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.

A336535 a(n) = (m(n)^2 + 3)*(m(n)^2 + 7)/32, where m(n) = 2*n - 1.

Original entry on oeis.org

1, 6, 28, 91, 231, 496, 946, 1653, 2701, 4186, 6216, 8911, 12403, 16836, 22366, 29161, 37401, 47278, 58996, 72771, 88831, 107416, 128778, 153181, 180901, 212226, 247456, 286903, 330891, 379756, 433846, 493521, 559153, 631126, 709836, 795691, 889111, 990528, 1100386, 1219141, 1347261, 1485226, 1633528, 1792671
Offset: 1

Views

Author

Jeff Brown, Jul 24 2020

Keywords

Comments

For m(n) = 3,5,11, and 181, the perfect numbers (A000396), 6, 28, 496, and 33550336 are produced, respectively. 3,5,11, and 181 are the numbers m(n) such that (m(n)^2+7) is a power of 2. cf A038198.
The unique primitive Pythagorean triple whose inradius T(n) and its long leg and hypotenuse are consecutive natural numbers is (2*T(n)+1, 2*T(n)*(T(n)+1), 2*T(n)*(T(n)+1)+1) and its semiperimeter is (T(n)+1)*(2*T(n)+1) where T(n) = A000217(n). - Miguel-Ángel Pérez García-Ortega, May 16 2025

Examples

			m(2) = 2*2-1 = 3 and (3^2+3)*(3^2+7)/32 = 6, so 6 is in the sequence.

References

  • David M. Burton, Elementary Number Theory, McGraw-Hill (2011), 25.

Links

Crossrefs

Cf. A000396, A038198.
Cf. A002061, A014206.
Cf. A000217, A058919, A383833, A383834.

Programs

Formula

From Stefano Spezia, Jul 25 2020: (Start)
O.g.f.: x*(1 + x + 8*x^2 + x^3 + x^4)/(1 - x)^5.
a(n) = (1 - n + n^2)*(2 - n + n^2)/2.
a(n) = A002061(n)*A014206(n-1)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)
a(n) = (A000217(n-1)+1)*(2*A000217(n-1)+1). - Miguel-Ángel Pérez García-Ortega, May 16 2025

A383833 Area of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

0, 6, 84, 546, 2310, 7440, 19866, 46284, 97236, 188370, 341880, 588126, 967434, 1532076, 2348430, 3499320, 5086536, 7233534, 10088316, 13826490, 18654510, 24813096, 32580834, 42277956, 54270300, 68973450, 86857056, 108449334, 134341746, 165193860, 201738390
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, May 11 2025

Keywords

Examples

			For n=1, the short leg is A002061(1) = 3 and the long leg is A212135(2) = 4 so the area is then a(1) = (3 * 4 )/2 = 6.

References

  • Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Links

Crossrefs

Cf. A000217, A002061, A058919, A383834, A336535.

Programs

  • Mathematica
    a=Table[(n(n+1))/2,{n,0,30}];Apply[Join,Map[{#(#+1)(2#+1)}&,a]]

Formula

a(n) = A000217(n) * (A000217(n) + 1) * (2*A000217(n) + 1).

A237516 Pyramidal centered square numbers.

Original entry on oeis.org

1, 15, 91, 325, 861, 1891, 3655, 6441, 10585, 16471, 24531, 35245, 49141, 66795, 88831, 115921, 148785, 188191, 234955, 289941, 354061, 428275, 513591, 611065, 721801, 846951, 987715, 1145341, 1321125, 1516411, 1732591, 1971105, 2233441, 2521135, 2835771, 3178981, 3552445, 3957891, 4397095, 4871881
Offset: 1

Views

Author

Kival Ngaokrajang, Feb 08 2014

Keywords

Comments

a(n) is sum of natural numbers filled in order-n diamond.
First differences give A173962.
The unique primitive Pythagorean triple whose inradius T(n) and its long leg and hypotenuse are consecutive natural numbers is (2*T(n)+1, 2*T(n)*(T(n)+1), 2*T(n)*(T(n)+1)+1) and its semiperimeter is (T(n)+1)*(2*T(n)+1) where T(n) = A002378(n). - Miguel-Ángel Pérez García-Ortega, Jun 05 2025

Links

Crossrefs

Cf. A000217, A001844, A002061, A173962.
Cf. A002378, A008514, A384288, A384566.

Programs

  • Mathematica
    Table[Sum[i, {i, 2n(n + 1) + 1}], {n, 0, 29}] (* Alonso del Arte, Feb 09 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{1,15,91,325,861},60] (* Harvey P. Dale, Apr 21 2018 *)
a=Table[(n(n+1)),{n,0,29}];Apply[Join,Map[{(#+1)(2#+1)}&,a]] (* Miguel-Ángel Pérez García-Ortega, Jun 05 2025 *)
  • PARI
    Vec(-x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5 + O(x^100)) \\ Colin Barker, Jan 17 2015

Formula

a(n) = 2*n^4 - 4*n^3 + 5*n^2 - 3*n + 1.
a(n) = Sum_{i = 1..(2*n*(n + 1) + 1)} i.
From Colin Barker, Jan 17 2015: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: -x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5. (End)
a(n) = A000217(A001844(n-1)). - Ivan N. Ianakiev, Jun 14 2015
a(n) = A002061(n)*A001844(n-1). - Bruce J. Nicholson, May 14 2017
a(n) = (A002378(n)+1)*(2*A002378(n)+1). - Miguel-Ángel Pérez García-Ortega, Jun 05 2025
E.g.f.: -1 + exp(x)*(1 + 7*x^2 + 8*x^3 + 2*x^4). - Elmo R. Oliveira, Aug 22 2025

A384288 Length of the long leg in the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

12, 84, 312, 840, 1860, 3612, 6384, 10512, 16380, 24420, 35112, 48984, 66612, 88620, 115680, 148512, 187884, 234612, 289560, 353640, 427812, 513084, 610512, 721200, 846300, 987012, 1144584, 1320312, 1515540, 1731660, 1970112, 2232384, 2520012, 2834580
Offset: 1

Views

Author

Miguel-Ángel Pérez García-Ortega, May 31 2025

Keywords

Examples

			Triangles begin:
  n=1:      5,   12,   13;
  n=2:     13,   84,   85;
  n=3:     25,  312,  313;
  ...
This sequence gives the middle column.

Links

Crossrefs

Cf. A002378 (inradius), A001844 (short leg), A008514 (sum of the legs), A237516 (semiperimeter), A384566 (area).

Formula

a(n) = 2 * A002378(n) * (A002378(n) + 1).

A384329 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000217(n) and its long leg and hypotenuse are consecutive natural numbers, n >= 0.

Original entry on oeis.org

-1, 0, 1, 1, 0, 1, 5, 12, 13, 11, 60, 61, 19, 180, 181, 29, 420, 421, 41, 840, 841, 55, 1512, 1513, 71, 2520, 2521, 89, 3960, 3961, 109, 5940, 5941, 131, 8580, 8581, 155, 12012, 12013, 181, 16380, 16381, 209, 21840, 21841, 239, 28560, 28561, 271, 36720, 36721, 305, 46512, 46513, 341, 58140, 58141
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, May 26 2025

Keywords

Comments

Row n = 0 and n = 1 are included by convention and correspond to the Pythagorean triples (-1)^2 + 0^2 = 1^2 and 1^2 + 0^2 = 1^2.

Examples

			  n=0:     -1,     0,     1;
  n=1:      1,     0,     1;
  n=2:      5,    12,    13;
  n=3:     11,    60,    61;
  ...

Links

Crossrefs

Cf. A000217, A165900 (short leg), A062392 (semiperimeter), A384498 (sum of the legs).

Programs

  • Mathematica
    a=Table[(n(n+1))/2,{n,0,18}];Apply[Join,Map[{2#-1,2#^2-2#,2#^2-2#+1}&,a]]

Formula

row(n) = (2*T(n) - 1, 2*T(n)*(T(n) - 1), 2*T(n)*(T(n) - 1) + 1) where T(n) = A000217(n).

A385022 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A002378(n) and its long leg and hypotenuse are consecutive natural numbers.

Original entry on oeis.org

3, 4, 5, 11, 60, 61, 23, 264, 265, 39, 760, 761, 59, 1740, 1741, 83, 3444, 3445, 111, 6160, 6161, 143, 10224, 10225, 179, 16020, 16021, 219, 23980, 23981, 263, 34584, 34585, 311, 48360, 48361, 363, 65884, 65885, 419, 87780, 87781, 479, 114720, 114721, 543, 147424, 147425
Offset: 1

Views

Author

Miguel-Ángel Pérez García-Ortega, Jun 15 2025

Keywords

Examples

			  n=1:      3,     4,     5;
  n=2:     11,    60,    61;
  n=3:     23,   264,   265;
  ...

Links

Crossrefs

Cf. A002378, A142463 (short leg), A385187 (area).

Programs

  • Mathematica
    a=Table[(n(n+1)),{n,1,16}];Apply[Join,Map[{2#-1,2#^2-2#,2#^2-2#+1}&,a]]

Formula

row(n) = (2*T(n) - 1, 2*T(n)*(T(n) - 1), 2*T(n)*(T(n) - 1) + 1) where T(n) = A002378(n).

A383957 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

7, 7, 17, 71, 449, 3697, 35377, 369799, 4095521, 47297537, 564278417, 6911822737, 86538816337, 1103803791601, 14305269324961, 187980077927431, 2500329797088481, 33615543666867361, 456277457385934801, 6246438372527004961, 86175353802778434481, 1197196443885744428881, 16738118900659230353761
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, May 16 2025

Keywords

Examples

			For n=1, the short leg is A383251(1,1) = 3 and the long leg is A383251(1,2) = 4 so the sum of the legs is then a(1) = 3 + 4 = 7.

Links

Crossrefs

Cf. A000108, A383251, A381483, A382114, A006007, A058919, A336535.

Programs

  • Mathematica
    a=Table[(2n)!/(n!(n+1)!),{n,0,23}];Apply[Join,Map[{2#^2+4#+1}&,a]]

Formula

a(n) = A383251(n,1) + A383251(n,2).
a(n) = 2*(A000108(n))^2 + 4*A000108(n) + 1.

A383958 Sum of the legs of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000108(n) and its long leg and hypotenuse are consecutive natural numbers.

Original entry on oeis.org

1, 1, 7, 49, 391, 3527, 34847, 368081, 4089799, 47278087, 564211231, 6911587591, 86537984287, 1103800819999, 14305258627199, 187980039148049, 2500329655657799, 33615543148288199, 456277455475379999, 6246438365457952199, 86175353776521952799, 1197196443787879360799, 16738118900293300099199
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, May 16 2025

Keywords

Examples

			For n=3, the short leg is A383615(3,1) = 3 and the long leg is A383615(3,2) = 4 so the sum of the legs is then a(3) = 3 + 4 = 7.

Links

Crossrefs

Cf. A000108, A383615, A131428, A383616, A381846, A001246.

Programs

  • Mathematica
    a=Table[(2n)!/(n!(n+1)!),{n,0,23}];Apply[Join,Map[{2#^2-1}&,a]]

Formula

a(n) = A383615(n,1) + A383615(n,2).
a(n) = 2*A000108(n)^2 - 1.
a(n) = 2*A001246(n) - 1.
Showing 1-8 of 8 results.

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