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.

User: Miguel-Ángel Pérez García-Ortega

Miguel-Ángel Pérez García-Ortega's wiki page.

Miguel-Ángel Pérez García-Ortega has authored 66 sequences. Here are the ten most recent ones:

A386291 Long leg 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

4, 4, 12, 60, 420, 3612, 35112, 368940, 4092660, 47287812, 564244824, 6911705164, 86538400312, 1103802305800, 14305263976080, 187980058537740, 2500329726373140, 33615543407577780, 456277456430657400, 6246438368992478580, 86175353789650193640
Offset: 0

Keywords

Crossrefs

Formula

a(n) = 2 * A000108(n) * (A000108(n) + 1).

A386201 Lengths of the long leg in the unique primitive Pythagorean triple whose inradius is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

12, 4, 24, 40, 112, 264, 684, 1740, 4512, 11704, 30504, 79600, 208012, 543924, 1422984, 3723720, 9746112, 25511224, 66782124, 174826300, 457682512, 1198198104, 3136874424, 8212364640, 21500121612, 56287841764, 147363147384, 385801185640, 1010039738512
Offset: 0

Keywords

Crossrefs

Cf. A000032, A380821 (short legs).

Formula

a(n) = 2 * A000032(n) * (A000032(n) + 1).

A385973 The hypotenuses of the triangles defined in A365577.

Original entry on oeis.org

5, 25, 481, 130561, 8589803521, 36893488138829168641, 680564733841876926889855726716117319681, 231584178474632390847141970017375815705859404597439251151988418800962722856961
Offset: 1

Keywords

Crossrefs

A385972 The long legs of the triangles defined in A365577.

Original entry on oeis.org

4, 24, 480, 130560, 8589803520, 36893488138829168640, 680564733841876926889855726716117319680, 231584178474632390847141970017375815705859404597439251151988418800962722856960
Offset: 1

Keywords

Crossrefs

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

Keywords

Examples

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

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).

A385187 Area of 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

1, 6, 330, 3036, 14820, 51330, 142926, 341880, 731016, 1433790, 2625810, 4547796, 7519980, 11957946, 18389910, 27475440, 40025616, 57024630, 79652826, 109311180, 147647220, 196582386, 258340830, 335479656, 430920600, 547983150, 690419106, 862448580, 1068797436, 1314736170, 1606120230
Offset: 1

Keywords

Comments

a(n) is a multiple of 6 for all n.

Examples

			For n=2, the short leg is A385022(2,1) = 11 and the long leg is A385022(2,2) = 60  so the area is then a(2) = (11 * 60)/2 = 330.
		

Crossrefs

Programs

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

Formula

a(n) = (A385022(n,1) * A385022(n,2))/2.
a(n) = A002378(n)*(A002378(n) - 1)*(2*A002378(n) - 1).

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

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.
		

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).

A384566 Area of 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

0, 30, 546, 3900, 17220, 56730, 153510, 360696, 762120, 1482390, 2698410, 4652340, 7665996, 12156690, 18654510, 27821040, 40469520, 57586446, 80354610, 110177580, 148705620, 197863050, 259877046, 337307880, 433080600, 550518150, 693375930, 865877796, 1072753500, 1319277570, 1611309630
Offset: 0

Keywords

Comments

a(n) is multiple of 6 for all n.

Examples

			For n=1, the short leg is A384288(1,1) = 5 and the long leg is A384288(1,2) = 12 so the area is then a(1) = (5 * 12 )/2 = 30.
		

Crossrefs

Programs

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

Formula

a(n) = (A384288(n,1) * A384288(n,2))/2.
a(n) = A002378(n)*(A002378(n) + 1)*(2*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

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;
  ...
		

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).

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

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.
		

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.