A384566 -id:A384566 - cp's OEIS frontend

A237516 Pyramidal centered square numbers.

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

Kival Ngaokrajang, Feb 08 2014

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

Colin Barker, Table of n, a(n) for n = 1..1000
José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.
Kival Ngaokrajang, Illustration for n = 1..6.
Eric Weisstein's World of Mathematics, Diamond.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A001844, A002061, A173962.

Cf. A002378, A008514, A384288, A384566.

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 *)

Vec(-x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5 + O(x^100)) \\ Colin Barker, Jan 17 2015

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

Miguel-Ángel Pérez García-Ortega, May 31 2025

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

José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

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

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

cp's OEIS Frontend

A237516 Pyramidal centered square numbers.

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