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 A262945 Number of ordered pairs (x,y) with x >= 0 and y >= 0 such that n - x^4 - 2*y^2 is a triangular number or a pentagonal number. 14
 1, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 4, 5, 2, 5, 4, 5, 7, 3, 1, 1, 4, 4, 6, 4, 1, 4, 4, 3, 5, 6, 5, 6, 4, 1, 1, 2, 5, 4, 5, 3, 3, 2, 1, 5, 4, 7, 9, 5, 4, 2, 2, 2, 5, 3, 2, 5, 2, 1, 3, 4, 3, 8, 4, 4, 5, 6, 3, 3, 3, 2, 7, 6, 1, 3, 3, 4, 7, 4, 6, 6, 7, 5, 2, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture: a(n) > 0 for every n = 0,1,2,..., and a(n) = 1 only for the following 55 values of n: 0, 26, 27, 32, 41, 42, 50, 65, 80, 97, 112, 122, 130, 160, 196, 227, 239, 272, 322, 371, 612, 647, 736, 967, 995, 1007, 1106, 1127, 1205, 1237, 1240, 1262, 1637, 1657, 1757, 2912, 2987, 3062, 3107, 3524, 3647, 3902, 5387, 5587, 5657, 6047, 6107, 11462, 13427, 14717, 15002, 17132, 20462, 30082, 35750. See also A262941, A262944, A262954 and A262955 for similar conjectures. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396. EXAMPLE a(26) = 1 since 26 = 2^4 + 2*0^2 + 4*5/2. a(32) = 1 since 32 = 0^4 + 2*4^2 + 0*1/2. a(41) = 1 since 41 = 1^4 + 2*3^2 + p_5(4), where p_5(n) denotes the pentagonal number n*(3*n-1)/2. a(196) = 1 since 196 = 1^4 + 2*5^2 + p_5(10). a(3524) = 1 since 3524 = 0^4 + 2*22^2 + 71*72/2. a(3647) = 1 since 3647 = 0^4 + 2*34^2 + p_5(30). a(6047) = 1 since 6047 = 5^4 + 2*39^2 + p_5(40). a(6107) = 1 since 6107 = 0^4 + 2*1^2 + 110*111/2. a(11462) = 1 since 11462 = 9^4 + 2*5^2 + 98*99/2. a(13427) = 1 since 13427 = 7^4 + 2*0^2 + 148*149/2. a(14717) = 1 since 14717 = 8^4 + 2*72^2 + 22*23/2. a(15002) = 1 since 15002 = 0^4 + 2*86^2 + 20*21/2. a(17132) = 1 since 17132 = 3^4 + 2*30^2 + p_5(101). a(20462) = 1 since 20462 = 0^4 + 2*26^2 + 195*196/2. a(30082) = 1 since 30082 = 11^4 + 2*63^2 + 122*123/2. a(35750) = 1 since 35750 = 0^4 + 2*44^2 + 252*253/2. MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[8n+1]]||(IntegerQ[Sqrt[24n+1]]&&Mod[Sqrt[24n+1]+1, 6]==0) Do[r=0; Do[If[SQ[n-x^4-2y^2], r=r+1], {x, 0, n^(1/4)}, {y, 0, Sqrt[(n-x^4)/2]}]; Print[n, " ", r]; Continue, {n, 0, 100}] CROSSREFS Cf. A000217, A000290, A000326, A000583, A262941, A262944, A262954, A262955. Sequence in context: A330406 A125954 A122443 * A309674 A270516 A099318 Adjacent sequences:  A262942 A262943 A262944 * A262946 A262947 A262948 KEYWORD nonn AUTHOR Zhi-Wei Sun, Oct 05 2015 STATUS approved

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Last modified April 6 08:27 EDT 2020. Contains 333268 sequences. (Running on oeis4.)