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A088534 Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y. 13
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

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

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-Fran├žois 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 September 19 23:31 EDT 2019. Contains 327207 sequences. (Running on oeis4.)