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 A270706 Number of ordered ways to write n as x^2*T(x) + y^2 + T(z), where x, y and z are integers with x nonzero, y positive and z nonnegative, and T(m) denotes the triangular number m*(m+1)/2. 2
 1, 2, 1, 2, 4, 2, 2, 4, 2, 3, 5, 2, 3, 4, 3, 6, 4, 2, 6, 5, 2, 4, 6, 2, 3, 7, 3, 5, 6, 4, 8, 5, 2, 5, 3, 5, 9, 7, 3, 5, 8, 3, 6, 5, 2, 8, 4, 2, 9, 6, 4, 7, 7, 4, 5, 7, 5, 9, 5, 3, 7, 4, 5, 12, 9, 4, 5, 8, 4, 6, 11, 3, 9, 5, 3, 10, 3, 4, 9, 6, 5, 11, 8, 5, 7, 9, 3, 5, 4, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 90, 438, 480, 7108. (ii) Let pen(x) = x*(3x+1)/2. Any natural number can be written as a*f(x)*g(x) + f(y) + g(z) with x, y and z integers, whenever (a,f(x),g(x)) is among the following ordered triples: (1,T(x),x^2), (1,T(x),pen(x)), (1,T(x),x*(5x+1)/2), (1,T(x),x*(5x+3)/2), (1,T(x),x*(3x+j)) (j = 1,2), (1,pen(x),3*T(x)), (1,pen(x),x*(7x+j)/2) (j = 1,3,5), (1,pen(x),x*(4x+1)), (2,T(x),x^2), (2,T(x),pen(x)), (2,T(x),x(5x+j)/2) (j = 1,3), (2,T(x),x*(3x+j)) (j = 1,2), (2,2*T(x),pen(x)), (2,pen(x),x(7x+j)/2) (j = 3,5), (k,x^2,pen(x)) (k = 1,2,3,4,5,8,11). (iii) Each natural number can be written as P(x,y,z) with x, y and z integers, where P(x,y,z) is either of the following polynomials: T(x)*x(5x+1)/2+T(y)+2*T(z), a*T(x)*pen(x)+pen(y)+pen(z) (a = 1,2,3,4), T(x)*pen(x)+pen(y)+3*pen(z), T(x)*pen(x)+pen(y)+4*pen(z), 2*T(x)*pen(x)+pen(y)+3*pen(z), pen(x)*x(5x+j)/2+pen(y)+3*pen(z) (j = 1,3), x(3x+2)*pen(x)+pen(y)+4*pen(z), pen(x)*x(7x+1)/2+pen(y)+pen(z), pen(x)*x(9x+7)/2+pen(y)+pen(z). See also A270594 and A270705 for some other similar conjectures. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Z.-W. Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113. Z.-W. Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396. Z.-W. Sun, On universal sums ax^2+by^2+f(z), aT_x+bT_y+f(z) and zT_x+by^2+f(z), preprint, arXiv:1502.03056 [math.NT], 2015. EXAMPLE a(1) = 1 since 1 = (-1)^2*T(-1) + 1^2 + T(0). a(3) = 1 since 3 = 1^2*T(1) + 1^2 + T(1). a(90) = 1 since 90 = 3^2*T(3) + 6^2 + T(0). a(438) = 1 since 438 = 4^2*T(4) + 5^2 + T(22). a(480) = 1 since 480 = 1^2*T(1) + 17^2 + T(19). a(7108) = 1 since 1^2*T(1) + 69^2 + T(68). MATHEMATICA TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]] Do[r=0; Do[If[x!=0&&TQ[n-y^2-x^3*(x+1)/2], r=r+1], {y, 1, Sqrt[n]}, {x, -1-Floor[(2(n-y^2))^(1/4)], (2(n-y^2))^(1/4)}]; Print[n, " ", r]; Continue, {n, 1, 90}] CROSSREFS Cf. A000217, A000290, A001318, A001082, A085787, A262813, A262815, A262816, A262827, A262941, A262944, A262945, A262954, A262955, A262956, A270469, A270488, A270516, A270533, A270559, A270566, A270594, A270616, A270705. Sequence in context: A302290 A145173 A270594 * A082793 A114929 A247321 Adjacent sequences:  A270703 A270704 A270705 * A270707 A270708 A270709 KEYWORD nonn AUTHOR Zhi-Wei Sun, Mar 21 2016 STATUS approved

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Last modified January 22 19:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)