A282720 -id:A282720 - cp's OEIS frontend

A282728 Irregular triangle read by rows: row n gives "digits" d_1, d_2, d_3, ... of the 3-expansion of A282720(n).

1, 3, 6, 2, 3, 2, 7, 4, 6, 4, -2, -7, 2, 2, 3, 2, 2, 8, 2, 4, 9, 6, -2, -1, 4, 4, 6, 4, -6, 2, 1, 4, -2, -4, -9, 4, -2, -2, -8, 2, 2, 2, 3, 2, 2, 2, 9, 2, 2, 4, 12, 2, 6, -2, 5, 2, 4, 6, 9, 6, -6, 4, 8, 6, -2, 0, -3, 6, 2, -4, -15, 4, 4, 4, 6, 4, -6, -2, 4, 15, 4, -6, 2, 0, 3

Offset: 1

N. J. A. Sloane, Mar 03 2017

			Triangle begins:
  1;
  3;
  6;
  2,  3;
  2,  7;
  4,  6;
  4, -2, -7;
  ...

Julien Leroy, Michel Rigo, and Manon Stipulanti, Behavior of Digital Sequences Through Exotic Numeration Systems, Electronic Journal of Combinatorics 24(1) (2017), #P1.44. See Lemma 8 p.5 and Table 1 p. 8.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, arXiv:1705.10065 [math.CO], 2017.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.
Manon Stipulanti, Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems, arXiv:1801.03287 [math.CO], 2018.

Cf. A282720.

add(v, w) = v = Vecrev(v); w = Vecrev(w); my(z = if (#v == #w, v+w, if (#v < #w, Vec(v,#w) + w, Vec(w, #v) + v))); Vecrev(z);
row(n) = if (n==0, return([])); if (n==1, return([1])); my(e = logint(n, 2), r = n-2^e); if (r <= 2^(e-1), add(2*digits(3^(e-1), 3), add(a(2^(e-1)+r), a(r))), add(4*digits(3^e, 3), add(-digits(2*3^(e-1), 3), add(-a(2^(e-1)+2^e-r), -a(2^e-r))))); \\ Michel Marcus, Feb 25 2025

More terms from Michel Marcus, Feb 25 2025

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.

Original entry on oeis.org

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

Andrey Zabolotskiy, Aug 06 2020

nonn

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.

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.

Cf. A336878 (second differences divided by 3), A178590, A178568.
Cf. A000290, A282720.
Cf. A047999.

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

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

cp's OEIS Frontend

A282728 Irregular triangle read by rows: row n gives "digits" d_1, d_2, d_3, ... of the 3-expansion of A282720(n).

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