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Number of right triangles of a given area required to form successively larger squares.
14

%I #72 Oct 31 2024 15:10:30

%S 2,14,34,62,98,142,194,254,322,398,482,574,674,782,898,1022,1154,1294,

%T 1442,1598,1762,1934,2114,2302,2498,2702,2914,3134,3362,3598,3842,

%U 4094,4354,4622,4898,5182,5474,5774,6082,6398,6722,7054,7394,7742,8098,8462,8834,9214

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

%C 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

%C 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

%C For n > 0, Hermite polynomial H_2(n) = 4*n^2 - 2. - _Vincenzo Librandi_, Aug 07 2010

%C 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

%C 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

%H Harry J. Smith, <a href="/A060626/b060626.txt">Table of n, a(n) for n=0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F a(n) = 8*n + a(n-1) + 4 with n > 0, a(0)=2. - _Vincenzo Librandi_, Aug 07 2010

%F G.f.: 2*(1 + 4*x - x^2)/(1-x)^3. - _Colin Barker_, Jun 28 2012

%F a(n) = 4*(n+1)^2 - 2 = 2*A056220(n+1). - _Bruce J. Nicholson_, Aug 31 2017

%F a(n) + a(n-1) + (n-1)^2 = (3*n + 1)^2 = A016777(n)^2. - _Ezhilarasu Velayutham_, May 23 2019

%F From _Elmo R. Oliveira_, Oct 31 2024: (Start)

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

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

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

%t Table[4*n*(n + 2) + 2, {n, 0, 100}] (* _Paolo Xausa_, Aug 08 2024 *)

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

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

%Y Cf. A132411, A348692.

%Y Twice Column 2 of array A188644.

%Y Subsequence of A016825.

%Y Equals disjoint union of A349496 and A349766.

%K nonn,easy,changed

%O 0,1

%A _Jason Earls_, Apr 13 2001

%E More terms from _James A. Sellers_, Apr 14 2001