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

Offset

0,1

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) = 4n^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).

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

Maple

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

Prog

(PARI) { for (n=0, 1000, write("b060626.txt", n, " ", 4*n^2 + 8*n + 2); ) } \\ Harry J. Smith, Jul 08 2009

Crossrefs

Cf. A007318, A091823, A002378, A348692.

Twice Column 2 of array A188644.

Subsequence of A016825.

Equals disjoint union of A349496 and A349766.

Sequence in context: A050591 A073535 A225292 * A096311 A034842 A324043

Adjacent sequences: A060623 A060624 A060625 * A060627 A060628 A060629

Keyword

nonn,easy

Author

Jason Earls, Apr 13 2001

Extensions

More terms from James A. Sellers, Apr 14 2001

Status

approved