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 A262944 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is a square or a pentagonal number. 14
 1, 2, 2, 2, 2, 3, 4, 3, 1, 3, 5, 3, 2, 2, 5, 5, 3, 3, 5, 5, 3, 6, 6, 3, 3, 8, 6, 5, 5, 3, 7, 5, 5, 3, 4, 4, 8, 9, 3, 5, 7, 6, 3, 5, 5, 7, 5, 3, 4, 5, 6, 6, 9, 4, 5, 7, 7, 5, 4, 4, 7, 6, 1, 5, 5, 7, 7, 7, 1, 6, 10, 8, 6, 3, 4, 3, 6, 4, 6, 9, 5, 7, 9, 3, 5, 8, 9, 8, 3, 3, 11, 10, 6, 6, 8, 12, 5, 6, 4, 7 (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, 9, 63, 69, 489, 714, 1089. (ii) For any positive integer n, there are integers x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is twice a square or twice a pentagonal number. (iii) For any positive integer n, there are integers x >= 0 and y > 0 such that n - 2*x^4 - y*(y+1)/2 is a square or a pentagonal number. See also A262941 and A262945 for similar conjectures. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396. EXAMPLE a(1) = 1 since 1 = 0^4 + 1*2/2 + p_5(0), where p_5(n) denotes the pentagonal number n*(3*n-1)/2. a(9) = 1 since 9 = 1^4 + 2*3/2 + p_5(2). a(63) = 1 since 63 = 0^4 + 7*8/2 + p_5(5). a(69) = 1 since 69 = 2^4 + 7*8/2 + 5^2. a(489) = 1 since 489 = 3^4 + 12*13/2 + p_5(15). a(714) = 1 since 714 = 4^4 + 18*19/2 + p_5(14). a(1089) = 1 since 1089 = 4^4 + 38*39/2 + p_5(8). MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[n]]||(IntegerQ[Sqrt[24n+1]]&&Mod[Sqrt[24n+1]+1, 6]==0) Do[r=0; Do[If[SQ[n-x^4-y(y+1)/2], r=r+1], {x, 0, n^(1/4)}, {y, 1, (Sqrt[8(n-x^4)+1]-1)/2}]; Print[n, " ", r]; Continue, {n, 1, 100}] CROSSREFS Cf. A000217, A000290, A000583, A001318, A160325, A254885, A262813, A262815, A262816, A262941, A262945. Sequence in context: A187758 A171931 A221530 * A322789 A069904 A164978 Adjacent sequences:  A262941 A262942 A262943 * A262945 A262946 A262947 KEYWORD nonn AUTHOR Zhi-Wei Sun, Oct 05 2015 STATUS approved

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Last modified October 17 16:51 EDT 2019. Contains 328120 sequences. (Running on oeis4.)