M. Jeremie Lafitte (Levitas) - cp's OEIS frontend

User: M. Jeremie Lafitte (Levitas)

M. Jeremie Lafitte (Levitas)'s wiki page.

M. Jeremie Lafitte (Levitas) has authored 3 sequences.

A000361 From a fractal set of positive Lebesgue measure, a self-replicating tiling with holes, the 4-reptile following the 2-reptile of Paul Levy.

1, 0, 2, 1, 1, 2, 5, 0, 10, 6, 3, 2, 19, 2, 10, 1, 5, 10, 89, 1, 170, 28, 7, 2, 71, 12, 170, 5, 25, 10, 21, 0, 42, 26, 51, 10, 1251, 38, 682, 6, 301, 170, 5833, 3, 2730, 120, 15, 2, 271, 56

Offset: 0

M. Jeremie Lafitte (Levitas)

Counting filled equal triangles along lines on the Mandelvyn Triangle.

Croft, Falconer and Guy, Unsolved Problems in Geometry, Springer-Verlag, 1991; Problem of least k such that there exists a non-simply-connected k-reptile.
M. Jeremie Lafitte (Levitas), Sur l'Effet Noa`h en Geometrie, rapport a l'INPI, Mars 1995.

M. Jeremie Lafitte (Levitas), Ensembles Auto-Similaires d'Interieur Non-Vide, Preprint Hiver 1997, Chaire de Geometrie, Departement de Mathematiques, Ecole Polytechnique Federale de Lausanne, Switzerland. [Cached copy, with permission]
M. Jeremie Lafitte (Levitas), Fractal triangle underlying A000360, A000361, A000876
M. Jeremie Lafitte (Levitas), Notes on A000360, A000361, A000876 [Based on a latex file sent by M. Jeremie Lafitte (Levitas) to NJAS in 1995 - see file of emails below]
M. Jeremie Lafitte (Levitas), Latex source for the pdf file [Sent by M. Jeremie Lafitte (Levitas) to NJAS in 1995 - see file of emails below]
M. Jeremie Lafitte (Levitas) and N. J. A. Sloane, Emails, 1995-2007 (The three sequences mentioned in this correspondence are now A000360, A000361, A000876)

Cf. A000360, A000876.

A000876 From a self-replicating tiling.

Original entry on oeis.org

1, 1, 0, 0, 3, 3, 0, 0, 1, 0, 1, 1, 2, 0, 7, 7, 0, 2, 1, 1, 0, 1, 2, 2, 3, 0, 3, 6, 10, 0, 13, 2, 9, 3, 10, 0, 15, 15, 0, 10, 3, 9, 2, 13, 0, 10, 6, 3, 0, 3, 2, 2, 5, 2, 25, 1, 42, 12, 7, 0, 7, 28, 170, 1, 217, 10, 37, 1

Offset: 0

M. Jeremie Lafitte (Levitas)

nonn

M. Jeremie Lafitte (Levitas), Ensembles Auto-Similaires d'Interieur Non-Vide, Preprint Hiver 1997, Chaire de Geometrie, Departement de Mathematiques, Ecole Polytechnique Federale de Lausanne, Switzerland. [Cached copy, with permission]
M. Jeremie Lafitte (Levitas), Fractal triangle underlying A000360, A000361, A000876
M. Jeremie Lafitte (Levitas), Notes on A000360, A000361, A000876 [Based on a latex file sent by M. Jeremie Lafitte (Levitas) to NJAS in 1995 - see file of emails below]
M. Jeremie Lafitte (Levitas), Latex source for the pdf file [Sent by M. Jeremie Lafitte (Levitas) to NJAS in 1995 - see file of emails below]
M. Jeremie Lafitte (Levitas) and N. J. A. Sloane, Emails, 1995-2007 (The three sequences mentioned in this correspondence are now A000360, A000361, A000876)

Cf. A000360, A000361.

A000360 Distribution of nonempty triangles inside a fractal rep-4-tile.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 0, 2, 2, 2, 1, 3, 1, 2, 1, 2, 2, 4, 1, 4, 3, 3, 1, 4, 2, 4, 2, 3, 2, 3, 0, 3, 3, 4, 2, 6, 3, 5, 2, 5, 4, 7, 2, 6, 4, 4, 1, 5, 3, 6, 3, 6, 4, 6, 1, 5, 4, 5, 2, 5, 2, 3, 1, 3, 3, 6, 2, 7, 5, 6, 2, 8, 5, 9, 4, 8, 5, 7, 1, 7, 6, 9, 4, 11, 6, 9, 3, 8, 6, 10, 3, 8, 5, 5, 1, 6, 4, 8, 4, 9, 6, 9, 2

Offset: 0

M. Jeremie Lafitte (Levitas)

a(n) = Running count of congruent nonempty triangles along lines perpendicular to the base of the Gosper-Lafitte triangle.
Also, a(n) = Sum of the coefficients of the terms with an even exponent in the Stern polynomial B(n+1,t), or in other words, the sum of the even-indexed terms (the leftmost is at index 0) of the irregular triangle A125184, starting from its second row. - Antti Karttunen, Apr 20 2017
Back in May 1995, it was proved that a(n) = modulo 3 mapping, (+1,-1,+0)/2, of the Stern-Brocot sequence A002487, dropping its 1st term. - M. Jeremie Lafitte (Levitas), Apr 23 2017

M. J. Lafitte, Sur l'Effet Noah en Géométrie, rapport à l'INPI, mars 1995.

T. D. Noe, Table of n, a(n) for n = 0..10000
S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007) 86-95.
M. J. Lafitte, Ensembles Auto-Similaires d'Intérieur Non-Vide, Preprint Hiver 1997, Chaire de Géometrie, Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Switzerland. [Cached copy, with permission]
M. J. Lafitte, Fractal triangle underlying A000360, A000361, A000876
M. J. Lafitte, Notes on A000360, A000361, A000876 [Based on a latex file sent by M. Jeremie Lafitte (Levitas) to NJAS in 1995 - see file of emails below]
M. J. Lafitte, Latex source for the pdf file [Sent by MJL to NJAS in 1995 - see file of emails below]
M. J. Lafitte and N. J. A. Sloane, Emails, 1995-2007 (The three sequences mentioned in this correspondence are now A000360, A000361, A000876)

Cf. A002487, A000361, A000876.
Cf. A001222, A057078, A125184, A284553, A284556, A284565 (bisection).
Cf. also mutual recurrence pair A287729, A287730.

import Data.List (transpose)
a000360 n = a000360_list !! n
a000360_list = 1 : concat (transpose
   [zipWith (+) a000360_list $ drop 2 a057078_list,
    zipWith (+) a000360_list $ tail a000360_list])
-- Reinhard Zumkeller, Mar 22 2013
(Scheme, with memoization-macro definec):
(define (A000360 n) (A000360with_prep_0 (+ 1 n)))
(definec (A000360with_prep_0 n) (cond ((<= n 1) n) ((even? n) (A284556 (/ n 2))) (else (+ (A000360with_prep_0 (/ (- n 1) 2)) (A000360with_prep_0 (/ (+ n 1) 2))))))
(definec (A284556 n) (cond ((<= n 1) 0) ((even? n) (A000360with_prep_0 (/ n 2))) (else (+ (A284556 (/ (- n 1) 2)) (A284556 (/ (+ n 1) 2))))))
;; Antti Karttunen, Apr 07 2017

a[0] = 1; a[n_?EvenQ] := a[n] = a[n/2] + a[n/2-1]; a[n_?OddQ] := a[n] = a[(n-1)/2] - Mod[(n-1)/2-1, 3] + 1; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Jan 20 2015, after Ralf Stephan *)

a(n) = if(n==0, 1, if(n%2, a((n - 1)/2) - ((n - 1)/2 - 1)%3 + 1, a(n/2) + a(n/2 - 1))); \\ Indranil Ghosh, Apr 20 2017

a(3n) = (A002487(3n+1) + 1)/2, a(3n+1) = (A002487(3n+2) - 1)/2, a(3n+2) = A002487(3n+3)/2. - M. Jeremie Lafitte (Levitas), Apr 23 2017
a(0) = 1, a(2n) = a(n) + a(n-1), a(2n+1) = a(n) + 1 - (n-1 mod 3). - Ralf Stephan, Oct 05 2003; Note: according to Ralf Stephan, this formula was found empirically. It follows from that found for the Stern-Brocot sequence A002487 and the first formula. - Antti Karttunen, Apr 21 2017, M. Jeremie Lafitte (Levitas), Apr 23 2017
From Antti Karttunen, Apr 07 2017: (Start)
Ultimately equivalent to the above formulae, we have:
a(n) = A001222(A284553(1+n)).
a(n) = A002487(1+n) - A284556(1+n).
a(n) = b(1+n), with b from a mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1), c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1). [c(n) = A284556(n), b(n)+c(n) = A002487(n).]
(End)

More terms from David W. Wilson, Aug 30 2000

Original relation to the Stern-Brocot sequence A002487 reformulated by M. Jeremie Lafitte (Levitas), Apr 23 2017

cp's OEIS Frontend

User: M. Jeremie Lafitte (Levitas)

A000361 From a fractal set of positive Lebesgue measure, a self-replicating tiling with holes, the 4-reptile following the 2-reptile of Paul Levy.

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A000876 From a self-replicating tiling.

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A000360 Distribution of nonempty triangles inside a fractal rep-4-tile.

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