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.

A359805 Irregular triangle T(n, k), n > 0, k = 1..A056137(A009023(n)), read by rows: the n-th row contains the numbers m < A009023(n) such that A009023(n)^2 + m^2 is a square.

Original entry on oeis.org

3, 6, 5, 9, 8, 12, 15, 20, 7, 10, 18, 21, 16, 24, 12, 15, 27, 9, 30, 40, 33, 24, 28, 14, 20, 36, 39, 48, 33, 42, 11, 25, 32, 45, 16, 60, 48, 51, 24, 21, 30, 54, 65, 40, 57, 36, 18, 39, 60, 13, 35, 63, 80, 66, 48, 56, 60, 69, 28, 40, 72, 20, 75, 78, 36, 56, 88, 100
Offset: 1

Views

Author

Rémy Sigrist, Mar 08 2023

Keywords

Comments

See A360020 for the corresponding hypotenuses.

Examples

			Triangle T(n, k) begins:
  n   A009023(n)  n-th row
  --  ----------  ---------
   1           4  3
   2           8  6
   3          12  5, 9
   4          15  8
   5          16  12
   6          20  15
   7          21  20
   8          24  7, 10, 18
   9          28  21
  10          30  16
  11          32  24
  12          35  12
  13          36  15, 27
  14          40  9, 30

Links

Crossrefs

Cf. A009023, A056137, A360020.

Programs

  • PARI
    { for (n=1, 105, for (m=1, n-1, if (issquare(n^2+m^2), print1 (m", ")))) }

A360020 Irregular triangle T(n, k), n > 0, k = 1..A056137(A009023(n)), read by rows: T(n, k) is the square root of A009023(n)^2 + A359805(n, k)^2.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 29, 25, 26, 30, 35, 34, 40, 37, 39, 45, 41, 50, 58, 55, 51, 53, 50, 52, 60, 65, 73, 65, 70, 61, 65, 68, 75, 65, 87, 80, 85, 74, 75, 78, 90, 97, 85, 95, 85, 82, 89, 100, 85, 91, 105, 116, 110, 102, 106, 109, 115, 100, 104, 120, 101
Offset: 1

Views

Author

Rémy Sigrist, Mar 08 2023

Keywords

Examples

			Triangle T(n, k) begins:
  n   A009023(n)  n-th row
  --  ----------  ----------
   1           4  5
   2           8  10
   3          12  13, 15
   4          15  17
   5          16  20
   6          20  25
   7          21  29
   8          24  25, 26, 30
   9          28  35
  10          30  34
  11          32  40
  12          35  37
  13          36  39, 45
  14          40  41, 50

Links

Crossrefs

Cf. A000196, A009023, A056137, A359805.

Programs

  • PARI
    { for (n=1, 99, for (m=1, n-1, if (issquare(n^2 + m^2, &h), print1 (h", ")))) }

Formula

T(n, k) = A000196(A009023(n)^2 + A359805(n, k)^2).

A156681 Consider all Pythagorean triangles A^2 + B^2 = C^2 with A < B < C; sequence gives values of B, sorted to correspond to increasing A (A009004).

Original entry on oeis.org

4, 12, 8, 24, 15, 12, 40, 24, 60, 16, 35, 84, 48, 20, 36, 112, 30, 63, 144, 24, 80, 180, 21, 48, 99, 28, 72, 220, 120, 264, 32, 45, 70, 143, 60, 312, 168, 36, 120, 364, 45, 96, 195, 420, 40, 72, 224, 480, 60, 126, 255, 44, 56, 180, 544, 288, 84, 120, 612, 48, 77, 105
Offset: 1

Views

Author

Ant King, Feb 17 2009

Keywords

Comments

The ordered sequence of B values is A009012(n) (allowing repetitions) and A009023(n) (excluding repetitions).

Examples

			As the first four Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (6,8,10) and (7,24,25), then a(1)=4, a(2)=12, a(3)=8 and a(4)=24.

References

  • Albert H. Beiler, Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
  • W. Sierpinski, Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Links

Crossrefs

Cf. A156682, A009004, A009012, A009023.

Programs

  • Mathematica
    PythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

Formula

a(n) = sqrt(A156682(n)^2 - A009004(n)^2).

A057099 Middle side of a Pythagorean triangle (ordered by the product of the sides).

Original entry on oeis.org

4, 8, 12, 12, 15, 16, 24, 24, 20, 21, 24, 40, 35, 30, 28, 36, 32, 48, 60, 36, 48, 45, 40, 63, 45, 44, 84, 42, 60, 48, 72, 80, 56, 70, 60, 52, 56, 72, 112, 55, 99, 60, 77, 64, 75, 84, 96, 80, 68, 120, 63, 72, 144, 120, 96, 76, 105, 90, 72, 80, 143, 126, 120, 90, 84, 108, 91
Offset: 1

Views

Author

Henry Bottomley, Aug 01 2000

Keywords

Examples

			a(1)=4 since 3*4*5=60 is smallest possible positive product

Crossrefs

Cf. A009012, A009023, A046084, A057096.

Programs

  • Mathematica
    maxShortLeg = 66; terms = 67;
r[a_] := {a, b, c} /. {ToRules[Reduce[a <= b < c && a^2+b^2 == c^2, {b, c}, Integers]]};
abc = r /@ Complement[Range[maxShortLeg], {1, 2, 4}] // Flatten[#, 1]&;
SortBy[abc, Times @@ # &][[;; terms, 2]] (* Jean-François Alcover, Nov 21 2019 *)

Formula

a(n) =A057096(n)/(A057098(n)*A057100(n)) =sqrt(A057100(n)^2-A057098(n)^2)

A227481 Number of squares in row n of the triangle in A069011.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Oct 11 2013

Keywords

Comments

a(A074235(n)) = 1; a(A009023(n)) > 1.

Links

Programs

  • Haskell
    a227481 = sum . map a010052 . a069011_row

Formula

a(n) = sum(A010052(A069011(n,k)): k=0..n).

A056137 Number of ways in which n can be the longer leg (middle side) of an integer-sided right triangle.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 0, 1, 0, 1, 2, 0, 0, 3, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0
Offset: 1

Views

Author

Henry Bottomley, Jun 15 2000

Keywords

Crossrefs

Cf. A009023, A055523.

Formula

a(n) = A046079(n) - A056138(n) = A046081(n) - A046080(n) - A056138(n).

A074235 Numbers that cannot be a long leg of an integer right triangle.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 22, 23, 25, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 81, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 98, 101, 102, 103, 106, 107, 109, 111
Offset: 1

Views

Author

Zak Seidov, Sep 18 2002

Keywords

Comments

The possible values of a long leg of an integer right triangle are in A009012.
A227481(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2013

Examples

			5 is a term because a^2 + 5^2 = c^2 has no solution for a < 5 with integers a, c.
13 is a term because a^2 + 13^2 = c^2 has no solution for a < 13 with integers a, c.

Links

Crossrefs

Cf. A009012.
Cf. A009023 (complement).

Programs

  • Haskell
    a074235 n = a074235_list !! (n-1)
a074235_list = filter ((== 1) . a227481) [1..]
-- Reinhard Zumkeller, Oct 11 2013

A167293 Long legs of Pythagorean triangles that are not divisible by any other long leg of a Pythagorean triangle.

Original entry on oeis.org

4, 15, 21, 35, 55, 77, 91, 99, 117, 143, 153, 171, 187, 209, 221, 247, 253, 299, 323, 325, 377, 391, 403, 425, 437, 475, 493, 527, 551, 575, 589, 621, 629, 667, 697, 703, 713, 725, 775, 779, 783, 817, 837, 851, 899, 925, 943, 957, 989, 999, 1023, 1025, 1073
Offset: 1

Views

Author

Gerald McGarvey, Nov 01 2009

Keywords

Comments

All long legs of Pythagorean triangles (A009023) are multiples of these values, so these values can be thought of as "primes" of the sequence of long legs.

Crossrefs

A009023

Programs

  • PARI
    llp = vector(60); np = 1; llp[np] = 4;
notdiv(k) = for(j=1,np,if(k%llp[j],1,return(0)));return(1);
isLongLeg(n) = local(b);b=0;for(k=1,n-1,if(issquare(k^2+n^2),b=1));return(b);
for(k=4,1175,if(notdiv(k),if(isLongLeg(k),np+=1;llp[np]=k)))
for(n=1,60,print1(llp[n],", "))

Extensions

Comments and PARI program corrected by Gerald McGarvey, Nov 03 2009
Showing 1-8 of 8 results.

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