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%I #77 May 26 2024 08:30:49
%S 1,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,
%T 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,
%U 52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74
%N Numbers of smaller squares into which a square may be dissected.
%C All even k > 2 are present by generalizing this corner+border construction, all odd k > 5 are present because k+3 can be obtained from k by splitting any single square into four, 1 is trivially present and k = 2, 3 & 5 are then fairly easily eliminated.
%C Also number of smaller similar triangles into which a triangle may be dissected. - _Lekraj Beedassy_, Nov 25 2003 [This isn't true; for example, an isosceles right triangle can be dissected in 2 and therefore into any positive integer number of smaller similar triangles. - _M. F. Hasler_, May 24 2024]
%C Also positive integers k such that there exist k integers x_1, x_2, ..., x_k, distinct or not, satisfying 1 = 1/(x_1)^2 + 1/(x_2)^2 + ... + 1/(x_k)^2. For example, the unique solution for k = 4 is 1 = 1/2^2 + 1/2^2 + 1/2^2 + 1/2^2 (see Hassan Tarfaoui link, Concours Général 1990). - _Bernard Schott_, Oct 05 2021
%C With the current definition, one could argue that the initial 1 is wrong and should be 0 instead: One cannot dissect a square into 1 smaller square: not dissecting it yields a(1) = 0 smaller squares. - _M. F. Hasler_, May 24 2024
%D A. Soifer, How Does One Cut A Triangle?, Chapter 2, CEME, Colorado Springs CO 1990.
%D Allan C. Wechsler and Michael Kleber, messages to math-fun mailing list, Sep 06, 2002.
%H Mr. Glaeser, <a href="http://www.lepetitarchimede.fr/pa/PA00p6-7+.jpg">Carrés</a>, Le Petit Archimède, no. 0, January 1973.
%H Murray Klamkin, <a href="/A074764/a074764.pdf">Review of "How Does One Cut a Triangle?" by Alexander Soifer</a>, Amer. Math. Monthly, October 1991, pp. 775-. [Annotated scanned copy of pages 775-777 only] See "Grand Problem 2".
%H Miklós Laczkovich, <a href="https://doi.org/10.1007/BF02122782">Tilings of polygons with similar triangles</a>, Combinatorica 10.3 (1990): 281-306.
%H Miklós Laczkovich. <a href="https://doi.org/10.1016/0012-365X(93)E0176-5">Tilings of triangles</a> Discrete mathematics 140.1 (1995): 79-94.
%H Miklós Laczkovich, <a href="https://doi.org/10.1007/PL00009359">Tilings of polygons with similar triangles, II</a>, Discrete & Computational Geometry 19.3 (1998): 411-425.
%H Alexander Soifer, <a href="https://doi.org/10.1007/978-0-387-74652-4">How Does One Cut a Triangle?</a>, Chapter 2, Springer-Verlag New York, 2009.
%H Hassan Tarfaoui, <a href="http://d.tarfaoui.free.fr/cg/1990/EX3/exobis.pdf">Concours Général 1990 - Exercice 3</a> (in French).
%H Andrzej Zak, <a href="http://home.agh.edu.pl/~zakandrz/Publikacje/zak.pdf">Dissection of a triangle into similar triangles</a>, Discrete & Computational Geometry 34.2 (2005): 295-312.
%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F {k : k != 2, 3, or 5}.
%F G.f. of characteristic function: x*(1 - x + x^3 - x^4 + x^5)/(1-x).
%F G.f.: (1 + 2*x -x^2 - x^3)/(1 - x)^2. - _Georg Fischer_, Aug 17 2021
%F a(n) = n + 3 for all n > 2. - _M. F. Hasler_, May 24 2024
%e 6 is a term of the sequence because a square can be dissected as follows:
%e +---+---+---+
%e |...|...|...|
%e +---+---+---+
%e |.......|...|
%e |.......+---+
%e |.......|...|
%e +-------+---+
%p gf:= x*(1 - x + x^3 - x^4 + x^5)/(1-x):
%p select(t-> coeftayl(gf, x=0, t)=1, [$1..100])[]; # _Alois P. Heinz_, Aug 17 2021
%t CoefficientList[Series[(1 + 2*x -x^2 - x^3)/(1 - x)^2, {x, 0, 20}], x] (* _Georg Fischer_, Aug 17 2021 *)
%t LinearRecurrence[{2,-1},{1,4,6,7},80] (* _Harvey P. Dale_, Oct 17 2021 *)
%o (PARI) A074764(n)=if(n>2, n+3, n^2) \\ _M. F. Hasler_, May 24 2024
%o (Python) A074764 = lambda n: n+3 if n>2 else n*n # _M. F. Hasler_, May 24 2024
%Y Cf. A005792.
%K nonn,easy,changed
%O 1,2
%A _Marc LeBrun_, Sep 06 2002
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