A108924 -id:A108924 - cp's OEIS frontend

A140966 a(n) = (5 + (-2)^n)/3.

2, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881, 1431655767, -2863311529, 5726623063

Offset: 0

Paul Curtz, Jul 27 2008

Inverse binomial transform of A048573.
This is an example of the case k=-1 of sequences with recurrences a(n) = k*a(n-1) + (k+3)*a(n-2) - (2*k+2)*a(n-3).
The case k=1 is covered, for example, by A097163, A135520, A136326, A136336, or A137208.
Sequences with k=2 are A094554 and A094555.
Sequences with k=3 are A084175, A108924, and A139818.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,2).

Cf. A048573.
Cf. A097163, A135520, A136326, A136336, A137208.
Cf. A094554, A094555.
Cf. A084175, A108924, A139818.
Cf. A000079, A001045, A078008, A083582, A083584, A163834.

[( 5+(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

(5+(-2)^Range[0,30])/3 (* or *) LinearRecurrence[{-1,2},{2,1},40] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=(5+(-2)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = -a(n-1) + 2*a(n-2).
G.f.: (2+3*x)/((1-x)*(1+2*x)).
a(n+1) - a(n) = (-1)^(n+1)*A000079(n).
a(n+3) = (-1)^n*A083582(n).
a(n+1) - 2*a(n) = -a(n+2).
a(n+1) - 3*a(n) = 5*(-1)^(n+1)*A078008(n) = (-1)^(n+1)*A001045(n-1).
a(2n+3) = -A083584(n), a(2n) = A163834(n). - Philippe Deléham, Feb 24 2014
E.g.f.: (5*exp(x) + exp(-2*x))/3. - Stefano Spezia, Jul 27 2024

Definition simplified by R. J. Mathar, Sep 11 2009

A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

Original entry on oeis.org

1, 10, 220, 3080, 52976, 818720, 13333440, 211474560, 3398520576, 54257082880, 869067996160, 13897453373440, 222420341682176, 3558236809994240, 56935698394234880, 910939899548958720, 14575288593717067776, 233202615903456460800

Offset: 0

Ralf Steiner, May 16 2020

Matrix {{2, 0}, {1, -1}} is [g_{-2}] given by Firstov in eq. (24).
These primitive Pythagorean triples are also given by Lee Price as (M_2)^n (3,4,5)^T (T for transposed), with M_2 = {{2, 1, 1}, {2, -2, 2}, {2, -1, 3}}.
For a primitive Pythagorean triangle (x, y, z) = (u^2-v^2, 2*u*v, u^2+v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction. Here:
x(n) = A084175(n+2).
y(n) = 4*(A084175(n+1) - A084175(n)) = A054881(n+2).
     = 2*A192382(n+1) = 4*A003683(n+1).
z(n) = A084175(n+2) + 2*A084175(n+1) - 4*A084175(n).
     = A108924(n+2)/2 = A084175(n+2) + 2*A139818(n+1).
     = A000302(n+1) + A139818(n+1).
u(n) = A000079(n+1) = 2^(n+1).
v(n) = A001045(n+1) = (2^(n+1) + (-1)^n)/3.
For the area A(n): Limit_{n -> oo} (3^3/(2^(4*n+7)))*A(n) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

			a(0) = 3*4/12 = 1 for the triangle (3, 4, 5).

G. C. Greubel, Table of n, a(n) for n = 0..825
V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
H. Lee Price, The Pythagorean Tree: A New Species, arXiv:0809.4324 [math.HO], 2008-2011
R. Steiner, Spezielle Folge primitiver pythagoräischer Dreiecke, researchgate.net, 2020
Index entries for linear recurrences with constant coefficients, signature (10,120,-320,-1024).

Cf. A000079, A000302, A001045, A003683, A054881, A084175, A108924, A139818, A192382.

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81: n in [0..40]]; // G. C. Greubel, Feb 18 2023

Table[(2^(2*n+1)*(2^(2*n+5) -3) + (-2)^n*(3*2^(2*n+3) -1))/3^4, {n,0,40}]

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81 for n in range(41)] # G. C. Greubel, Feb 18 2023

a(n) = ( 2^(4*n+6) - 3*2^(2*n+1) - 3*(-2)^(3*n+3) - (-2)^n )/3^4.
G.f.: 1 / ((1 + 2*x)*(1 - 4*x)*(1 + 8*x)*(1 - 16*x)). - Colin Barker, Jun 11 2020
E.g.f.: (1/81)*(24*exp(-8*x) - exp(-2*x) - 6*exp(4*x) + 64*exp(16*x)). - G. C. Greubel, Feb 18 2023

cp's OEIS Frontend

A140966 a(n) = (5 + (-2)^n)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions