A349766 Numbers of the form 2*t^2 - 4 when t > 1 is a term in A001541.

14, 574, 19598, 665854, 22619534, 768398398, 26102926094, 886731088894, 30122754096398, 1023286908188734, 34761632124320654, 1180872205318713598, 40114893348711941774, 1362725501650887306814, 46292552162781456489998, 1572584048032918633353214, 53421565080956452077519374

Offset

1,1

Comments

Equivalently: integers k such that k$ / (k/2+1)! and k$ / (k/2+2)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information).

The 3 subsequences of A349081 are A035008, A139098 and this one.

Links

Table of n, a(n) for n=1..17.

Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009, No. 35, Vol. 2, pages 346-352.

Index to sequences related to Olympiads.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Formula

a(n) = 2*(cosh(2*n*arcsinh(1)))^2 - 4.

a(n) = 16*A001110(n) - 2. - Hugo Pfoertner, Dec 04 2021

Sum_{n>=1} 1/a(n) = (1 - 1/sqrt(2))/4. - Amiram Eldar, Dec 11 2025

From Elmo R. Oliveira, Apr 12 2026: (Start)

a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) for n >= 4.

G.f.: 2*x*(7 + 42*x - x^2)/((1 - x)*(1 - 34*x + x^2)). (End)

E.g.f.: 2 - 3*exp(x) + exp(17*x)*cosh(12*sqrt(2)*x). - Stefano Spezia, Apr 12 2026

Example

A001541(1) = 3, then for t = 3, 2*t^2-4 = 14; also for k = 14, 14$ / 8! = 1309248519599593818685440000000^2 and 14$ / 9! = 436416173199864606228480000000^2. Hence, 14 is a term.

Maple

with(orthopoly):
sequence = (2*T(n, 3)^2-4, n=1..20);

Mathematica

(2*#^2 - 4) & /@ LinearRecurrence[{6, -1}, {3, 17}, 17] (* Amiram Eldar, Dec 04 2021 *)
LinearRecurrence[{35, -35, 1}, {14, 574, 19598}, 17] (* Ray Chandler, Mar 01 2024 *)

Prog

(PARI) a(n) = my(t=subst(polchebyshev(n), 'x, 3)); 2*t^2-4; \\ Michel Marcus, Dec 04 2021

Crossrefs

Subsequence of A060626.

Cf. A000178, A001110, A001541, A035008, A139098, A348692, A349079, A349080, A349081.

Sequence in context: A002010 A166185 A361695 * A269478 A269645 A382847

Adjacent sequences: A349763 A349764 A349765 * A349767 A349768 A349769

Keyword

nonn,easy

Author

Bernard Schott, Dec 04 2021

Status

approved