A046080 -id:A046080 - cp's OEIS frontend

A055524 Longest other side of a Pythagorean triangle with n as length of one of the three sides (in fact n is a leg and a(n) the hypotenuse).

5, 5, 13, 10, 25, 17, 41, 26, 61, 37, 85, 50, 113, 65, 145, 82, 181, 101, 221, 122, 265, 145, 313, 170, 365, 197, 421, 226, 481, 257, 545, 290, 613, 325, 685, 362, 761, 401, 841, 442, 925, 485, 1013, 530, 1105, 577, 1201, 626, 1301, 677, 1405, 730, 1513, 785

Offset: 3

Henry Bottomley, May 22 2000

Paolo Xausa, Table of n, a(n) for n = 3..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055525, A055526, A055527.

A055524[n_] := (3*n^2-(-1)^n*(n^2-2)+6)/8; Array[A055524, 100, 3] (* or *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {5, 5, 13, 10, 25, 17}, 100] (* Paolo Xausa, Feb 29 2024 *)

Vec(-x^3*(2*x^5+x^4-5*x^3-2*x^2+5*x+5)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 15 2014

a(n) = sqrt(n^2+A055523(n)^2). a(2k) = k^2+1, a(2k+1) = k^2+(k+1)^2.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: -x^3*(2*x^5+x^4-5*x^3-2*x^2+5*x+5) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 15 2014
a(n) = (3*n^2+6-(n^2-2)*(-1)^n)/8. - Luce ETIENNE, Jul 11 2015

A063014 Number of solutions to n^2 = b^2 + c^2 (with c >= b >= 0).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 5, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 2

Offset: 0

Henry Bottomley, Jul 26 2001

nonn

			a(0)=1 since 0^2 = 0^2 + 0^2;
a(5)=2 since 5^2 = 0^2 + 5^2 = 3^2 + 4^2;
a(25)=3 since 25^2 = 0^2 + 25^2 = 7^2 + 24^2 = 15^2 + 20^2.

Felix Huber, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000161, A046080.

Column k=2 of A255212.

with(NumberTheory):
A063014 := n -> nops(SumOfSquares(n^2));
seq(A063014(n), n = 0 .. 100); # Felix Huber, Jun 01 2024

a(0) = 1; a(n) = A046080(n) + 1 for n > 0. [amended by Georg Fischer, Jan 25 2020]

a(n) = A000161(n^2). - Christian Krause, Dec 08 2022

A224921 Number of Pythagorean triples (a, b, c) with a^2 + b^2 = c^2 and 0 < a < b < c < n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 8, 9, 9, 9, 10, 11, 11, 11, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 20, 21, 22, 23, 23, 24, 24, 24, 25, 25, 26, 27, 27, 27, 27, 31, 31, 31, 32, 32, 33, 33, 33

Offset: 1

Reiner Moewald, Apr 19 2013

nonn

a(n+1) > a(n) iff n is in A009003. - Benoit Cloitre, Dec 08 2021

Reiner Moewald and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..500 from Reiner Moewald)

Cf. A156685. Essentially partial sums of A046080.

Cf. A009003.

a046080:= 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:
ListTools:-PartialSums(map(a046080, [$0..100])); # Robert Israel, Jul 18 2016

b[0] = b[1] = 0; b[n_] := With[{fi = Select[FactorInteger[n], Mod[#[[1]], 4] == 1&][[All, 2]]}, (Times @@ (2*fi + 1) - 1)/2];
Table[b[n], {n, 0, 100}] // Accumulate (* Jean-François Alcover, Feb 27 2019 *)

a(n)=sum(a=1,n-3,sum(b=a+1,sqrtint((n-1)^2-a^2), issquare(a^2+b^2))) \\ Charles R Greathouse IV, Apr 29 2013

A055525 Shortest other side of a Pythagorean triangle having n as length of one of the three sides.

Original entry on oeis.org

4, 3, 3, 8, 24, 6, 12, 6, 60, 5, 5, 48, 8, 12, 8, 24, 180, 12, 20, 120, 264, 7, 7, 10, 36, 21, 20, 16, 480, 24, 44, 16, 12, 15, 12, 360, 15, 9, 9, 40, 924, 33, 24, 528, 1104, 14, 168, 14, 24, 20, 28, 72, 33, 33, 76, 40, 1740, 11, 11, 960, 16, 48, 16, 88, 2244, 32, 92, 24

Offset: 3

Henry Bottomley, May 22 2000

nonn

Robert G. Wilson v, Table of n, a(n) for n = 3..10000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055526, A055527.

a[n_] := Block[{a, c, k = 1, n2 = n^2}, While[ If[ k > n, !IntegerQ[c = Sqrt[n2 + k^2]], !IntegerQ[c = Sqrt[n2 + k^2]] && !IntegerQ[a = Sqrt[n2 - k^2]]], k++; If[k == n, k++]]; If[ IntegerQ@ c, k, Sqrt[n2 - a^2]]]; (* Robert G. Wilson v, Feb 23 2024 *)

From Robert G. Wilson v, Feb 23 2024: (Start)
sqrt(2*(n-1)) < a(n) < n^2/2.
If n = k*m, then a(n) <= k*a(m). (End)

A055522 Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).

Original entry on oeis.org

6, 6, 30, 24, 84, 60, 180, 120, 330, 210, 546, 336, 840, 504, 1224, 720, 1710, 990, 2310, 1320, 3036, 1716, 3900, 2184, 4914, 2730, 6090, 3360, 7440, 4080, 8976, 4896, 10710, 5814, 12654, 6840, 14820, 7980, 17220, 9240, 19866, 10626, 22770, 12144, 25944

Offset: 3

Henry Bottomley, May 22 2000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055523, A055524, A055525, A055526, A055527.

seq(piecewise(n mod 2 = 0,n*(n^2-4)/8,n*(n^2-1)/4),n=3..60); # C. Ronaldo

Table[n*(3*(n^2 - 2) - (n^2 + 2)*(-1)^n)/16, {n, 3, 50}] (* Wesley Ivan Hurt, Apr 27 2017 *)

a(n) = n*A055523(n)/2.
a(2k) = k*(k+1)*(k-1), a(2k+1) = k*(k+1)*(2k+1).
O.g.f.: 6*x^3*(x+1+x^2)/((1-x)^4*(1+x)^4). a(2k+1)=A055112(k). a(2k)=A007531(k+1). [R. J. Mathar, Aug 06 2008]
a(n) = n*(3*(n^2-2)-(n^2+2)*(-1)^n)/16. - Luce ETIENNE, Jul 17 2015

A055526 Shortest hypotenuse of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

5, 5, 13, 10, 25, 10, 15, 26, 61, 13, 85, 50, 17, 20, 145, 30, 181, 25, 29, 122, 265, 25, 65, 170, 45, 35, 421, 34, 481, 40, 55, 290, 37, 39, 685, 362, 65, 41, 841, 58, 925, 55, 51, 530, 1105, 50, 175, 130, 85, 65, 1405, 90, 73, 65, 95, 842, 1741, 61, 1861, 962, 65

Offset: 3

Henry Bottomley, May 22 2000

nonn

Smallest k>n such that the squarefree part of k+n equals the squarefree part of k-n - Benoit Cloitre, May 26 2002

Paolo Xausa, Table of n, a(n) for n = 3..1000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055525, A055527.

core[n_] := core[n] = Times @@ Map[#[[1]]^Mod[#[[2]], 2] &, FactorInteger[n]];
A055526[n_] := Block[{k = n}, While[core[++k+n] != core[k-n]]; k];
Array[A055526, 100, 3] (* Paolo Xausa, Feb 29 2024 *)

for(n=3,105,s=n+1; while(abs(core(s+n)-core(s-n))>0,s++); print1(s,","))

a(n) = sqrt(n^2+A055527(n)^2).

A247588 Number of integer-sided acute triangles with largest side n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, 52, 56, 63, 67, 73, 80, 84, 90, 96, 104, 111, 117, 126, 132, 140, 147, 154, 165, 172, 183, 189, 198, 210, 219, 229, 237, 247, 260, 270, 282, 292, 302

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(3) = 3 because there are 3 integer-sided acute triangles with largest side 3: (1,3,3); (2,3,3); (3,3,3).

Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A046080, A002623, A224921, A247586, A247587, A247589.

tr_a:=proc(n) local a,b,t,d;t:=0:
for a to n do
for b from max(a,n+1-a) to n do
d:=a^2+b^2-n^2:
if d>0 then t:=t+1 fi
od od;
t; end;

a[ n_] := Length @ FindInstance[ n >= b >= a >= 1 && n < b + a && n^2 < b^2 + a^2, {a, b}, Integers, 10^9]; (* Michael Somos, May 24 2015 *)

a(n) = sum(j=0, n*(1 - sqrt(2)/2), n - j - floor(sqrt(2*j*n - j^2))); \\ Michel Marcus, Oct 07 2014

{a(n) = sum(j=0, n - sqrtint(n*n\2) - 1, n - j - sqrtint(2*j*n - j*j))}; /* Michael Somos, May 24 2015 */

a(n) = Sum_{j=0..floor(n*(1 - sqrt(2)/2))} (n - j - floor(sqrt(2*j*n - j^2))). - Anton Nikonov, Oct 06 2014

a(n) = (1/8)*(-4*ceiling((n - 1)/sqrt(2)) + 4*n^2 - A000328(n) + 1), n > 1. - Mats Granvik, May 23 2015

A054435 Smallest area of a Pythagorean triangle with n as length of one of the three sides.

Original entry on oeis.org

6, 6, 6, 24, 84, 24, 54, 24, 330, 30, 30, 336, 54, 96, 60, 216, 1710, 96, 210, 1320, 3036, 84, 84, 120, 486, 294, 210, 216, 7440, 384, 726, 240, 210, 270, 210, 6840, 270, 180, 180, 840, 19866, 726, 486, 12144, 25944, 336, 4116, 336, 540, 480, 630, 1944, 726

Offset: 3

Henry Bottomley, May 22 2000

nonn

Cf. A009112, A046079, A046080, A046081, A054436, A055522, A055523, A055524, A055525, A055526, A055527.

A054436 Smallest area of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

6, 6, 30, 24, 84, 24, 54, 120, 330, 30, 546, 336, 60, 96, 1224, 216, 1710, 150, 210, 1320, 3036, 84, 750, 2184, 486, 294, 6090, 240, 7440, 384, 726, 4896, 210, 270, 12654, 6840, 1014, 180, 17220, 840, 19866, 726, 540, 12144, 25944, 336, 4116, 3000, 1734, 1014

Offset: 3

Henry Bottomley, May 22 2000

nonn

Cf. A009112, A046079, A046080, A046081, A054435, A055522, A055523, A055524, A055525, A055526, A055527.

readlib(issqr): for a from 3 to 80 do for b from 1 by 1 while not issqr(a^2+b^2) do od: printf("%d, ",a*b/2) od: # C. Ronaldo

a[n_] := For[k = 1, True, k++, If[IntegerQ[Sqrt[n^2+k^2]], Return[n k/2]]];
a /@ Range[3, 100] (* Jean-François Alcover, Feb 14 2020 *)

a(n) = n*A055527(n)/2.

A181786 Number of inequivalent solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 3, 0, 2, 3, 3, 0, 6, 2, 3, 1, 2, 1, 8, 1, 3, 3, 4, 0, 10, 2, 5, 3, 4, 3, 8, 0, 5, 6, 6, 2, 11, 3, 6, 1, 8, 2, 12, 1, 6, 8, 8, 1, 15, 3, 8, 3, 7, 4, 20, 0, 6, 10, 9, 2, 16, 5, 9, 3, 9, 4, 15, 3, 15, 8, 10, 0, 22, 5, 11, 6, 9, 6, 18, 2, 11, 11, 14, 3, 21, 6, 13, 1, 12, 8, 31, 2

Offset: 0

T. D. Noe, Nov 12 2010

nonn

Note that a(n)=0 for n=0 and the n in A094958.
Also note that a(2n)=a(n), e.g., a(1000)=a(500)=a(250)=a(125)=14. - Zak Seidov, Mar 02 2012
a(n) is the number of distinct parallelepipeds each one having integer diagonal n and integer sides. - César Eliud Lozada, Oct 26 2014

Samuel Harkness, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Zak Seidov)

Cf. A016725, A016727, A046080, A181787, A181788

nn=100; t=Table[0,{nn}]; Do[n=Sqrt[a^2+b^2+c^2]; If[n<=nn && IntegerQ[n], t[[n]]++], {a,nn}, {b,a,nn}, {c,b,nn}]; Prepend[t,0]

cp's OEIS Frontend

A055524 Longest other side of a Pythagorean triangle with n as length of one of the three sides (in fact n is a leg and a(n) the hypotenuse).

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A063014 Number of solutions to n^2 = b^2 + c^2 (with c >= b >= 0).

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A224921 Number of Pythagorean triples (a, b, c) with a^2 + b^2 = c^2 and 0 < a < b < c < n.

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A055525 Shortest other side of a Pythagorean triangle having n as length of one of the three sides.

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A055522 Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).

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A055526 Shortest hypotenuse of a Pythagorean triangle with n as length of a leg.

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A247588 Number of integer-sided acute triangles with largest side n.

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A054435 Smallest area of a Pythagorean triangle with n as length of one of the three sides.

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A054436 Smallest area of a Pythagorean triangle with n as length of a leg.