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 A000360 Distribution of nonempty triangles inside a fractal rep-4-tile. 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS 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 REFERENCES M. J. Lafitte, Sur l'Effet Noah en Géométrie, rapport à l'INPI, mars 1995. LINKS 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) FORMULA 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) MATHEMATICA a = 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 *) PROG (Haskell) 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 (PARI) 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 CROSSREFS Cf. A002487, A000361, A000876. Cf. A001222, A057078, A125184, A284553, A284556, A284565 (bisection). Cf. also mutual recurrence pair A287729, A287730. Sequence in context: A335062 A047917 A144569 * A023556 A238783 A044944 Adjacent sequences: A000357 A000358 A000359 * A000361 A000362 A000363 KEYWORD nonn,nice,easy,eigen,look AUTHOR EXTENSIONS 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 STATUS approved

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Last modified March 31 15:44 EDT 2023. Contains 361668 sequences. (Running on oeis4.)