cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A211152 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant >= 2n.

Original entry on oeis.org

4, 80, 428, 1428, 3604, 7924, 14988, 26244, 43072, 66844, 98804, 142652, 197584, 268800, 357608, 465888, 596964, 756036, 942160, 1163608, 1420248, 1717344, 2056952, 2451248, 2891420, 3392528, 3958576
Offset: 1

Views

Author

Clark Kimberling, Apr 05 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = -n; b = n; z1 = 27;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, -2*n^2, m}]
Table[c1[n,2*n^2]-c1[n,2n-1],{n,1,z1}] (* A211152 *)
%/4  (* integers *)

A211153 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and determinant >= 3n.

Original entry on oeis.org

0, 20, 184, 804, 2284, 5408, 10780, 20084, 33664, 53840, 81124, 119156, 167872, 231096, 309816, 409128, 527088, 673112, 842776, 1049352, 1285760, 1561932, 1879908, 2247884, 2662000, 3134360, 3662256, 4264796, 4926892, 5673180
Offset: 1

Views

Author

Clark Kimberling, Apr 05 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = -n; b = n; z1 = 30;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, -2*n^2, m}]
Table[c1[n,2*n^2]-c1[n,3n-1],{n,1,z1}] (* A211153 *)
%/4 (* integers *)

A278846 Number of unimodular 2 X 2 matrices having entries in {0,1,...,n} with no entry repeated.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 8, 40, 48, 80, 88, 152, 160, 232, 264, 304, 344, 448, 480, 608, 648, 720, 784, 944, 968, 1104, 1176, 1304, 1376, 1576, 1616, 1840, 1944, 2080, 2184, 2352, 2424, 2688, 2816, 2984, 3072, 3368, 3440, 3760, 3896, 4064, 4224, 4576, 4664, 4984, 5120
Offset: 0

Views

Author

Indranil Ghosh, Nov 29 2016

Keywords

Comments

a(n) mod 8 = 0.

Links

Crossrefs

Cf. A210000 (where the matrix entries can be repeated).

Programs

  • Maple
    df:= proc(n) local count, c,d,q,av,bc,a,b;
  count:= 0:
  for d from 1 to n-1 do
      av:= {$1..n-1} minus {d};
      for q in [-1,1] do
        bc:= n*d+q;
        for b in numtheory:-divisors(bc) intersect av do
          c:= bc/b;
          if c < b and member(c,av) then count:=count+8 fi;
  od od od;
  count
end proc:
ListTools:-PartialSums(map(df, [$0..100])); # Robert Israel, Nov 29 2016
  • Mathematica
    df[n_] := Module[{count = 0, c, d, q, av, bc, a, b}, Do[av = Range[n - 1]  ~Complement~ {d}; Do[bc = n d + q; Do[c = bc/b; If[c < b && MemberQ[av, c], count += 8], {b, Divisors[bc] ~Intersection~ av}], {q, {-1 , 1}}], {d, 1, n - 1}]; count];
df /@ Range[0, 100] // Accumulate (* Jean-François Alcover, Jul 29 2020, after Robert Israel *)
  • Python
    def a(n):
    s=0
    for a in range(0,n+1):
        for b in range(0,n+1):
            for c in range(0,n+1):
                for d in range(0,n+1):
                    if (a!=b  and a!=d and b!=d and c!=a and c!=b and c!=d):
                        if abs(a*d-b*c)==1:
                            s+=1
    return s
print([a(n) for n in range(0, 52)]) # Indranil Ghosh, Nov 29 2016

A279725 Number of 3 X 3 matrices having all terms in {0,1,...,n} with |det| = 1.

Original entry on oeis.org

0, 168, 2022, 15090, 53160, 196962, 409956, 1096368, 2062140, 4070796, 6674010, 12603174, 18410352, 31642836, 45306438, 67301682, 93747984, 142196892, 183799392, 267038772, 342684960, 458663640, 582535842, 793793994, 963867732, 1266864846, 1550198598, 1957887150, 2357651670, 3015489714
Offset: 0

Views

Author

Indranil Ghosh, Jan 04 2017

Keywords

Comments

a(n) is always even.
a(n) mod 6 = 0.

Examples

			For n=2, a few of the possible matrices are [0,0,1,0,1,0,1,0,0], [0,0,1,0,1,0,1,0,1], [0,0,1,0,1,0,1,0,2], [1,0,0,0,1,1,2,0,1], [1,0,0,0,1,1,2,1,0], [1,0,0,0,1,1,2,1,2], [2,2,1,2,1,2,1,0,2], [2,2,1,2,1,2,1,1,0], [2,2,1,2,1,2,1,1,1], [2,2,1,2,1,2,1,2,0], .... There are 2022 possibilities.
Here each of the matrices is defined as M=[a,b,c,d,e,f,g,h,i] where a=M[1][1], b=M[1][2], c=M[1][3], d=M[2][1], e=M[2][2], f=M[2][3], g=M[3][1], h=M[3][2] and i=M[3][3].
So, for n=2, a(n)=2022.

Links

Crossrefs

Cf. A210000.

Programs

  • Sage
    import itertools
def a(n):
    ans, W = 0, itertools.product(range(n+1), repeat=9)
    for w in W:
        if abs(Matrix(ZZ, 3, 3, w).det())==1: ans += 1
    return ans  # Robin Visser, May 01 2025

Extensions

More terms from Robin Visser, May 01 2025

A280059 Number of 2 X 2 matrices having all elements in {-n,..,0,..,n} with determinant = permanent.

Original entry on oeis.org

1, 45, 225, 637, 1377, 2541, 4225, 6525, 9537, 13357, 18081, 23805, 30625, 38637, 47937, 58621, 70785, 84525, 99937, 117117, 136161, 157165, 180225, 205437, 232897, 262701, 294945, 329725, 367137, 407277, 450241, 496125
Offset: 0

Views

Author

Indranil Ghosh, Dec 25 2016

Keywords

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    Table[16*(n+1)^3 - 28*(n+1)^2 + 16*(n+1) - 3, {n,0,50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 45, 225, 637}, 50] (* G. C. Greubel, Dec 25 2016 *)
  • PARI
    for(n=0, 50, print1(16*(n+1)^3 - 28*(n+1)^2 + 16*(n+1) - 3, ", ")) \\ G. C. Greubel, Dec 25 2016

Formula

a(n) = 16*(n+1)^3 - 28*(n+1)^2 + 16*(n+1) - 3 for n>0.
From G. C. Greubel, Dec 25 2016: (Start)
G.f.: (1 + 41*x + 51*x^2 + 3*x^3)/(1 - x)^4.
E.g.f.: (1 + 44*x + 68*x^2 + 16*x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A281090 Number of 2 X 2 matrices with all elements in {0,...,n} and prime permanent.

Original entry on oeis.org

0, 1, 27, 85, 139, 307, 399, 765, 1043, 1517, 1889, 3021, 3523, 5299, 6269, 7671, 9209, 12729, 14179, 18995, 21307, 24991, 28303, 36261, 39307, 47541, 52833, 61173, 67113, 82125, 86601, 104655, 114695, 128069, 139213, 156653, 165819, 194591, 209753, 230835, 245457, 283887
Offset: 0

Views

Author

Indranil Ghosh, Jan 20 2017

Keywords

Examples

			For n = 4, a few of the possible matrices are [0,1;3,3], [0,1;3,4], [0,2;1,0], [0,2;1,1], [0,2;1,2], [2,0;1,1], [2,0;2,1], [2,0;3,1], [2,0;4,1], [2,1;0,1], [4,3;1,1], [4,3;1,2], [4,3;1,4], [4,3;3,1], [4,3;3,2], [3,2;2,3], [3,2;4,1], [3,2;4,3], [3,3;0,1], [3,3;1,0], ... There are 139 possibilities. So, a(4) = 139.

Links

Crossrefs

Cf. A210000, A281315.

Programs

  • Python
    from sympy import isprime
def t(n):
    s=0
    for a in range(0, n+1):
        for b in range(0, n+1):
            for c in range(0, n+1):
                for d in range(0, n+1):
                    if isprime(a*d+b*c)==True:
                        s+=1
    return s
for i in range(0, 152):
    print(f"{i} {t(i)}")

A209974 a(n) = A209973(n)/4.

Original entry on oeis.org

0, 0, 3, 5, 9, 13, 19, 25, 33, 39, 51, 61, 69, 81, 99, 107, 123, 139, 157, 175, 191, 203, 233, 255, 271, 291, 327, 345, 369, 397, 421, 451, 483, 503, 551, 575, 599, 635, 689, 713, 745, 785, 821, 863, 903, 927, 993, 1039, 1071, 1113, 1173
Offset: 0

Views

Author

Clark Kimberling, Mar 16 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000.
Cf. A000010, A126246.

Programs

  • Mathematica
    (See the Mathematica section at A210000.)

Formula

Apparently, a(n) = a(n-1) + 2*A126246(n) - A000010(n) for n >= 2. - Pontus von Brömssen, Jun 28 2021

A209983 (A209982)/2.

Original entry on oeis.org

0, 10, 26, 58, 90, 154, 186, 282, 346, 442, 506, 666, 730, 922, 1018, 1146, 1274, 1530, 1626, 1914, 2042, 2234, 2394, 2746, 2874, 3194, 3386, 3674, 3866, 4314, 4442, 4922, 5178, 5498, 5754, 6138, 6330, 6906, 7194, 7578, 7834
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)

A209985 (A209984)/4.

Original entry on oeis.org

0, 1, 23, 39, 71, 103, 151, 199, 263, 311, 407, 487, 551, 647, 791, 855, 983, 1111, 1255, 1399, 1527, 1623, 1863, 2039, 2167, 2327, 2615, 2759, 2951, 3175, 3367, 3607, 3863, 4023, 4407, 4599, 4791, 5079, 5511, 5703, 5959
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)

A209987 (A209986)/8.

Original entry on oeis.org

0, 0, 3, 22, 30, 46, 66, 90, 106, 154, 170, 210, 250, 298, 322, 402, 434, 498, 546, 618, 650, 770, 810, 898, 978, 1058, 1106, 1250, 1298, 1410, 1490, 1610, 1674, 1874, 1938, 2034, 2130, 2274, 2346, 2586, 2650
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)
Previous Showing 81-90 of 100 results. Next

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