A349766 -id:A349766 - cp's OEIS frontend

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

Jason Earls, Apr 13 2001

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

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

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.

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

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

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

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)

More terms from James Sellers, Apr 14 2001

A349081 Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!2!...*k! is the superfactorial of k.

Original entry on oeis.org

8, 14, 16, 32, 48, 72, 96, 128, 160, 200, 240, 288, 336, 392, 448, 512, 574, 576, 648, 720, 800, 880, 968, 1056, 1152, 1248, 1352, 1456, 1568, 1680, 1800, 1920, 2048, 2176, 2312, 2448, 2592, 2736, 2888, 3040, 3200, 3360, 3528, 3696, 3872, 4048, 4232, 4416, 4608, 4800, 5000

Offset: 1

Bernard Schott, Dec 01 2021

nonn

This sequence is the union of three infinite and disjoint subsequences:
-> Numbers k = 8t^2 > 0 (A139098); for these numbers, m_1 = k/2 - 1 = 4t^2-1 < m_2 = k/2 = 4t^2  (see example for k = 8).
-> Numbers k = 8t*(t+1) (A035008); for these numbers, m_1 = k/2 = 4t(t+1) < m_2 =  k/2 + 1 = (2t+1)^2 (see example for k = 16).
-> Even numbers of the form 2t^2-4, t>1 in A001541 (A349766); for these numbers, m_1 = k/2 + 1 = t^2 - 1 <  m_2 = k/2 + 2 = t^2 (see example for k = 14).
See A348692 for further information.

			For k = 8, 8$ / 2! is not a square, but m_1 = 3 because 8$ / 3! = 29030400^2 and m_2 = 4 because 8$ / 4! = 14515200^2.
For k = 14, m_1 = 8 because 14$ / 8! = 1309248519599593818685440000000^2 and m_2 = 9 because 14$ / 9! = 436416173199864606228480000000^2.
For k = 16, m_1 = 8 because 16$ / 8! = 6848282921689337839624757371207680000000000^2 and m_2 = 9 because 16$ / 9! = 2282760973896445946541585790402560000000000^2.

Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009, pages 346-352.

Cf. A001541, A035008, A139098, A348692, A349080.

Subsequence of A349079.

Do[j=0;l=1;g=BarnesG[k+2];While[j<2&&l<=k,If[IntegerQ@Sqrt[g/l!],j++];l++];If[j==2,Print@k],{k,5000}] (* Giorgos Kalogeropoulos, Dec 02 2021 *)

sf(n) = prod(k=2, n, k!); \\ A000178
isok(m) = if (!(m%2), my(s=sf(m)); #select(issquare, vector(4, k, s/(m/2+k-2)!), 1) == 2); \\ Michel Marcus, Dec 04 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A349081 Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!2!...*k! is the superfactorial of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A349081 Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!*2!*...*k! is the superfactorial of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A349081 Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!2!...*k! is the superfactorial of k.