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A000050 Number of positive integers <= 2^n of form x^2 + y^2.
(Formerly M0715 N0265)
6
1, 2, 3, 5, 9, 16, 29, 54, 97, 180, 337, 633, 1197, 2280, 4357, 8363, 16096, 31064, 60108, 116555, 226419, 440616, 858696, 1675603, 3273643, 6402706, 12534812, 24561934, 48168461, 94534626, 185661958, 364869032, 717484560, 1411667114, 2778945873, 5473203125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Seth A. Troisi, Table of n, a(n) for n = 0..50 (terms 0..35 from N. J. A. Sloane)

P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, arXiv:math/0204332 [math.NT], 2002.

P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, Math. Comp. 73 (2004), no. 245, 451-473.

D. Shanks and L. P. Schmid, Variations on a theorem of Landau. Part I, Math. Comp., 20 (1966), 551-569.

Seth A. Troisi, C++ program

Index entries for sequences related to populations of quadratic forms

EXAMPLE

There are 5 integers <= 2^3 of the form x^2 + y^2. The five (x,y) pairs (x <= y) are (0,1), (1,1), (0,2), (1,2), (2,2) and give the integers 1, 2, 4, 5, 8, respectively. So a(3) = 5. - Seth A. Troisi, Apr 27 2022

MATHEMATICA

(* This program is not suitable for a large number of terms *) a[0] = 1; a[n_] := a[n] = (For[cnt = 0; k = 2^(n-1)+1, k <= 2^n, k++, If[SquaresR[2, k] > 0, cnt++]]; cnt + a[n-1]); Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 26}] (* Jean-Fran├žois Alcover, Mar 20 2014 *)

PROG

(Haskell)

isqrt = a000196

issquare = a010052

a000050 n = foldl f 0 [1..2^n]

  where f i j = if a000050' j > 0 then i + 1 else i

a000050' k = foldl f 0 (h k)

  where f i y = g y + i

          where g y = issquare (k - y^2)

        h k = [0..isqrt k]

-- James Spahlinger, Oct 09 2012

CROSSREFS

Cf. A001481.

Sequence in context: A255071 A103285 A000049 * A050253 A198518 A182558

Adjacent sequences:  A000047 A000048 A000049 * A000051 A000052 A000053

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 7 08:05 EDT 2022. Contains 357270 sequences. (Running on oeis4.)