cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A055527 Shortest other leg of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

4, 3, 12, 8, 24, 6, 12, 24, 60, 5, 84, 48, 8, 12, 144, 24, 180, 15, 20, 120, 264, 7, 60, 168, 36, 21, 420, 16, 480, 24, 44, 288, 12, 15, 684, 360, 52, 9, 840, 40, 924, 33, 24, 528, 1104, 14, 168, 120, 68, 39, 1404, 72, 48, 33, 76, 840, 1740, 11, 1860, 960, 16, 48, 72
Offset: 3

Views

Author

Henry Bottomley, May 22 2000

Keywords

Comments

From Alex Ratushnyak, Mar 30 2014: (Start)
Least positive k such that n^2 + k^2 is a square.
For odd n, a(n) <= 4*triangular((n-1)/2), because n^2 + (4 * triangular((n-1)/2))^2 = ((n^2+1)/2) ^ 2, which is a perfect square since n is odd.
For n = 4*k+2, a(n) <= 8*triangular(k), because (4k+2)^2 + (4*k*(k+1))^2 = (4*k^2 + 4*k + 2)^2. (End)

Links

Crossrefs

Cf. A000290, A000217, A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055525, A055526.
See A082183 for a similar sequence involving triangular numbers.

Programs

  • Mathematica
    Table[k = 1; While[! IntegerQ[Sqrt[n^2 + k^2]], k++]; k, {n, 3, 100}] (* T. D. Noe, Apr 02 2014 *)

Formula

a(n) = sqrt(A055526(n)^2-n^2) = 2*A054436/n.

A055523 Longest other leg of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

4, 3, 12, 8, 24, 15, 40, 24, 60, 35, 84, 48, 112, 63, 144, 80, 180, 99, 220, 120, 264, 143, 312, 168, 364, 195, 420, 224, 480, 255, 544, 288, 612, 323, 684, 360, 760, 399, 840, 440, 924, 483, 1012, 528, 1104, 575, 1200, 624, 1300, 675, 1404, 728, 1512, 783
Offset: 3

Views

Author

Henry Bottomley, May 22 2000

Keywords

Links

Crossrefs

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055524, A055525, A055526, A055527.

Programs

  • Maple
    seq(`if`(n::even, (n/2-1)*(n/2+1), (n-1)*(n+1)/2), n=3..100); # Robert Israel, Dec 16 2014
  • Mathematica
    a[n_Integer/;n>=3]:=(3 (n^2-2)+(-1)^(n+1) (n^2+2))/8 (* Todd Silvestri, Dec 16 2014 *)
  • PARI
    Vec(x^3*(x^3-3*x-4)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 15 2014

Formula

a(n) = 2*A055522(n)/n = sqrt(A055524(n)^2-n^2).
a(2k) = (k-1)*(k+1), a(2k+1) = 2k*(k+1).
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: x^3*(x^3-3*x-4) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 15 2014
a(n) = (3*(n^2-2)+(-1)^(n+1)*(n^2+2))/8. - Todd Silvestri, Dec 16 2014
E.g.f.: 1 + (3*x^2/8 + 3*x/8 - 3/4)*exp(x) + (-x^2/8 + x/8 - 1/4)*exp(-x). - Robert Israel, Dec 16 2014

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).

Original entry on oeis.org

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

Views

Author

Henry Bottomley, May 22 2000

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

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

Views

Author

Henry Bottomley, May 22 2000

Keywords

Crossrefs

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

Programs

  • Maple
    seq(piecewise(n mod 2 = 0,n*(n^2-4)/8,n*(n^2-1)/4),n=3..60); # C. Ronaldo
  • Mathematica
    Table[n*(3*(n^2 - 2) - (n^2 + 2)*(-1)^n)/16, {n, 3, 50}] (* Wesley Ivan Hurt, Apr 27 2017 *)

Formula

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

Views

Author

Henry Bottomley, May 22 2000

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Henry Bottomley, May 22 2000

Keywords

Crossrefs

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

Views

Author

Henry Bottomley, May 22 2000

Keywords

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 3, 1, 1, 4, 2, 1, 3, 3, 1, 3, 1, 3, 4, 1, 3, 5, 1, 1, 4, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 4, 3, 1, 3, 3, 5, 4, 1, 1, 9, 1, 1, 5, 4, 4, 4, 1, 3, 4, 3, 1
Offset: 1

Views

Author

Henry Bottomley, Jun 15 2000

Keywords

Crossrefs

Cf. A009004, A055525, A055527.

Programs

  • PARI
    a(n)=my(b);sum(c=n+2,n^2\2+1,issquare(c^2-n^2,&b) && nCharles R Greathouse IV, Jul 07 2013

Formula

a(n) = A046079(n) - A056137(n) = A046081(n) - A046080(n) - A056137(n).
Showing 1-8 of 8 results.

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