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

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

1,2

Comments

All even n>2 are present by generalizing this corner+border construction, all odd n>5 are present because n+3 can be obtained from n by splitting any single square into four, 1 is trivially present and n=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

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

References

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.

Links

Table of n, a(n) for n=1..71.

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).

Formula

n # 2, 3 or 5.

G.f. of characteristic function: x*(1 - x + x^3 - x^4 + x^5)/(1-x).

G.f.: (1 + 2*x -x^2 - x^3)/(1 - x)^2. - Georg Fischer, Aug 17 2021

Example

6 is a term of the sequence because:
+---+---+---+
|...|...|...|
+---+---+---+
|.......|...|
|.......+---+
|.......|...|
+-------+---+

Maple

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

Mathematica

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 *)

Crossrefs

Cf. A005792.

Sequence in context: A234948 A123860 A122817 * A101087 A138887 A216845

Adjacent sequences: A074761 A074762 A074763 * A074765 A074766 A074767

Keyword

nonn,easy

Author

Marc LeBrun, Sep 06 2002

Status

approved