A005792 -id:A005792 - cp's OEIS frontend

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, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Marc LeBrun, Sep 06 2002

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.
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]
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
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

			6 is a term of the sequence because a square can be dissected as follows:
  +---+---+---+
  |...|...|...|
  +---+---+---+
  |.......|...|
  |.......+---+
  |.......|...|
  +-------+---+

A. Soifer, How Does One Cut A Triangle?, Chapter 2, CEME, Colorado Springs CO 1990.
Allan C. Wechsler and Michael Kleber, messages to math-fun mailing list, Sep 06, 2002.

Mr. Glaeser, Carrés, Le Petit Archimède, no. 0, January 1973.
Murray Klamkin, Review of "How Does One Cut a Triangle?" by Alexander Soifer, Amer. Math. Monthly, October 1991, pp. 775-. [Annotated scanned copy of pages 775-777 only] See "Grand Problem 2".
Miklós Laczkovich, Tilings of polygons with similar triangles, Combinatorica 10.3 (1990): 281-306.
Miklós Laczkovich. Tilings of triangles Discrete mathematics 140.1 (1995): 79-94.
Miklós Laczkovich, Tilings of polygons with similar triangles, II, Discrete & Computational Geometry 19.3 (1998): 411-425.
Alexander Soifer, How Does One Cut a Triangle?, Chapter 2, Springer-Verlag New York, 2009.
Hassan Tarfaoui, Concours Général 1990 - Exercice 3 (in French).
Andrzej Zak, Dissection of a triangle into similar triangles, Discrete & Computational Geometry 34.2 (2005): 295-312.
Index to sequences related to Olympiads and other Mathematical competitions.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005792, A014544 (cubes).

gf:= x*(1 - x + x^3 - x^4 + x^5)/(1-x):
select(t-> coeftayl(gf, x=0, t)=1, [$1..100])[];  # Alois P. Heinz, Aug 17 2021

CoefficientList[Series[(1 + 2*x -x^2 - x^3)/(1 - x)^2, {x, 0, 20}], x] (* Georg Fischer, Aug 17 2021 *)
LinearRecurrence[{2,-1},{1,4,6,7},80] (* Harvey P. Dale, Oct 17 2021 *)

A074764(n)=if(n>2, n+3, n^2) \\ M. F. Hasler, May 24 2024

A074764 = lambda n: n+3 if n>2 else n*n # M. F. Hasler, May 24 2024

{k : k != 2, 3, or 5}.
G.f. of characteristic function: x*(1 - x + x^3 - x^4 + x^5)/(1-x).
G.f.: x*(1 + 2*x -x^2 - x^3)/(1 - x)^2. - Georg Fischer, Aug 17 2021
a(n) = n + 3 for all n > 2. - M. F. Hasler, May 24 2024
E.g.f.: exp(x)*(3 + x) - x^2/2 - 3*(x + 1). - Stefano Spezia, Sep 17 2024

cp's OEIS Frontend

A074764 Numbers of smaller squares into which a square may be dissected.

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