A336877 The harmonic function on the Sierpinski gasket with vertices 0, 1, w = (-1)^(1/3) defined by the values a(0) = 0, a(1) = 1, a(w) = -1.

0, 1, 5, 12, 25, 41, 60, 85, 125, 168, 205, 245, 300, 361, 425, 504, 625, 749, 840, 925, 1025, 1128, 1225, 1337, 1500, 1669, 1805, 1944, 2125, 2321, 2520, 2761, 3125, 3492, 3745, 3965, 4200, 4429, 4625, 4836, 5125, 5417, 5640, 5857, 6125, 6408, 6685, 7013

Offset: 0

Author

Andrey Zabolotskiy, Aug 06 2020

Keywords

• nonn

Comments

The harmonic functions on the Sierpiński gasket are fully defined by their values at the corners of the triangle: 0, 1, w. A harmonic function can be restricted to the interval at the real axis [0; 1] and then extended to all nonnegative real arguments. The function with a(0) = 0, a(1) = 1, a(w) = -1 yields integer values at integer arguments.

If we replace 3 by d+1 on the right side of the first line in the Formula section, we'll obtain the analog for the d-dimensional Sierpiński gasket for d>1. The case d=1 gives the squares A000290, the case d=0 gives A282720.

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 0..8192
• A. A. Kirillov, A Tale of Two Fractals, Birkhäuser, 2013, doi:10.1007/978-0-8176-8382-5. See chapter 3, in particular Table 3.1.

Crossrefs

Cf. A336878 (second differences divided by 3), A178590, A178568.

Cf. A000290, A282720.

Cf. A047999.

Programs

Python

def a(d, m=6):
    chi = [0, 1]
    for p in range(m):
        chi += [(d+1)*chi[k]+2*chi[2**p]-chi[2**p-k] for k in range(1, 2**p+1)]
    return chi
chi = a(2)
print(chi)
d2chi3 = [(chi[k+1]-2*chi[k]+chi[k-1])//3 for k in range(1, len(chi)-1)]
print(d2chi3) # A336878

Formula

a(2^p+n) - 2*a(2^p) + a(2^p-n) = 3 * a(n).

a(2*n) = 5 * a(n).

a(n+1) - a(n) = A178590(2n+1) [discovered by Sequence Machine]; more generally, the 1st differences of the analogous sequence with given d (see comment above) is the bisection of the (d+1)-th row of A178568. - Andrey Zabolotskiy, Oct 07 2021