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A270705 Number of ordered ways to write n as x^2*pen(x) + pen(y) + pen(z) with pen(x) = x*(3x+1)/2 and pen(y) <= pen(z), where x, y and z are integers ("pen" stands for "pentagonal"). 2
1, 2, 5, 5, 6, 4, 3, 4, 4, 4, 3, 2, 3, 3, 6, 4, 4, 4, 3, 3, 3, 4, 6, 5, 6, 5, 5, 8, 8, 9, 7, 5, 7, 6, 7, 9, 7, 10, 5, 5, 9, 6, 12, 7, 8, 6, 3, 10, 6, 5, 7, 5, 8, 7, 8, 9, 5, 9, 8, 7, 5, 7, 7, 5, 6, 6, 5, 4, 6, 4, 8, 5, 9, 6, 3, 7, 5, 8, 8, 8, 8, 6, 6, 6, 6, 6, 8, 3, 1, 4, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Conjecture: (i) Any natural number can be written as a*x^2*pen(x) + b*pen(y) + c*pen(z) with x, y and z integers, provided that (a,b,c) is among the following ordered triples: (j,1,k) (j = 1,2; k = 1,2,3,4), (1,2,3), (3,1,4) and (4,1,3).

(ii) Every n = 0,1,2,... can be expressed as x^2*pen(x) + T(y) + T(z) with x, y and z integers, where T(m) denotes the triangular number m*(m+1)/2. Also, for each (a,b) = (1,2),(1,4),(2,2), any natural number can be written as a*x^2*T(x) + b*T(y) + T(z) with x, y and z integers.

(iii) Each natural number can be written as x^2*P(x) + pen(y) + pen(z) with x, y and z integers, where P(x) is either of the following polynomials: a*T(x) (a = 1,2,3,4,5), x*(5x+3)/2, x*(3x+1), x*(3x+2), x*(7x+1)/2, x*(4x+1), x*(4x+3), x*(9x+5)/2, x*(5x+3), x*(11x+9)/2, x*(13x+5)/2, x*(17x+9)/2, 3x*(3x+2), x*(11x+2).

See also A270594 and A270706 for other similar conjectures.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 0..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(88) = 1 since 88 = 1^2*pen(1) + pen(-5) + pen(-6).

MATHEMATICA

pen[x_]:=pen[x]=x(3x+1)/2

pQ[n_]:=pQ[n]=IntegerQ[Sqrt[24n+1]]

Do[r=0; Do[If[pQ[n-pen[y]-x^2*pen[x]], r=r+1], {y, -Floor[(Sqrt[12n+1]+1)/6], (Sqrt[12n+1]-1)/6}, {x, -1-Floor[(2(n-pen[y])/3)^(1/4)], (2(n-pen[y])/3)^(1/4)}]; Print[n, " ", r]; Continue, {n, 0, 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, A270706.

Sequence in context: A082084 A094236 A205444 * A073101 A235526 A130851

Adjacent sequences:  A270702 A270703 A270704 * A270706 A270707 A270708

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 21 2016

STATUS

approved

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Last modified November 14 17:12 EST 2018. Contains 317210 sequences. (Running on oeis4.)