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

Author

Kival Ngaokrajang, Feb 08 2014

Keywords

• nonn
• easy
• changed

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).

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