A028401 The (2^n+1)-th triangular number (cf. A000217).

3, 6, 15, 45, 153, 561, 2145, 8385, 33153, 131841, 525825, 2100225, 8394753, 33566721, 134242305, 536920065, 2147581953, 8590131201, 34360131585, 137439739905, 549757386753, 2199026401281, 8796099313665, 35184384671745

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of types of Boolean functions of n variables under a certain group.

Also the number of ordered decompositions of 2^n into 3 nonnegative integers (e.g., 2 = 0+0+2 = 0+2+0 = 2+0+0 = 1+1+0 = 1+0+1 = 0+1+1). - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000
• Daniel Poveda Parrilla, Illustration of initial terms
• I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
• Index entries for sequences related to Boolean functions
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Equals 2*A036562(n-4) - 1, n > 3.

Cf. A000217.

Programs

Mathematica

Drop[#, 2] &@ CoefficientList[Series[3 x^2*(1 - 5 x + 5 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* Michael De Vlieger, Jul 08 2019 *)

Python

def A028401(n): return ((m:=1<2 else 3 # Chai Wah Wu, Jul 11 2024

Formula

From Ralf Stephan, Aug 23 2003: (Start)

a(n) = (3/8)*2^n + (1/32)*4^n + 1.

a(n) = 3*A007581(n-2) = (3/4)*A060919(n-1). (End)

a(n) = (2^n+4)*(2^n+8)/32. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007

G.f.: 3*x^2*(1-5*x+5*x^2)/((1-x)*(1-2*x)*(1-4*x)). - Colin Barker, Mar 09 2012

a(n) = a(n-1) + 3*A000217(2^(n-3)) for n > 2. - Daniel Poveda Parrilla, Dec 27 2016

E.g.f.: (32*exp(x) + 12*exp(2*x) + exp(4*x) - 45 - 60*x)/32. - Stefano Spezia, Jul 11 2024

Extensions

More terms from Vladeta Jovovic, Feb 24 2000

Simpler definition from Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007