A270060 -id:A270060 - cp's OEIS frontend

A100073 Number of representations of n as the difference of two positive squares.

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0, 1, 2, 2, 0, 2, 1, 1, 0, 2, 2, 1, 0, 1, 1, 3, 0, 1, 3, 1, 0, 2, 1, 1, 0, 2, 2, 2, 0, 1, 2, 1, 0, 3, 2, 2, 0, 1, 1, 2, 0, 1, 3, 1, 0, 3, 1, 2, 0, 1, 3, 2, 0, 1, 2, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0, 2, 4, 1, 0, 3, 1, 1, 0, 1, 2, 4

Offset: 1

T. D. Noe, Nov 02 2004

Note that for odd n, a(n) = 1 iff n is a prime, or a prime squared.
A decomposition n = a^2 - b^2 = (a-b)(a+b) = d*(n/d) is given for each divisor d less than (as to exclude b = 0) but having the same parity as n/d. For even n this implies that d and n/d must be even, i.e., 4 | n. This leads to the given formula, a(n) = floor(numdiv(n)/2) for odd n, floor(numdiv(n/4)/2) for n = 4k, 0 else. - M. F. Hasler, Jul 10 2018
a(n) is the number of self-conjugate partitions of n into parts of 2 different sizes, i.e., the order of the set of partitions obtained by the intersection of the partitions in A000700 and A002133. See A270060. - R. J. Mathar, Jun 15 2022

			a(15) = 2 because 15 = 16 - 1 = 64 - 49.

Robert Israel, Table of n, a(n) for n = 1..10000
A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 5.
Christian Aebi and Grant Cairns, Lattice equable quadrilaterals III: tangential and extangential cases, Integers (2023) Vol. 23, #A48.

Cf. A056924 (number of divisors of n that are less than sqrt(n)), A016825 (numbers not the difference of two squares), A034178 (number of representations of n as the difference of two squares).

Cf. A000700, A002133.

A100073:= proc(n)
  if n::odd then floor(numtheory:-tau(n)/2)
  elif (n/2)::odd then 0
  else floor(numtheory:-tau(n/4)/2)
  fi
end proc:
map(A100073, [$1..200]); # Robert Israel, Jul 10 2018

nn=150; a=Table[0, {nn}]; Do[y=x-1; While[d=x^2-y^2; d<=nn&&y>0, a[[d]]++; y-- ], {x, 1+nn/2}]; a

a(n) = if (n % 2, ceil((numdiv(n)-1)/2), if (!(n%4),  ceil((numdiv(n/4)-1)/2), 0)); \\ Michel Marcus, Mar 07 2016

A100073(n)=if(bittest(n,0),numdiv(n)\2,!bittest(n,1),numdiv(n\4)\2) \\ or shorter: a(n)=if(n%4!=2,numdiv(n\4^!(n%2))\2) \\ - M. F. Hasler, Jul 10 2018

a(n) = A056924(n) for odd n, a(n) = A056924(n/4) if 4|n, otherwise a(n) = 0.

A028247 Number of T-frame polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 10, 19, 28, 44, 60, 86, 110, 146, 182, 233, 278, 343, 403, 490, 557, 664, 749, 879, 978, 1132, 1237, 1435, 1551, 1771, 1905, 2168, 2296, 2608, 2758, 3101, 3256, 3655, 3798, 4274, 4419, 4936, 5087, 5670, 5809, 6472, 6602, 7339, 7462, 8271

Offset: 1

Anne Fontaine (fonta(AT)hvcc.edu), Hudson Valley Community College, Troy NY 12180

nonn

A T-frame is a polyomino whose boundary word has the form x^a y^b x^c y^d x^-e y^-f x^g y^-h, where a, b, c, d, e, f, g, h are positive integers. The boundary word is determined by moving counterclockwise around the boundary of the polyomino. The symbols x and y represent unit steps to the right and up, respectively, while x^-1 and y^-1 represent steps to the left and down. - David Radcliffe, Jan 31 2023
Equivalently, polyominoes which are integral rectangles with integral notches cut from two adjacent corners; or right-angled octagons with integral sides, and as you traverse the perimeter counterclockwise you encounter turns in the order LLLLRLLR. - Allan C. Wechsler, from seqfans mailing list, Jan 31 2023.
For 2 <= n <= 28, a(2n) < a(2n+1); for 29 <= n <= 99, a(2n) > a(2n+1). - Don Reble from seqfans email, Jan 31 2023.

			The a(6) = 6 polyominoes are:
   OOO   OOO   OOOO   OOOO  OOOOO  OOOOO
    O     OO    O      OO    O       O
    O     O     O
    O

John Mason, Table of n, a(n) for n = 1..1000

Cf. A270060 (L frame), A360419 (U frame), A360420 (Z frame).

B(k,x) = sum(j=1, k, x^j/(1-x^j))
seq(n) = Vec(sum(k=2, n, (x^k/(1-x^k)) * (B(k-1, x + O(x^(1+n-k)))^2 + B(k-1, x^2 + O(x^(1+n-k))))/2, O(x*x^n)), -n) \\ Andrew Howroyd, Feb 08 2023

G.f.: Sum_{k>=2} (x^k/(1-x^k)) * (B(k-1, x)^2 + B(k-1, x^2))/2 where B(k,x) = Sum_{j=1..k} x^j/(1-x^j). - Andrew Howroyd, Feb 08 2023

a(1)-a(3) and terms a(32) and beyond from Allan C. Wechsler and John Mason, Feb 03 2023

A360419 a(n) = the number of U-frame polyominoes with n cells, reduced for symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 16, 24, 37, 50, 71, 93, 121, 151, 192, 231, 285, 338, 398, 470, 548, 626, 723, 827, 924, 1056, 1175, 1314, 1454, 1629, 1763, 1985, 2138, 2356, 2540, 2820, 2976, 3305, 3491, 3834, 4039, 4441, 4613, 5103, 5291, 5775, 5999, 6572

Offset: 1

John Mason, Feb 06 2023

nonn

A U-frame polyomino has a perimeter that forms a self-avoiding polygon such that as you traverse the perimeter counterclockwise you encounter turns in the order LLLLLLRR.

			a(5)=1 because of:
  OO
  O
  OO
The a(7) = 5 polyominoes are:
  O
  O     O O             O
  O O   O O   O OO   O  O   O   O
  OOO   OOO   OOOO   OOOO   OOOOO

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A028247, A270060, A360420, A360421.

B(n,k,x) = sum(j=k, n, x^j/(1 - x^j), O(x*x^n))
seq(n) = Vec(sum(k=1, (n-2)\3, x^k*(B(n-k, k+1, x)^2 + B((n-k)\2, k+1, x^2))/(1-x^k), O(x*x^n))/2, -n) \\ Andrew Howroyd, Feb 07 2023

G.f.: Sum_{k>=1} (x^k/(1 - x^k)) * (B(k+1, x)^2 + B(k+1, x^2))/2 where B(k, x) = Sum_{j>=k} x^j/(1 - x^j). - Andrew Howroyd, Feb 07 2023

A360420 a(n) = the number of Z-frame polyominoes with n cells, reduced for symmetry.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 10, 19, 27, 43, 58, 85, 105, 143, 175, 226, 266, 334, 386, 475, 534, 641, 717, 854, 933, 1092, 1187, 1385, 1482, 1713, 1820, 2091, 2203, 2511, 2637, 2998, 3110, 3516, 3647, 4118, 4226, 4761, 4865, 5461, 5576, 6221, 6319, 7088, 7138, 7953

Offset: 1

John Mason, Feb 06 2023

nonn

A Z-frame polyomino has a perimeter that forms a self-avoiding polygon such that as you traverse the perimeter counterclockwise you encounter turns in the order LLLRLLLR.

			a(4)=1 because of:
  OO
 OO

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Andrew Howroyd, Formula and PARI Program, 2023.
John Mason, Examples

Cf. A028247, A270060, A360419, A360421.

seq(50) \\ See Links - Andrew Howroyd, Feb 08 2023

a(n) <= A028247(n). - Andrew Howroyd, Feb 08 2023

Terms a(19) and beyond from Andrew Howroyd, Feb 08 2023

A067627 Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 1, 3, 2, 1, 6, 1, 3, 7, 2, 5, 9, 2, 1, 8, 11, 2, 1, 13, 14, 1, 3, 19, 15, 3, 5, 27, 19, 1, 11, 34, 22, 2, 1, 15, 49, 23, 2, 1, 27, 59, 28, 3, 3, 39, 78, 30, 1, 5, 60, 93, 34, 3, 11, 82, 118, 36, 1, 18, 115, 140, 41, 3, 1, 30, 155, 170, 42, 2, 1, 48

Offset: 1

Naohiro Nomoto, Feb 02 2002

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

			Triangle turned on its side begins:
1.1.1.2.1.2.1.2.2..2..1..3..1..2..2....etc A038548
....1.1.3.3.6.7.9.11.14.15.19.22.23....etc A270060
..........1.1.3.5..8.13.19.27.34.49....etc
...................1..1..3..5.11.15....etc

Cf. A000700, A000701, A046682, A060177. Diagonals give A038548. row sums give A046682.

compareL := proc(L1,L2)
    if nops(L1) < nops(L2) then
        -1 ;
    elif nops(L1) > nops(L2) then
        1;
    else
        for i from 1 to nops(L1) do
            if op(i,L1) > op(i,L2) then
                return 1 ;
            elif op(i,L1) < op(i,L2) then
                return -1 ;
            end if;
        end do:
        0 ;
    end if;
end proc:
A067627 := proc(n,k)
    local a,p,s,pc ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(s) = k then
            pc := combinat[conjpart](p) ;
            if compareL(p,pc) <= 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc:
for n from 1 to 30 do
for k from A003056(n) to 1 by -1 do
    printf("%4d,",A067627(n,k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, May 08 2019

More terms from R. J. Mathar, May 08 2019

A360421 a(n) = the number of X-frame polyominoes with n cells, reduced for symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 7, 20, 49, 109, 216, 414, 723, 1228, 1958, 3065, 4580, 6734

Offset: 1

John Mason, Feb 06 2023

An X-frame polyomino has a perimeter that forms a self-avoiding polygon such that as you traverse the perimeter counterclockwise you encounter turns in the order LLRLLRLLRLLR.

			a(5) = 1 because of:
    O
   OOO
    O
The a(7) = 7 polyominoes are:
    O      O      O       O     O     O     O
   OOOOO  OOOOO  OOOOO  OOOOO  OOOO  OOOO  OOOO
    O       O       O     O     O      O    OO
                                O      O

Cf. A028247, A270060, A360419, A360420.

cp's OEIS Frontend

A100073 Number of representations of n as the difference of two positive squares.

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A028247 Number of T-frame polyominoes with n cells.

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A360419 a(n) = the number of U-frame polyominoes with n cells, reduced for symmetry.

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A360420 a(n) = the number of Z-frame polyominoes with n cells, reduced for symmetry.

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A067627 Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.

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Programs

Maple

Extensions

A360421 a(n) = the number of X-frame polyominoes with n cells, reduced for symmetry.

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Author

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Comments

Examples

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