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) = 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).
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)
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 *)
PROG
(PARI) { for (n=0, 1000, write("b060626.txt", n, " ", 4*n^2 + 8*n + 2); ) } \\ Harry J. Smith, Jul 08 2009
KEYWORD
nonn,easy
AUTHOR
Jason Earls, Apr 13 2001
EXTENSIONS
More terms from James A. Sellers, Apr 14 2001
STATUS
approved