, 78, 79, 143, 153, 325, 494, 949, 1079, 3328, 4335, 5609, 7949, 7967, 8888, 9665. %C A270566 Conjecture verified for n up to 10^11. - _Mauro Fiorentini_, Jul 24 2023 %H A270566 Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 %H A270566 Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113. %H A270566 Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396. %H A270566 Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283. %e A270566 a(24) = 1 since 24 = 2^4 + (-2)*(3*(-2)+1)/2 + (-1)*(7*(-1)+1)/2. %e A270566 a(78) = 1 since 78 = 1^4 + 7*(3*7+1)/2 + 0*(7*0+1)/2. %e A270566 a(143) = 1 since 143 = 1^4 + 6*(3*6+1)/2 + (-5)*(7*(-5)+1)/2. %e A270566 a(494) = 1 since 494 = 4^4 + (-7)*(3*(-7)+1)/2 + (-7)*(7*(-7)+1)/2. %e A270566 a(949) = 1 since 949 = 4^4 + 0*(3*0+1)/2 + 14*(7*14+1)/2. %e A270566 a(1079) = 1 since 1079 = 0^4 + 25*(3*25+1)/2 + 6*(7*6+1)/2. %e A270566 a(3328) = 1 since 3328 = 0^4 + 38*(3*38+1)/2 + 18*(7*18+1)/2. %e A270566 a(4335) = 1 since 4335 = 2^4 + 49*(3*49+1)/2 + 14*(7*14+1)/2. %e A270566 a(5609) = 1 since 5609 = 0^4 + (-61)*(3*(-61)+1)/2 + 4*(7*4+1)/2. %e A270566 a(7949) = 1 since 7949 = 3^4 + 43*(3*43+1)/2 + 38*(7*38+1)/2. %e A270566 a(7967) = 1 since 7967 = 7^4 + (-61)*(3*(-61)+1)/2 + 2*(7*2+1)/2. %e A270566 a(8888) = 1 since 8888 = 0^4 + (-77)*(3*(-77)+1)/2 + 3*(7*3+1)/2. %e A270566 a(9665) = 1 since 9665 = 3^4 + 73*(3*73+1)/2 + 21*(7*21+1)/2. %t A270566 (* From _Zhi-Wei Sun_, Start *) %t A270566 pQ[n_] := pQ[n] = IntegerQ[Sqrt[24 n + 1]]; %t A270566 Do[r = 0; Do[If[pQ[n - x^4 - y (7 y + 1)/2], r = r + 1], {x, 0, n^(1/4)}, {y, -Floor[(Sqrt[56 (n - x^4) + 1] + 1)/14], (Sqrt[56 (n - x^4) + 1] - 1)/14}]; Print[n, " ", r]; Continue, {n, 0, 80}] %t A270566 (* From _Zhi-Wei Sun_, End *) %t A270566 A270566[n_] := Length@Solve[x >= 0 && n == x^4 + y*(3 y + 1)/2 + z*(7 z + 1)/2, {x, y, z}, Integers]; %t A270566 Array[A270566, 25, 0] (* _JungHwan Min_, Mar 19 2016 *) %Y A270566 Cf. A001318, A000583, A262813, A262815, A262816, A262827, A262941, A262944, A262945, A262954, A262955, A262956, A270469, A270488, A270516, A270533, A270559. %K A270566 nonn %O A270566 0,2 %A A270566 _Zhi-Wei Sun_, Mar 19 2016 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE