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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.

%I A336877 #27 Nov 19 2021 10:55:52
%S A336877 0,1,5,12,25,41,60,85,125,168,205,245,300,361,425,504,625,749,840,925,
%T A336877 1025,1128,1225,1337,1500,1669,1805,1944,2125,2321,2520,2761,3125,
%U A336877 3492,3745,3965,4200,4429,4625,4836,5125,5417,5640,5857,6125,6408,6685,7013
%N 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.
%C A336877 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.
%C A336877 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.
%H A336877 Andrey Zabolotskiy, <a href="/A336877/b336877.txt">Table of n, a(n) for n = 0..8192</a>
%H A336877 A. A. Kirillov, <a href="https://www.math.upenn.edu/~kirillov/MATH480-F07/tf.pdf">A Tale of Two Fractals</a>, Birkhäuser, 2013, <a href="https://doi.org/10.1007/978-0-8176-8382-5">doi:10.1007/978-0-8176-8382-5</a>. See chapter 3, in particular Table 3.1.
%F A336877 a(2^p+n) - 2*a(2^p) + a(2^p-n) = 3 * a(n).
%F A336877 a(2*n) = 5 * a(n).
%F A336877 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
%o A336877 (Python)
%o A336877 def a(d, m=6):
%o A336877     chi = [0, 1]
%o A336877     for p in range(m):
%o A336877         chi += [(d+1)*chi[k]+2*chi[2**p]-chi[2**p-k] for k in range(1, 2**p+1)]
%o A336877     return chi
%o A336877 chi = a(2)
%o A336877 print(chi)
%o A336877 d2chi3 = [(chi[k+1]-2*chi[k]+chi[k-1])//3 for k in range(1, len(chi)-1)]
%o A336877 print(d2chi3) # A336878
%Y A336877 Cf. A336878 (second differences divided by 3), A178590, A178568.
%Y A336877 Cf. A000290, A282720.
%Y A336877 Cf. A047999.
%K A336877 nonn
%O A336877 0,3
%A A336877 _Andrey Zabolotskiy_, Aug 06 2020