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A046080 a(n) = number of integer-sided right triangles with hypotenuse n. 43
0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,25

COMMENTS

a(n)=0 for n in A004144. - Lekraj Beedassy, May 14 2004

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, New York: Dover, pp. 116-117, 1966.

LINKS

Stanislav Sykora, Table of n, a(n) for n = 1..20000

Ron Knott, Pythagorean Triples and Online Calculators

F. Richman, Pythagorean Triples

A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 7.

Eric Weisstein's World of Mathematics, Pythagorean Triple

FORMULA

Let n = 2^e_2 * product_i p_i^f_i * product_j q_j^g_j where p_i == 1 mod 4, q_j == 3 mod 4; then a(n) = (1/2)*(product_i (2*f_i + 1) - 1). - Beiler, corrected

8*a(n) + 4 = A046109(n) for n > 0. - Ralf Stephan, Mar 14 2004

a(A084647(k)) = 3. - Jean-Christophe Hervé, Dec 01 2013

a(A084648(k)) = 4. - Jean-Christophe Hervé, Dec 01 2013

a(A084649(k)) = 5. - Jean-Christophe Hervé, Dec 01 2013

a(n) = A063725(n^2) / 2. - Michael Somos, Mar 29 2015

MAPLE

f:= proc(n) local F, t;

  F:= select(t -> t[1] mod 4 = 1, ifactors(n)[2]);

  1/2*(mul(2*t[2]+1, t=F)-1)

end proc:

map(f, [$1..100]); # Robert Israel, Jul 18 2016

MATHEMATICA

a[1] = 0; a[n_] := With[{fi = Select[ FactorInteger[n], Mod[#[[1]], 4] == 1 & ][[All, 2]]}, (Times @@ (2*fi+1)-1)/2]; Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Feb 06 2012, after first formula *)

PROG

(PARI) a(n)={my(m=0, k=n, n2=n*n, k2, l2);

while(1, k=k-1; k2=k*k; l2=n2-k2; if(l2>k2, break); if(issquare(l2), m++)); return(m)} \\ brute force, Stanislav Sykora, Mar 18 2015

(PARI) {a(n) = if( n<1, 0, sum(k=1, sqrtint(n^2 \ 2), issquare(n^2 - k^2)))}; /* Michael Somos, Mar 29 2015 */

(PARI) a(n) = {my(f = factor(n/(2^valuation(n, 2)))); (prod(k=1, #f~, if ((f[k, 1] % 4) == 1, 2*f[k, 2] + 1, 1)) - 1)/2; } \\ Michel Marcus, Mar 08 2016

CROSSREFS

First differs from A083025 at n=65.

Cf. A006339, A024362, A046079, A046081, A024362, A009000.

A088111 gives records; A088959 gives where records occur.

Cf. A063725.

Partial sums: A224921.

Sequence in context: A088950 A267113 A083025 * A170967 A035227 A049340

Adjacent sequences:  A046077 A046078 A046079 * A046081 A046082 A046083

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified September 26 01:25 EDT 2017. Contains 292500 sequences.