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A253900 a(n) is the number of squares of the form x^2 + x + n^2 for 0 <= x <= n^2. 1
1, 2, 2, 3, 3, 2, 4, 4, 2, 4, 4, 3, 6, 4, 2, 4, 8, 4, 4, 4, 2, 6, 6, 3, 6, 4, 4, 8, 4, 2, 6, 12, 4, 4, 4, 2, 6, 12, 4, 5, 5, 4, 8, 4, 4, 8, 8, 4, 6, 6, 2, 8, 8, 2, 4, 4, 4, 12, 12, 6, 6, 8, 4, 4, 4, 4, 16, 8, 2, 4, 8, 8, 12, 6, 2, 6, 12, 4, 4, 8, 4, 8, 8, 3, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Properties of the sequence:

Of the first 1000 terms, 70.5% are powers of 2 (see the table below). We observe repeated terms a(n) = a(n+1) for n = 2, 4, 7, 10, 18, 19, 22, 26, 33, 34, 40, 44, 46, 49, 52, 55, ....

The following table lists statistics of a(n) for n=1..1000.

-------------------------------

| a(n)  | frequency |    %    |

-------------------------------

|   1   |      1    |   0.1%  |

|   2   |     61    |   6.1%  |

|   3   |      9    |   0.9%  |

|   4   |    235    |  23.5%  |

|   5   |      2    |   0.2%  |

|   6   |     72    |   7.2%  |

|   7   |      1    |   0.1%  |

|   8   |    266    |  26.6%  |

|   9   |     12    |   1.2%  |

|  10   |      6    |   0.6%  |

|  12   |    116    |  11.6%  |

|  14   |      1    |   0.1%  |

|  16   |    130    |  13.0%  |

|  18   |     10    |   1.0%  |

|  20   |     11    |   1.1%  |

|  24   |     45    |   4.5%  |

|  27   |      1    |   0.1%  |

|  32   |     12    |   1.2%  |

|  36   |      5    |   0.5%  |

|  40   |      1    |   0.1%  |

|  48   |      2    |   0.2%  |

|  54   |      1    |   0.1%  |

-------------------------------

| TOTAL |   1000    | 100.0%  |

-------------------------------

Based on the results in the table and the computing of Jon E. Schoenfield through n=3500, is it possible to determine an approximation of the probability p(a(n)= power of 2)?

Conjecture: the probability that a(n) is a power of 2 is such that 0.703 < p(a(n)=2^p) < 0.705.

The integers n such that a(n)=2 are 2, 3, 6, 9, 15, 21, 30, 36, 51, 54, 69, ... Is this A040040? - Michel Marcus, Jan 22 2015

LINKS

Michel Lagneau, Table of n, a(n) for n = 1..1000

EXAMPLE

a(7) = 4 because the 4 squares of the form x^2 + x + 7^2 are 49, 121, 289, 2401 for x = 0, 8, 15, 48, respectively.

a(8) = 4 because the 4 squares of the form x^2 + x + 8^2 are 64, 196, 484, 4096 for x = 0, 11, 20, 63, respectively.

MATHEMATICA

lst={}; Do[k=0; Do[If[IntegerQ[Sqrt[x^2+x+n^2]], k=k+1], {x, 0, n^2}]; AppendTo[lst, k], {n, 1, 100}]; lst

PROG

(PARI) a(n) = sum(x=0, n^2, issquare(x^2 + x + n^2)); \\ Michel Marcus, Jan 21 2015

CROSSREFS

Sequence in context: A039643 A288887 A154258 * A327487 A105496 A167618

Adjacent sequences:  A253897 A253898 A253899 * A253901 A253902 A253903

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jan 18 2015

STATUS

approved

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Last modified August 13 22:57 EDT 2020. Contains 336473 sequences. (Running on oeis4.)