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 A000161 Number of partitions of n into 2 squares. 50
 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,26 COMMENTS Number of ways of writing n as a sum of 2 (possibly zero) squares when order does not matter. Number of similar sublattices of square lattice with index n. Let Pk = the number of partitions of n into k nonzero squares. Then we have A000161 = P0 + P1 + P2, A002635 = P0 + P1 + P2 + P3 + P4, A010052 = P1, A025426 = P2, A025427 = P3, A025428 = P4. - Charles R Greathouse IV, Mar 08 2010, amended by M. F. Hasler, Jan 25 2013 a(A022544(n))=0; a(A001481(n))>0; a(A125022(n))=1; a(A118882(n))>1. - Reinhard Zumkeller, Aug 16 2011 REFERENCES J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 339 LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy] H. Bottomley, Illustration of initial terms R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996. J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps). Michael Gilleland, Some Self-Similar Integer Sequences E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 84. M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228. [From Ant King, Oct 05 2010] FORMULA a(n) = card { { a,b } c N | a^2+b^2 = n }. - M. F. Hasler, Nov 23 2007 Let f(n)= the number of divisors of n that are congruent to 1 modulo 4 minus the number of its divisors that are congruent to 3 modulo 4, and define delta(n) to be 1 if n is a perfect square and 0 otherwise. Then a(n)=1/2 (f(n)+delta(n)+delta(1/2 n)). - Ant King, Oct 05 2010 EXAMPLE 25 = 3^2+4^2 = 5^2, so a(25) = 2. MAPLE A000161 := proc(n) local i, j, ans; ans := 0; for i from 0 to n do for j from i to n do if i^2+j^2=n then ans := ans+1 fi od od; RETURN(ans); end; [ seq(A000161(i), i=0..50) ]; A000161 := n -> nops( numtheory[sum2sqr](n) ); # M. F. Hasler, Nov 23 2007 MATHEMATICA Length[PowersRepresentations[ #, 2, 2]] &/@Range[0, 150] (* Ant King, Oct 05 2010 *) PROG (PARI) A000161(n)=sum(i=0, n, sum(j=0, i, if(i^2+j^2-n, 0, 1))) (PARI) A000161(n)=sum(i=0, sqrtint(n>>1), issquare(n-i^2)) \\ M. F. Hasler, Nov 23 2007 (PARI) a(n)=sum(k=sqrtint((n-1)\2)+1, sqrtint(n), issquare(n-k^2)) \\ Charles R Greathouse IV, Mar 21 2014 (Haskell) a000161 n =    sum \$ map (a010052 . (n -)) \$ takeWhile (<= n `div` 2) a000290_list a000161_list = map a000161 [0..] -- Reinhard Zumkeller, Aug 16 2011 CROSSREFS Cf. A002654, A001481, A025427, A025428, A063725, A025426, A000290. Cf. A000925, A247367. Equivalent sequences for other numbers of squares: A010052 (1), A000164 (3), A002635 (4), A000174 (5). Sequence in context: A056973 A107782 A086017 * A060398 A253242 A260649 Adjacent sequences:  A000158 A000159 A000160 * A000162 A000163 A000164 KEYWORD nonn,core,easy,nice,changed AUTHOR STATUS approved

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