A100073 -id:A100073 - cp's OEIS frontend

A270060 Number of incomplete rectangles of area n.

0, 0, 1, 1, 3, 3, 6, 7, 9, 11, 14, 15, 19, 22, 23, 28, 30, 34, 36, 41, 42, 51, 49, 57, 55, 68, 64, 75, 71, 84, 79, 95, 89, 106, 92, 116, 104, 127, 116, 134, 121, 150, 130, 160, 143, 172, 148, 188, 156, 193, 177, 209, 177, 226, 185, 231, 210, 246, 207, 269, 218, 272, 239, 287, 238, 312, 250, 317, 279, 320, 271, 359, 283, 355, 316

Offset: 1

Stanislav Mikusek, Mar 09 2016

nonn

An incomplete rectangle is a six-sided figure obtained when two rectangles with different widths are coupled together so that two of the edges form a straight line.
In other words, this shape is a rectangle from which a smaller rectangle has been removed from one corner.
Incomplete rectangles which differ by a rotation and/or reflection are not counted as different.
Also the number of integer partitions of n into parts of 2 distinct sizes, where any integer partition and its conjugate are considered equivalent.  For example a(8)=7 counts (7,1), (6,2), (6,1,1), (5,3), (5,1,1,1), (4,2,2), and (3,3,2).
The unit squares composing the incomplete rectangle can be viewed as the boxes of a Ferrers diagram of an integer partition of n with 2 different sizes of rows.  A002133(n) counts all Ferrers diagrams with 2 different sizes of rows. A100073(n) counts all self-conjugate Ferrers diagrams with 2 different sizes of rows since these Ferrers diagrams look like a square with a smaller square removed from the corner.  Thus a(n)=(A002133(n)+A100073(n))/2. Lara Pudwell, Apr 03 2016

			n = 3
.___.
| ._|
|_|
.
n = 4
._____.
| .___|
|_|
.
n = 5
._______. ._____. ._____.
| ._____| |   ._| | .___|
|_|       |___|   | |
                  |_|
.
The three solutions for n = 6:
XXXXX
X
.....
XXXX
XX
.....
XXXX
X
X
.....

Cf. A038548 (number of complete rectangles of area n), A002133, A100073, A067627.

Maple
```
# see A067627(n,k=2).
```

/* rectangle : LL = long side, SS = short side
removed corner : L = long side, S = short side */
{
int a[100];
int LL,SS,L,S,area;
for(area:=1;area<=100;area++){
a[area]:=0;
};
for(LL:=1;LL<=100;LL++){
for(SS:=1;SS<=LL;SS++){
  for(L:=1;L<=LL;L++){
    for(S:=1;S<=LL;S++){
      area=LL*SS-L*S;
      if( area>=1 && area<=100 ){
        if( L>=S || LSS ){
          a[area]++;
        };
      };
    };
  };
};
};
for(area:=1;area<=100;area++){
  print a[area];
};
}

a(n)=(A002133(n)+A100073(n))/2. See the integer partition comment above. Lara Pudwell, Apr 03 2016

G.f.: sum(sum(x^(i+j)/(2*(1-x^i)*(1-x^j))+x^(i^2-j^2)/2,j=1..i-1),i=1..infinity). See the integer partition comment above. Lara Pudwell, Apr 03 2016

A306102 Numbers that are the difference of two positive squares in at least two ways.

Original entry on oeis.org

15, 21, 24, 27, 32, 33, 35, 39, 40, 45, 48, 51, 55, 56, 57, 60, 63, 64, 65, 69, 72, 75, 77, 80, 81, 84, 85, 87, 88, 91, 93, 95, 96, 99, 104, 105, 108, 111, 112, 115, 117, 119, 120, 123, 125, 128, 129, 132, 133, 135, 136, 140, 141, 143, 144, 145, 147, 152, 153, 155, 156

Offset: 1

Geoffrey B. Campbell (Geoffrey.Campbell(AT)anu.edu.au), Jul 10 2018

nonn

Numbers n such that A100073(n) >= 2; see there for more information and formulas.

In sequence A058957 the smaller square is allowed to be zero, therefore it lists all squares > 4 (m^2 - 0^2 = ((m^2+1)/2)^2 - ((m^2-1)/2)^2 if odd, = (m^2/4+1)^2 - (m^2/4-1)^2 if even) in addition to the terms given here, which already comprise squares (64, 144, ...) having more representations than these "trivial" ones. - M. F. Hasler, Jul 11 2018

Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory group on LinkedIn, July 8, 2018.

Cf. A100073, A058957, A056924, A000290.

Contains A306103 and A306104 as subsequences.

Select[Range@156, Length@ FindInstance[x^2 - y^2 == # && x>y>0, {x,y}, Integers, 2] == 2 &] (* Giovanni Resta, Jul 10 2018 *)

select( is(n)=A100073(n)>1, [1..200]) \\ M. F. Hasler, Jul 10 2018

A306102 = { n = 2k+1 | A056924(n) > 1 } U { n = 4k | A056924(n/4) > 1 }. - M. F. Hasler, Jul 10 2018

A306103 Numbers that are the difference of two positive squares in at least three ways.

Original entry on oeis.org

45, 48, 63, 72, 75, 80, 96, 99, 105, 112, 117, 120, 128, 135, 144, 147, 153, 160, 165, 168, 171, 175, 176, 180, 189, 192, 195, 200, 207, 208, 216, 224, 225, 231, 240, 243, 245, 252, 255, 256, 261, 264, 272, 273, 275, 279, 280, 285, 288, 297, 300

Offset: 1

Geoffrey B. Campbell and M. F. Hasler, Jul 10 2018

nonn

Numbers n such that A100073(n) >= 3; see there for more information & formulas.

			48 = 7^2 - 1^2 = 8^2 - 4^2 = 13^2 - 11^2.

Metin Sariyar, Table of n, a(n) for n = 1..10000
Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory Group on LinkedIn, July 8, 2018.

Cf. A100073, A058957, A056924.

Subsequence of A306102. Contains A306104 as a subsequence.

Select[Range[300], Length[FindInstance[x^2 - y^2 == # && x>y>0, {x,y}, Integers, 3 ]] == 3 &] (* Giovanni Resta, Jul 10 2018 *)

PARI
```
select( is(n)=A100073(n)>2, [1..300])
```

A306103 = { n = 2k+1 | A056924(n) > 2 } U { n = 4k | A056924(n/4) > 2 }.

A306104 Numbers that are the difference of two positive squares in at least four ways.

Original entry on oeis.org

96, 105, 120, 135, 144, 160, 165, 168, 189, 192, 195, 216, 224, 225, 231, 240, 255, 264, 273, 280, 285, 288, 297, 312, 315, 320, 336, 345, 351, 352, 357, 360, 375, 384, 385, 399, 400, 405, 408, 416, 420, 429, 432, 435, 440, 441, 448, 455, 456, 459, 465, 480, 483, 495

Offset: 1

Geoffrey B. Campbell and M. F. Hasler, Jul 10 2018

nonn

Numbers n such that A100073(n) >= 4; see there for more information & formulas.

			96 = 10^2 - 2^2 = 11^2 - 5^2 = 14^2 - 10^2 = 25^2 - 23^2.

Metin Sariyar, Table of n, a(n) for n = 1..10000
Geoffrey Campbell, Numbers that are the difference of two squares in two or more ways, Number Theory Group on LinkedIn, July 8, 2018.

Cf. A100073, A058957, A056924.

Subsequence of A306103, A306102 and A058957.

Select[Range@495, Length@ FindInstance[x^2 - y^2 == # && x>y>0, {x, y}, Integers, 4] == 4 &] (* Giovanni Resta, Jul 10 2018 *)

PARI
```
select( is(n)=A100073(n)>3, [1..500])
```

A306104 = { n = 2k+1 | A056924(n) > 3 } U { n = 4k | A056924(n/4) > 3 }.

A368041 a(n) is the least number k such that k^2 can be written as the difference of two positive squares in exactly n ways.

Original entry on oeis.org

1, 3, 8, 16, 12, 64, 128, 24, 512, 1024, 48, 4096, 72, 60, 32768, 65536, 192, 144, 524288, 384, 2097152, 4194304, 120, 16777216, 432, 1536, 134217728, 576, 3072, 1073741824, 2147483648, 240, 1152, 17179869184, 12288, 68719476736, 137438953472, 360, 1728, 1099511627776

Offset: 0

Ilya Gutkovskiy, Dec 09 2023

nonn

Index of first occurrence of n in A046079.

All the terms are of the form 2^m * A147516(k), m >= 0, k >= 1. - Amiram Eldar, Nov 08 2024

			a(2) = 8: 8^2 = 10^2 - 6^2 = 17^2 - 15^2.

Amiram Eldar, Table of n, a(n) for n = 0..999

Cf. A025487, A046079, A068314, A071571, A094191, A100073, A122842, A147516.

a(n) = min(A122842(n+1), 2*A071571(n)). - Jon E. Schoenfield, Dec 09 2023

a(26)-a(29) from Michel Marcus, Dec 09 2023

a(30)-a(39) from Jon E. Schoenfield, Dec 09 2023

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

A334078 a(n) is the smallest positive integer that can be expressed as the difference of two positive squares in at least n ways.

Original entry on oeis.org

3, 15, 45, 96, 192, 240, 480, 480, 720, 960, 1440, 1440, 2880, 2880, 2880, 3360, 5040, 5040, 6720, 6720, 10080, 10080, 10080, 10080, 20160, 20160, 20160, 20160, 20160, 20160, 30240, 30240, 40320, 40320, 40320, 40320, 60480, 60480, 60480, 60480, 80640, 80640

Offset: 1

Ilya Gutkovskiy, Apr 13 2020

nonn

Eric Weisstein's World of Mathematics, Square Number

Cf. A000290, A024352, A048610, A068314, A094191, A100073, A214771, A334077.

A368073 Number of representations of n as the difference of two positive square pyramidal numbers.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Ilya Gutkovskiy, Dec 10 2023

nonn

			a(25) = 2: 25 = 30 - 5 = 55 - 30.

Eric Weisstein's World of Mathematics, Square Pyramidal Number

Cf. A000330, A100073, A296338, A368072.

cp's OEIS Frontend

A270060 Number of incomplete rectangles of area n.

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A306102 Numbers that are the difference of two positive squares in at least two ways.

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A306103 Numbers that are the difference of two positive squares in at least three ways.

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A306104 Numbers that are the difference of two positive squares in at least four ways.

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A368041 a(n) is the least number k such that k^2 can be written as the difference of two positive squares in exactly n ways.

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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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A334078 a(n) is the smallest positive integer that can be expressed as the difference of two positive squares in at least n ways.

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A368073 Number of representations of n as the difference of two positive square pyramidal numbers.

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