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 A088534 Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y. 14
 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,50 REFERENCES B. C. Berndt, "On a certain theta-function in a letter of Ramanujan from Fitzroy House", Ganita 43 (1992), 33-43. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 Oscar Marmon, Hexagonal Lattice Points on Circles, arXiv:math/0508201 [math.NT], 2005. FORMULA a(A003136(n)) > 0; a(A034020(n)) = 0; a(A118886(n)) > 1; a(A198772(n)) = 1; a(A198773(n)) = 2; a(A198774(n)) = 3; a(A198775(n)) = 4; a(A198799(n)) = n and a(m) <> n for m < A198799(n). - Reinhard Zumkeller, Oct 30 2011, corrected by M. F. Hasler, Mar 05 2018 In the prime factorization of n, let S_1 be the set of distinct prime factors p_i for which p_i == 1 (mod 3), let S_2 be the set of distinct prime factors p_j for which p_j == 2 (mod 3), and let M be the exponent of 3. Then n = 3^M * (Product_{p_i in S_1} p_i ^ e_i) * (Product_{p_j in S_2} p_j ^ e_j), and the number of solutions for x^2 + xy + y^2 = n, 0 <= x <= y is floor((Product_{p_i in S_1} (e_i + 1) + 1) / 2) if all e_j are even and 0 otherwise. E.g. a(1729) = 4 since 1729 = 7^1*13^1*19^1 and floor(((1+1)*(1+1)*(1+1)+1)/2) = 4. - Seth A. Troisi, Jul 02 2020 EXAMPLE From M. F. Hasler, Mar 05 2018: (Start) a(0) = a(1) = 1 since 0 = 0^2 + 0*0 + 0^2 and 1 = 0^2 + 0*1 + 1^2. a(2) = 0 since 2 cannot be written as x^2 + xy + y^2. a(49) = 2 since 49 = 0^2 + 0*7 + 7^2 = 3^2 + 3*5 + 5^2. (End) MATHEMATICA a[n_] := Sum[Boole[i^2 + i*j + j^2 == n], {i, 0, n}, {j, 0, i}]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jun 20 2018 *) PROG (PARI) a(n)=sum(i=0, n, sum(j=0, i, if(i^2+i*j+j^2-n, 0, 1))) (PARI) A088534(n, d)=sum(x=0, sqrt(n\3), sum(y=max(x, sqrtint(n-x^2)\2), sqrtint(n-2*x^2), x^2+x*y+y^2==n&&(!d||!printf("%d", [x, y]))))\\ Set 2nd arg = 1 to print all decompositions, with 0 <= x <= y. - M. F. Hasler, Mar 05 2018 (Haskell) a088534 n = length    [(x, y) | y <- [0..a000196 n], x <- [0..y], x^2 + x*y + y^2 == n] a088534_list = map a088534 [0..] -- Reinhard Zumkeller, Oct 30 2011 (Julia) function A088534(n)     n % 3 == 2 && return 0     M = Int(round(2*sqrt(n/3)))     count = 0     for y in 0:M, x in 0:y         n == x^2 + y^2 + x*y && (count += 1)     end     return count end A088534list(upto) = [A088534(n) for n in 0:upto] A088534list(104) |> println # Peter Luschny, Mar 17 2018 CROSSREFS Cf. A003136, A034020, A000196. Cf. A118886 (indices of values > 1), A198772 (indices of 1's), A198773 (indices of 2's), A198774 (indices of 3's), A198775 (indices of 4's), A198799 (index of 1st term = n). Sequence in context: A308264 A065335 A230264 * A178602 A216279 A025441 Adjacent sequences:  A088531 A088532 A088533 * A088535 A088536 A088537 KEYWORD nonn AUTHOR Benoit Cloitre, Nov 16 2003 EXTENSIONS Edited by M. F. Hasler, Mar 05 2018 STATUS approved

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Last modified August 3 01:47 EDT 2021. Contains 346429 sequences. (Running on oeis4.)