A245676 -id:A245676 - cp's OEIS frontend

A006074 Number of polyaboloes (or polytans): number of different shapes that can be formed with n congruent isosceles right triangles, identifying mirror images.

1, 3, 4, 14, 30, 107, 318, 1116, 3743, 13240, 46476, 166358, 596638, 2158829, 7839845, 28616815, 104814161, 385269397, 1420242629, 5249877583, 19452536934, 72237904034

Offset: 1

N. J. A. Sloane

Also called supertangrams: a generalization of tangrams.

Martin Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 151 (but beware errors).
T. H. O'Beirne, New Scientist, 266 (Dec 21 1961), p. 752.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ed Pegg, Jr., Illustrations of polyforms
Andrew Clarke, Polyaboloes
Andrew Clarke, Illustration of initial terms
Michael Keller, Counting Polyforms
Henri Picciotto, Geometric Puzzles
Miroslav Vicher, Polyforms
Eric Weisstein's World of Mathematics, Polyabolo
Wikimedia, Polyabolo

Cf. A151519 (distinguishing mirror images), A245676 (number of convex polyaboloes). - George Sicherman, Nov 25 2017

Corrected values for a(8) and a(9), found by Aaron N. Siegel and confirmed by a Japanese puzzlist named Taro, reported by Michael Keller (Wgreview(AT)aol.com), Sep 02 2000
One more term from Vladeta Jovovic, Aug 11 2007
Link updated by William Rex Marshall, Dec 16 2009
Modified the definition (what is a "half-square"?) and added a(13), by George Sicherman, Apr 04 2012
a(14) from Juris Cernenoks, Sep 06 2012
a(15) from George Sicherman, Aug 02 2013
a(16)-a(20) from John Mason, Jan 07 2022
a(21) from Aaron N. Siegel, May 17 2022
a(22) from Aaron N. Siegel, Jun 07 2022

A093709 Characteristic function of squares or twice squares.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0

Offset: 0

Michael Somos, Apr 11 2004

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Partial sums of a(n) for n >= 1 are A071860(n+1). - Jaroslav Krizek, Oct 18 2009
For n > 0, this is also the number of different triangular polyabolos that can be formed from n congruent isosceles right triangles (illustrated at A245676). - Douglas J. Durian, Sep 10 2017

			G.f. = 1 + q + q^2 + q^4 + q^8 + q^9 + q^16 + q^18 + q^25 + q^32 + q^36 + q^49 + ...

Robert Israel, Table of n, a(n) for n = 0..10000
S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 1.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. - From _N. J. A. Sloane_, Feb 23 2009
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for characteristic functions

Cf. A000203, A000593, A001227, A028982, A053866, A092869.

A := Basis( ModularForms( Gamma1(8), 1/2), 104); A[1] + A[2]; /* Michael Somos, Jan 01 2015 */

seq(`if`(issqr(n) or issqr(n/2),1,0), n=0..100); # Robert Israel, Apr 05 2016

Table[Boole[IntegerQ[Sqrt[n]] || IntegerQ[Sqrt[2*n]]], {n, 0, 104}] (* Jean-François Alcover, Dec 05 2013 *)
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors[ n]] || OddQ [ Length @ Divisors[ 2 n]]]]; (* Michael Somos, Jan 01 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}]; (* Michael Somos, Jan 01 2015 *)

PARI
```
{a(n) = issquare(n) || issquare(2*n)};
```

Expansion of psi(q^4) * f(-q^3, -q^5) / f(-q, -q^7) in powers of q where psi(), f() are Ramanujan theta functions.
Expansion of f(-q^3, -q^5)^2 / psi(-q) in powers of q where psi(), f() are Ramanujan theta functions. - Michael Somos, Jan 01 2015
Euler transform of period 8 sequence [ 1, 0, -1, 1, -1, 0, 1, -1, ...].
G.f. A(x) satisfies A(x^2) = (A(x) + A(-x)) / 2. a(2*n) = a(n).
Given g.f. A(x), then A(x) / A(x^2) = 1 + x*A092869(x^2).
Given g.f. A(x), then B(x) = A(x^2) / A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 + v - 2(u + u^2)*v + 2*(u*v)^2.
Multiplicative with a(0) = a(2^e) = 1, a(p^e) = 1 if e even, 0 otherwise.
a(n) = A053866(n) unless n=0. Characteristic function of A028982 union 0.
G.f.: (theta_3(q) + theta_3(q^2)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(2*k^2)).
Dirichlet g.f.: zeta(2*s) * (1 + 2^-s).
For n>0: a(n) = A010052(n) + A010052(A004526(n))*A059841(n). - Reinhard Zumkeller, Nov 14 2009
a(n) = A000035(A000203(n)) = A000035(A000593(n)) = A000035(A001227(n)), if n>0. - Omar E. Pol, Apr 05 2016
Sum_{k=1..n} a(k) ~ (1 + 1/sqrt(2)) * sqrt(n). - Vaclav Kotesovec, Oct 16 2020

A292146 Number of different convex quadrilaterals that can be formed from n congruent isosceles right triangles. Reflections are not counted as different.

Original entry on oeis.org

0, 2, 2, 5, 2, 5, 3, 9, 2, 5, 2, 11, 2, 6, 4, 13, 3, 7, 2, 11, 4, 5, 3, 19, 2, 5, 4, 12, 2, 10, 3, 17, 4, 6, 4, 16, 2, 5, 4, 19, 3, 10, 2, 11, 6, 6, 3, 27, 3, 7, 4, 11, 2, 10, 4, 20, 4, 5, 2, 22, 2, 6, 7, 21, 4, 10, 2, 12, 4, 10, 3, 28, 3, 5, 6, 11, 4, 10

Offset: 1

Douglas J. Durian, Sep 09 2017

nonn

Illustrated with other convex polyabolos in A245676.

			For n=2, there is a square and a parallelogram.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Andrew Howroyd, Convex Quadrilaterals formed from Polyabolos

Strictly less than A245676.

\\ here b is A100073
b(n) = if(n%2, floor(numdiv(n)/2), if(n%4, 0, floor(numdiv(n/4)/2)));
d(n) = my(t); sum(k=1, floor(sqrt((n-1)/2)), issquare(n+2*k^2,&t) && t>2*k);
a(n) = 2*b(n) + d(n) + if(n%2, 0, 2*numdiv(n/2) + b(n/2)) + if(n%4, 0, ceil(numdiv(n/4)/2)); \\ Andrew Howroyd, Sep 16 2017

Terms a(33) and beyond from Andrew Howroyd, Sep 16 2017

A292147 Number of distinct convex pentagons that can be formed from n congruent isosceles right triangles. Reflections are not counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 0, 1, 2, 4, 1, 3, 5, 5, 2, 3, 2, 6, 5, 2, 8, 11, 1, 3, 7, 6, 7, 6, 7, 13, 7, 4, 8, 8, 5, 5, 13, 14, 8, 8, 4, 8, 14, 5, 17, 22, 5, 5, 7, 8, 14, 8, 14, 18, 13, 4, 11, 18, 7, 10, 22, 18, 15, 7, 9, 10, 19, 9, 16, 29, 8, 9, 12, 10, 23, 9, 19

Offset: 1

Douglas J. Durian, Sep 10 2017

nonn

Illustrated with other convex polyabolos in A245676.

			The smallest convex pentagonal polyabolo is for n=5.

Douglas J. Durian, Table of n, a(n) for n = 1..750

Strictly less than A245676.

a(33) and beyond from Douglas J. Durian, Jan 24 2020

A292148 Number of distinct convex hexagons that can be formed from n congruent isosceles right triangles. Reflections are not counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 1, 2, 2, 4, 1, 4, 2, 6, 5, 5, 5, 10, 2, 6, 6, 8, 8, 10, 7, 14, 7, 10, 6, 13, 12, 12, 13, 19, 8, 16, 6, 15, 22, 17, 16, 25, 9, 16, 10, 23, 21, 22, 18, 27, 23, 18, 16, 29, 21, 29, 20, 35, 18, 27, 18, 23, 39, 28, 24, 43, 21, 32, 19

Offset: 1

Douglas J. Durian, Sep 13 2017

nonn

Illustrated with other convex polyabolos in A245676.

			The smallest convex hexagonal polyabolo that is asymmetric is for n = 9.

Douglas J. Durian, Table of n, a(n) for n = 1..750

Strictly less than A245676.

a(33) and beyond from Douglas J. Durian, Jan 24 2020

A292149 Number of distinct convex heptagons that can be formed from n congruent isosceles right triangles. Reflections are not counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 1, 3, 2, 1, 3, 2, 3, 1, 4, 6, 3, 3, 3, 3, 8, 1, 8, 11, 3, 4, 5, 8, 9, 5, 11, 8, 11, 6, 10, 13, 8, 9, 11, 17, 13, 6, 14, 11, 21, 8, 15, 25, 13, 12, 17, 20, 22, 12, 18, 24, 22, 13, 27, 27

Offset: 1

Douglas J. Durian, Sep 13 2017

nonn

Illustrated with other convex polyabolos in A245676.

			The smallest convex heptagonal polyabolos are for n = 15 (symmetrical) and n = 17 (asymmetrical).

Douglas J. Durian, Table of n, a(n) for n = 1..750

Strictly less than A245676.

a(33) and beyond from Douglas J. Durian, Jan 25 2020

A292150 Number of distinct convex octagons that can be formed from n congruent isosceles right triangles. Reflections are not counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 1, 1, 1, 1, 5, 0, 3, 1, 1, 1, 3, 3, 5, 1, 1, 2, 8, 1, 3, 2, 7, 3, 3, 5, 4, 2, 8, 3, 12, 2, 4, 7, 7, 3, 6, 6, 15, 3, 5, 8, 12, 7, 9, 5, 15, 4, 12, 11, 11, 7, 6

Offset: 1

Douglas J. Durian, Sep 13 2017

nonn

Illustrated with other convex polyabolos in A245676.

			The smallest convex octagonal polyabolos are for n = 14 and n = 20; both are symmetrical.

Douglas J. Durian, Table of n, a(n) for n = 1..750

Strictly less than A245676.

a(33) and beyond from Douglas J. Durian, Jan 25 2020

cp's OEIS Frontend

A006074 Number of polyaboloes (or polytans): number of different shapes that can be formed with n congruent isosceles right triangles, identifying mirror images.

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A093709 Characteristic function of squares or twice squares.

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A292146 Number of different convex quadrilaterals that can be formed from n congruent isosceles right triangles. Reflections are not counted as different.

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A292147 Number of distinct convex pentagons that can be formed from n congruent isosceles right triangles. Reflections are not counted as distinct.

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Author

Keywords

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Examples

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Extensions

A292148 Number of distinct convex hexagons that can be formed from n congruent isosceles right triangles. Reflections are not counted as distinct.

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Author

Keywords

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Examples

Links

Crossrefs

Extensions

A292149 Number of distinct convex heptagons that can be formed from n congruent isosceles right triangles. Reflections are not counted as distinct.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A292150 Number of distinct convex octagons that can be formed from n congruent isosceles right triangles. Reflections are not counted as distinct.

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Extensions