A009112 -id:A009112 - cp's OEIS frontend

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

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).

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.

A118903 Areas of rectangles with integer sides and diagonal.

Original entry on oeis.org

12, 48, 60, 108, 120, 168, 192, 240, 300, 360, 420, 432, 480, 540, 588, 660, 672, 768, 960, 972, 1008, 1080, 1092, 1200, 1260, 1440, 1452, 1500, 1512, 1680, 1728, 1848, 1920, 1980, 2028, 2160, 2352, 2448, 2640, 2688, 2700, 2772, 2940, 3000

Offset: 1

Giovanni Resta, May 05 2006

nonn

			12 = 3*4 and 3^2+4^2=5^2.

Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A009112, A118901, A118904, A118905.

Twice A009112.

A177021 Numbers which are the area of exactly three Pythagorean triangles.

Original entry on oeis.org

840, 3360, 7560, 10920, 13440, 21000, 30240, 31920, 41160, 43680, 53760, 68040, 84000, 98280, 101640, 120960, 127680, 141960, 164640, 166320, 174720, 189000, 215040, 242760, 272160, 273000, 286440, 287280, 303240, 336000, 370440, 393120, 406560, 444360

Offset: 1

Claudio Meller, on a suggestion by Antonio Roldán, Dec 08 2010

nonn

The triangles need not be primitive. Number of terms less than 10^n: 0, 0, 1, 3, 14, 53, ....
13123110 is the smallest number which is the area of three primitive Pythagorean triangles, (1380,19019,19069)(3059,8580,9109) and (4485,5852,7373); this triple was found by Charles L. Shedd in 1945.
From Sture Sjöstedt, Dec 06 2016: (Start)
840 = 3*5*7*8; p=3, q=8, q-p=5, r=7 is a solution to p^2 - pq + q^2 = r^2. If r is a prime number in the sequence 7, 13, 19, ..., there are three Pythagorean triangles with the same area and at least one of them is primitive.
10920 = 7*8*13*15; p=7, q=15, q-p=8, r=13.
x^2 + 3*y^2 = 4*r^2 where r is a prime number in the sequence 7, 13, 19, ... gives lattice points that can be used to find solutions to p^2 - pq + q^2 = r^2. p, q, (q-p) and r are the y-coordinates in the first quadrant. (End)

			a(1) = 840 is the area of {15,112,113}, {24,70,74} & {40,42,58}.
a(2) = 3360 is the area of {30,224,226}, {48,140,148} & {80,84,116}.
a(3) = 7560 is the area of {45,336,339}, {72,210,222} & {120,126,174}.

Morton Cohen, Charles Lutwidge Dodgson (Lewis Carroll), b. Jan. 27, 1832, d. Jan. 14, 1898, A Brief Biography, Vintage Books, ISBN 978-0-679-74562-4 (26 November 1996).

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A009112, A177063.

lst = {}; m = 2; While[ m < 10^3, n = 1; While[ n < m, If[ IntegerQ@ Sqrt[ m^2 + n^2], a = m*n/2; If[a < 10^6, AppendTo[ lst, a], n = m]]; n++ ]; m++ ]; Union@ Flatten@ Select[ Split@ Sort@ lst, Length@ # == 3 &]

A177021 = { n | A177063(n)=3 }. - M. F. Hasler, Dec 09 2010

Extended and edited by Robert G. Wilson v, Dec 08 2010

a(28)-a(34) from Giovanni Resta, Aug 16 2017

A147778 Positive integers of the form uv(u^2 - v^2) where u, v are coprime integers.

Original entry on oeis.org

6, 24, 30, 60, 84, 120, 180, 210, 240, 330, 336, 504, 546, 630, 720, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2016, 2184, 2310, 2340, 2520, 2574, 2730, 3036, 3360, 3570, 3696, 3900, 3960, 4080, 4290, 4620, 4896, 4914, 5016, 5280, 5544, 5610, 5814

Offset: 1

Max Alekseyev, Nov 12 2008

nonn

Terms with even u or v form A024365. Squarefree terms form A147779.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of: A003273, A009112, A073120.

N:= 10^5:
A:= {}:
for v from 1 to floor((N/2)^(1/3)) do
   for u from v+1 do
      if igcd(u,v) = 1 then
        t:= u*v*(u^2-v^2);
        if t > N then break fi;
        A:= A union {t};
      fi
    od
od:
A;
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 06 2015

A228873 a(n) = F(n) * F(n+1) * F(n+2) * F(n+3), the product of four consecutive Fibonacci numbers, A000045.

Original entry on oeis.org

6, 30, 240, 1560, 10920, 74256, 510510, 3495030, 23965920, 164237040, 1125770256, 7715953440, 52886430870, 362487682830, 2484530961360, 17029219589256, 116720030923320, 800010932051760, 5483356663145790, 37583485265670630, 257601041359736256

Offset: 1

T. D. Noe, Sep 24 2013

nonn

Mohanty and Mohanty prove in Corollary 2.5 that these numbers are Pythagorean. The number a(n) is primitive Pythagorean if F(n) and F(n+1) have opposite parity. Every third number, starting at a(1) = 6, is not primitive Pythagorean.

Since a(n) = F(n+1)*F(n+2)*(F(n+2)^2 - F(n+1)^2), a(n) is in A073120. - Robert Israel, Apr 06 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, The Fibonacci Quarterly, Vol. 28, No. 1 (1990), pp. 31-42.
H.-J. Seiffert, Problem B-1020, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 44, No. 4 (2006), p. 278; Two Fibonacci Identities, Solution to Problem B-1020 by Harris Kwong, ibid., Vol. 45, No. 2 (2007), pp. 182-184.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000045 (Fibonacci numbers), A228874 (similar sequence for Lucas numbers).

Cf. A009112 (Pythagorean numbers), A024365, A073120.

[Fibonacci(n)*Fibonacci(n+1)*Fibonacci(n+2)*Fibonacci(n+3): n in [1..30]]; // Vincenzo Librandi, Oct 04 2013

seq(mul(combinat:-fibonacci(i),i=n..n+3),n=1..30); # Robert Israel, Apr 06 2015

Table[Fibonacci[n] Fibonacci[n+1] Fibonacci[n+2] Fibonacci[n+3], {n, 25}]
CoefficientList[Series[-6/((x - 1) (x^2 + 3 x + 1) (x^2 - 7 x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2013 *)
Times@@@Partition[Fibonacci[Range[30]],4,1] (* Harvey P. Dale, Dec 23 2013 *)
LinearRecurrence[{5,15,-15,-5,1},{6,30,240,1560,10920},30] (* Harvey P. Dale, Jul 24 2021 *)

G.f.: -6*x/((x-1)*(x^2+3*x+1)*(x^2-7*x+1)). - Alois P. Heinz, Oct 02 2013
a(n+5) = 5*a(n+4)+15*a(n+3)-15*a(n+2)-5*a(n+1)+a(n). - Robert Israel, Apr 06 2015
a(n) = 2 * A000217(A059840(n+2)). - Diego Rattaggi, Jan 27 2021
Sum_{n>=1} 1/a(n) = (12-5*sqrt(5))/4. - Diego Rattaggi, Aug 16 2021
a(n) = 3 * Sum_{k=1..n} 2^(n-k)*F(k)^2*F(k+1)*F(k+2) (Seiffert, 2006). - Amiram Eldar, Jan 11 2022

A101450 Areas of (not necessarily primitive) Pythagorean triangles which are palindromes.

Original entry on oeis.org

6, 46464, 48384, 63336, 65856, 66066, 474474, 666666, 4053504, 4228224, 4244424, 4383834, 4405044, 6693966, 42066024, 42455424, 42666624, 43177134, 48455484, 49899894, 60366306, 480333084, 610262016, 656161656, 4242662424, 4365995634, 4409449044, 4475775744

Offset: 1

Zak Seidov, Jan 19 2005

Compare areas of primitive Pythagorean triangles which are palindromes, A101439.

			6 is a member as it is a palindromic number and is area of the Pythagorean triangle (3, 4, 5).
46464 is a member as it is a palindromic number and is area of the Pythagorean triangle (264, 352, 440).
63336 is a member as it is a palindromic number and is area of the Pythagorean triangle with (377, 336, 505).
666666 is a member as it is a palindromic number and is area of the Pythagorean triangle(693, 1924, 2045).

Sean A. Irvine, Table of n, a(n) for n = 1..100

Cf. A101439.

Cf. A002113, A009112.

for(n=1,oo, is_A009112(a=A002113(n)) && print1(a", ")) \\ or better:
A101450_row(n) = select(is_A009112, A002113_row(n)) \\ row n = the n-digit terms - M. F. Hasler, Jun 06 2024

Intersection of A002113 and A009112. - M. F. Hasler, Jun 06 2024

a(16), a(17), a(19)-a(23) from Robert G. Wilson v, Jan 31 2005

Missing terms inserted and entry revised by Sean A. Irvine, May 13 2025

A147779 Squarefree positive integers of the form uv(u^2-v^2) for some integer u,v.

Original entry on oeis.org

6, 30, 210, 330, 546, 2310, 2730, 3570, 4290, 5610, 6090, 6630, 7854, 8970, 9690, 10374, 10626, 13566, 18354, 19866, 22134, 25806, 26970, 39270, 43890, 51330, 51414, 52026, 54834, 56730, 59334, 66990, 68034, 71610, 72930, 74046, 75174

Offset: 1

Max Alekseyev, Nov 12 2008

nonn

Cf. A003273, A006991, A009112, A073120, A147778.

Squarefree terms of A147778. Squarefree terms of A073120. Squarefree terms of A009112.

Terms of A006991 (primitive congruent numbers) corresponding to right triangles with integer sides.

cp's OEIS Frontend

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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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.

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A118903 Areas of rectangles with integer sides and diagonal.

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A177021 Numbers which are the area of exactly three Pythagorean triangles.

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A147778 Positive integers of the form u*v*(u^2 - v^2) where u, v are coprime integers.

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A228873 a(n) = F(n) * F(n+1) * F(n+2) * F(n+3), the product of four consecutive Fibonacci numbers, A000045.

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Magma

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A101450 Areas of (not necessarily primitive) Pythagorean triangles which are palindromes.

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A147778 Positive integers of the form uv(u^2 - v^2) where u, v are coprime integers.

A147779 Squarefree positive integers of the form uv(u^2-v^2) for some integer u,v.