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

1,2

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

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).

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

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 *)

Prog

(Small Basic)
For n = 0 to 50
a = 0
For n1 = 1 to 2*n*(n+1)+1
  a = a + n1
EndFor
TextWindow.Write(a+", ")
EndFor
(PARI) Vec(-x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5 + O(x^100)) \\ Colin Barker, Jan 17 2015

Crossrefs

Cf. A000217, A001844, A002061, A173962.

Cf. A002378, A008514, A384288, A384566.

Sequence in context: A048630 A326148 A035163 * A020242 A020255 A367243

Adjacent sequences: A237513 A237514 A237515 * A237517 A237518 A237519

Keyword

nonn,easy

Author

Kival Ngaokrajang, Feb 08 2014

Status

approved