A060626 Number of right triangles of a given area required to form successively larger squares.

2, 14, 34, 62, 98, 142, 194, 254, 322, 398, 482, 574, 674, 782, 898, 1022, 1154, 1294, 1442, 1598, 1762, 1934, 2114, 2302, 2498, 2702, 2914, 3134, 3362, 3598, 3842, 4094, 4354, 4622, 4898, 5182, 5474, 5774, 6082, 6398, 6722, 7054, 7394, 7742, 8098, 8462, 8834, 9214

Offset: 0

Author

Jason Earls, Apr 13 2001

Keywords

• nonn
• easy

Comments

a(n) = number of row of Pascal's triangle in which three consecutive entries appear in the ratio n : n+1 : n+2 (valid for n = 0 if you consider a position of -1 to have value 0). E.g., entries in the ratio 1:2:3 appear in row 14 (1001, 2002, 3003); entries in the ratio 2:3:4 appear in row 34 (927983760, 1391975640, 1855967520); and so on. (The position within the row is given by A091823). - Howard A. Landman, Mar 08 2004

a(n)*(a(n)+1) is an oblong number (Cf. A002378) with the property that the product with the oblong numbers n*(n+1) or (n+1)*(n+2) both are again oblong numbers. Example: For n=3 we have (62*63)*(3*4) = 216*217 and (62*63)*(4*5) = 279*280. - Herbert Kociemba, Apr 13 2008

For n > 0, Hermite polynomial H_2(n) = 4*n^2 - 2. - Vincenzo Librandi, Aug 07 2010

The identity (4*n^2-2)^2 - (n^2-1)*(4*n)^2 = 4 can be written as a(n+1)^2 - A132411(n+2)*A008586(n+2)^2 = 4. - Vincenzo Librandi, Jun 16 2014

Equivalently: positive integers k congruent to 2 mod 4 (A016825) such that k$ / (k/2+1)! is a square when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692, A349496 and A349766 for further information). Integers k multiple of 4 such that that k$ / (k/2+1)! is a square are in A035008. - Bernard Schott, Dec 05 2021

Links

• Harry J. Smith, Table of n, a(n) for n=0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000178, A002378, A007318, A008586, A016777, A035008, A056220, A091823.

Cf. A132411, A348692.

Twice Column 2 of array A188644.

Subsequence of A016825.

Equals disjoint union of A349496 and A349766.

Programs

Maple

for n from 0 to 80 do printf(`%d,`,4*n^2+8*n+2) od:

Mathematica

Table[4*n*(n + 2) + 2, {n, 0, 100}] (* Paolo Xausa, Aug 08 2024 *)

PARI

a(n) = { 4*n^2 + 8*n + 2 } \\ Harry J. Smith, Jul 08 2009

Formula

a(n) = 4*n^2 + 8*n + 2.

a(n) = 8*n + a(n-1) + 4 with n > 0, a(0)=2. - Vincenzo Librandi, Aug 07 2010

G.f.: 2*(1 + 4*x - x^2)/(1-x)^3. - Colin Barker, Jun 28 2012

a(n) = 4*(n+1)^2 - 2 = 2*A056220(n+1). - Bruce J. Nicholson, Aug 31 2017

a(n) + a(n-1) + (n-1)^2 = (3*n + 1)^2 = A016777(n)^2. - Ezhilarasu Velayutham, May 23 2019

From Elmo R. Oliveira, Oct 31 2024: (Start)

E.g.f.: 2*exp(x)*(1 + 6*x + 2*x^2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Extensions

More terms from James Sellers, Apr 14 2001