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.

Previous Showing 91-100 of 100 results.

A209989 (A209988)/4.

Original entry on oeis.org

0, 0, 13, 29, 91, 123, 171, 219, 315, 363, 459, 539, 667, 763, 907, 971, 1163, 1291, 1435, 1579, 1835, 1931, 2171, 2347, 2539, 2699, 2987, 3131, 3515, 3739, 3931, 4171, 4555, 4715, 5099, 5291, 5675, 5963, 6395, 6587, 6971
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.)

A209991 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant in {0,1}.

Original entry on oeis.org

1, 13, 38, 79, 136, 209, 302, 407, 536, 681, 846, 1015, 1240, 1441, 1678, 1951, 2240, 2505, 2854, 3151, 3552, 3945, 4326, 4687, 5216, 5657, 6110, 6615, 7192, 7649, 8342, 8831, 9472, 10105, 10702, 11407, 12272, 12857, 13526, 14279, 15224
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Crossrefs

Cf. A059306, A171503, A210000.

Programs

  • Mathematica
    a = 0; b = n; z1 = 40;
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, 0, 1}]
Table[c1[n, 1], {n, 0, z1}]    (* A209991 *)

Formula

A059306 + A171503.

A209993 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant in {-1,0,1}.

Original entry on oeis.org

1, 16, 45, 94, 159, 248, 349, 478, 623, 792, 973, 1182, 1423, 1672, 1933, 2238, 2559, 2888, 3261, 3630, 4063, 4504, 4925, 5374, 5935, 6456, 6957, 7534, 8159, 8728, 9453, 10062, 10767, 11480, 12141, 12942, 13855, 14584, 15325, 16174, 17183
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A000010, A018804, A210000.

Programs

  • Maple
    pillai:= proc(n) local i; add(igcd(i,n),i=1..n) end proc:
T:= 16: R:= 1,16:
for n from 2 to 50 do
  v:= 1 +  4*n + 8*numtheory:-phi(n) + 4*pillai(n);
  T:= T + v;
  R:= R,T;
od:
R; # Robert Israel, Jan 07 2024
  • Mathematica
    a = 1; b = n; z1 = 40;
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, 0, 1}]
Table[c1[n, 1], {n, 0, z1}]   (* A209992 *)

Formula

For n > 1, a(n) - a(n-1) = 1 + 4 * n + 8 * A000010(n) + 4 * A018804(n). - Robert Israel, Jan 07 2024

A209994 Number of 2 X 2 matrices with all elements in {1,2,...,n} and determinant in {-1,0,1}.

Original entry on oeis.org

0, 1, 10, 31, 60, 105, 154, 223, 300, 393, 490, 607, 748, 889, 1034, 1215, 1404, 1593, 1818, 2031, 2300, 2569, 2810, 3071, 3436, 3753, 4042, 4399, 4796, 5129, 5610, 5967, 6412, 6857, 7242, 7759, 8380, 8809, 9242, 9775, 10460
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 1; b = n; z1 = 40;
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, -m, m}]
Table[c1[n, 1], {n, 0, z1}]    (* A209994 *)

A209995 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant in {0,1,...,n}.

Original entry on oeis.org

1, 13, 50, 120, 244, 410, 681, 981, 1431, 1948, 2623, 3315, 4373, 5323, 6588, 8009, 9706, 11290, 13535, 15503, 18233, 20912, 23879, 26725, 30910, 34443, 38556, 42887, 48041, 52519, 58888, 63994, 70647, 77056, 83981, 91064, 100373
Offset: 0

Views

Author

Clark Kimberling, Mar 18 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 0; b = n; z1 = 40;
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, 0, m}]
Table[c1[n, n], {n, 0, z1}]   (* A209995 *)

A209997 (A209990)/8.

Original entry on oeis.org

0, 0, 1, 8, 16, 51, 59, 83, 99, 123, 159, 199, 215, 263, 287, 359, 391, 455, 479, 551, 623, 671, 711, 799, 831, 991, 1039, 1111, 1159, 1271, 1343, 1463, 1527, 1607, 1671, 1887, 1935, 2079, 2151, 2247, 2391
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Crossrefs

Cf. A210000, A209990.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)

A211069 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant in [-n,n].

Original entry on oeis.org

1, 14, 57, 150, 309, 574, 921, 1444, 2091, 2952, 3919, 5314, 6709, 8534, 10603, 13102, 15593, 18962, 22133, 26332, 30569, 35346, 40097, 46690, 52553, 59394, 66603, 75094, 82833, 93282, 102181, 113422, 124411, 136412, 148613
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 1; b = n; z1 = 35;
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, -n, m}]
Table[c1[n, n], {n, 1, z1}]  (* A211069 *)

A211070 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant d satisfying -n < d < n.

Original entry on oeis.org

1, 10, 43, 118, 263, 474, 831, 1258, 1893, 2652, 3685, 4748, 6375, 7918, 9931, 12216, 15015, 17654, 21395, 24726, 29253, 33822, 39011, 43906, 50995, 57210, 64431, 71968, 81075, 88962, 100159, 109346, 121107, 132636, 145097
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 1; b = n; z1 = 35;
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, -n, m}]
Table[c1[n, n - 1] - c1[n, -n], {n, 1, z1}] (* A211070 *)

A211147 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and nonnegative determinant.

Original entry on oeis.org

1, 57, 377, 1345, 3553, 7737, 14937, 26177, 42945, 66681, 99193, 142209, 198241, 269049, 357593, 466433, 598401, 756281, 944057, 1164289, 1421601, 1719161, 2061081, 2451329, 2895489, 3396729, 3960569, 4591937, 5296289, 6077881
Offset: 0

Views

Author

Clark Kimberling, Apr 03 2012

Keywords

Comments

It appears that all terms are of the form 8*k + 1.
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, 0, m}]
Table[c1[n, 2 n^2], {n, 0, z1}]

A281550 Number of 2 X 2 matrices with all elements in 0..n such that the sum of the elements is prime.

Original entry on oeis.org

0, 10, 46, 114, 234, 458, 826, 1370, 2090, 3010, 4174, 5658, 7534, 9930, 12954, 16662, 21074, 26242, 32246, 39182, 47186, 56386, 66874, 78798, 92290, 107434, 124282, 142942, 163550, 186266, 211250, 238626, 268526, 301134, 336610, 375086, 416678, 461454, 509434, 560662, 615182, 673106
Offset: 0

Views

Author

Indranil Ghosh, Jan 23 2017

Keywords

Examples

			For n = 4, a few of the possible matrices are [0,4;2,1], [0,4;3,0], [0,4;3,4], [0,4;4,3], [1,0;0,1], [1,0;0,2], [1,0;0,4], [1,0;1,0], [1,0;1,1], [1,0;1,3], [2,2;3,0], [2,2;3,4], [2,2;4,3], [2,3;0,0], [2,3;0,2], [3,4;3,3], [3,4;4,0], [3,4;4,2], [4,0;0,1], [4,0;0,3], [4,0;1,0], ... There are 234 possibilities.
Here each of the matrices M is defined as M = [a,b;c,d] where a = M[1][1], b = M[1][2], c = M[2][1], d = M[2][2]. So, a(4) = 234.

Links

Crossrefs

Cf. A210000, A281090, A281315.

Programs

  • PARI
    a(n)=my(X=Pol(vector(n+1,i,1))+O('x^(4*n)),Y=X^4,s); forprime(p=2,4*n, s+=polcoeff(Y,p)); s \\ Charles R Greathouse IV, Feb 15 2017
  • 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+b+c+d)==True:
                        s+=1
    return s
for i in range(0, 201):
    print(str(i)+" "+str(t(i)))

Formula

a(n) = Sum_{p prime} Sum_{k=0..4} (-1)^k * binomial(4, k) * binomial(p+3-k*(n+1), 3). - David Radcliffe, Jun 13 2025
Previous Showing 91-100 of 100 results.

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