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.

Showing 1-10 of 100 results. Next

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.

Original entry on oeis.org

0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413
Offset: 0

Views

Author

Clark Kimberling, Apr 27 2012

Keywords

Comments

Each sequence in the following guide counts 4-tuples
(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}. The notation "m div" means that m divides every term of the sequence.
A211058 ... wx <= yz
A211787 ... wx <= 2yz
A211795 ... wx < 2yz
A211797 ... wx > 2yz
A211809 ... wx >= 2yz
A211812 ... wx <= 3yz
A211917 ... wx < 3yz
A211918 ... wx > 3yz
A211919 ... wx >= 3yz
A211920 ... 2wx < 3yz
A211921 ... 2wx <= 3yz
A211922 ... 2wx > 3yz
A211923 ... 2wx >= 3yz
A212019 ... wx = 2yz ..... 2 div
A212020 ... wx = 3yz ..... 2 div
A212021 ... 2wx = 3yz .... 2 div
A212047 ... wx = 4yz
A212048 ... 3wx = 4yz .... 2 div
A212049 ... wx = 5yz ..... 2 div
A212050 ... 2wx = 5yz .... 2 div
A212051 ... 3wx = 5yz .... 2 div
A212052 ... 4wx = 5yz .... 2 div
A209978 ... wx = yz + 1 .. 2 div
A212053 ... wx <= yz + 1
A212054 ... wx > yz + 1
A212055 ... wx <= yz + 2
A212056 ... wx > yz + 2
A197168 ... wx = yz + 2 .. 2 div
A061201 ... w = xyz
A212057 ... w < xyz
A212058 ... w >= xyz
A212059 ... w = xyz - 1
A212060 ... w = xyz - 2
A212061 ... wx = (yz)^2
A212062 ... w^2 = xyz
A212063 ... w^2 < xyz
A212064 ... w^2 >= xyz
A212065 ... w^2 <= xyz
A212066 ... w^2 > xyz
A212067 ... w^3 = xyz
A002623 ... w = 2x + y + z
A006918 ... w = 2x + 2y + z
A000601 ... w = x + 2y + 3z (except for initial 0's)
A212068 ... 2w = x + y + z
A212069 ... 3w = x + y + z (w = average{x,y,z})
A212088 ... 3w < x + y + z
A212089 ... 3w >= x + y + z
A212090 ... w < x + y + z
A000332 ... w >= x + y + z
A212145 ... w < 2x + y + z
A001752 ... w >= 2x + y + z
A001400 ... w = 2x +3y + 4z
A005900 ... w = -x + y + z
A192023 ... w = -x + y + z + 2
A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div
A212087 ... w^2 + x^2 = y^2 + z^2
A212092 ... w^2 < x^2 + y^2 + z^2
A212093 ... w^2 <= x^2 + y^2 + z^2
A212094 ... w^2 > x^2 + y^2 + z^2
A212095 ... w^2 >= x^2 + y^2 + z^2
A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div
A212097 ... w^3 < x^3 + y^3 + z^3
A212098 ... w^3 <= x^3 + y^3 + z^3
A212099 ... w^3 > x^3 + y^3 + z^3
A212100 ... w^3 >= x^3 + y^3 + z^3
A212101 ... wx^2 = yz^2
A212102 ... 1/w = 1/x + 1/y + 1/z
A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}
A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.
A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.
A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.
A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.
A212133 ... median(w,x,y,z) = mean(w,x,y,z)
A212134 ... median(w,x,y,z) <= mean(w,x,y,z)
A212135 ... median(w,x,y,z) > mean(w,x,y,z)
A212241 ... wx + yz > n
A212243 ... 2wx + yz = n
A212244 ... w = xyz - n
A212245 ... w = xyz - 2n
A212246 ... 2w = x + y + z - n
A212247 ... 3w = x + y + z + n
A212249 ... 3w < x + y + z + n
A212250 ... 3w >= x + y + z + n
A212251 ... 3w = x + y + z + n + 1
A212252 ... 3w = x + y + z + n + 2
A212254 ... w = x + 2y + 3z - n
A212255 ... w^2 = mean(x^2, y^2, z^2)
A212256 ... 4/w = 1/x + 1/y +1/z + 1/n
In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent. In the list below, the same is true for expressions involving more than one relation.
A000332 ... w < x <= y < z .... C(n,4)
A000914 ... w < x <= y < z .... Stirling 1st kind
A000914 ... w < x <= y >= z ... Stirling 1st kind
A050534 ... w < x < y >= z .... tritriangular
A001296 ... w <= x <= y >= z .. 4-dim pyramidal
A006322 ... x < x > y >= z
A002418 ... w < x >= y < z
A050534 ... w < x >=y >= z
A212415 ... w < x >= y <= z
A001296 ... w < x >= y <= z
A212246 ... w <= x > y <= z
A006322 ... w <= x >= y <= z
A212501 ... w > x < y >= z
A212503 ... w < 2x and y < 2z ..... A (note below)
A212504 ... w < 2x and y > 2z ..... A
A212505 ... w < 2x and y >= 2z .... A
A212506 ... w <= 2x and y <= 2z ... A
A212507 ... w < 2x and y <= 2z .... B
A212508 ... w < 2x and y < 3z ..... C
A212509 ... w < 2x and y <= 3z .... C
A212510 ... w < 2x and y > 3z ..... C
A212511 ... w < 2x and y >= 3z .... C
A212512 ... w <= 2x and y < 3z .... C
A212513 ... w <= 2x and y <= 3z ... C
A212514 ... w <= 2x and y > 3z .... C
A212515 ... w <= 2x and y >= 3z ... C
A212516 ... w > 2x and y < 3z ..... C
A212517 ... w > 2x and y <= 3z .... C
A212518 ... w > 2x and y > 3z ..... C
A212519 ... w > 2x and y >= 3z .... C
A212520 ... w >= 2x and y < 3z .... C
A212521 ... w >= 2x and y <= 3z ... C
A212522 ... w >= 2x and y > 3z .... C
A212523 ... w + x < y + z
A212560 ... w + x <= y + z
A212561 ... w + x = 2y + 2z
A212562 ... w + x < 2y + 2z ....... B
A212563 ... w + x <= 2y + 2z ...... B
A212564 ... w + x > 2y + 2z ....... B
A212565 ... w + x >= 2y + 2z ...... B
A212566 ... w + x = 3y + 3z
A212567 ... 2w + 2x = 3y + 3z
A212570 ... |w - x| = |x - y| + |y - z|
A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div
A212572 ... |w - x| <= |x - y| + |y - z| .. B
A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div
A212574 ... |w - x| >= |x - y| + |y - z| .. B
A212575 ... 2|w - x| = |x - y| + |y - z|
A212576 ... |w - x| = 2|x - y| + 2|y - z|
A212577 ... |w - x| = 2|x - y| - |y - z|
A212578 ... 2|w - x| = |x - y| - |y - z|
A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}
A212692 ... w = |x - y| + |y - z| ............... 2 div
A212568 ... w < |x - y| + |y - z| ............... 2 div
A212573 ... w <= |x - y| + |y - z| .............. 2 div
A212574 ... w > |x - y| + |y - z|
A212575 ... w >= |x - y| + |y - z|
A212676 ... w + x = |x - y| + |y - z| ......... H
A212677 ... w + y = |x - y| + |y - z|
A212678 ... w + x + y = |x - y| + |y - z|
A006918 ... w + x + y + z = |x - y| + |y - z| . H
A212679 ... |x - y| = |y - z| ................. H
A212680 ... |x - y| = |y - z| + 1 ..............H 2 div
A212681 ... |x - y| < |y - z| ................... 2 div
A212682 ... |x - y| >= |y - z|
A212683 ... |x - y| = w + |y - z| ............... 2 div
A212684 ... |x - y| = n - w + |y - z|
A212685 ... |w - x| = w + |y - z|
A186707 ... |w - x| < w + |y - z| ... (Note D)
A212714 ... |w - x| >= w + |y - z| .......... H . 2 div
A212686 ... 2*|w - x| = n + |y - z| ............. 4 div
A212687 ... 2*|w - x| < n + |y - z| ......... B
A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div
A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div
A212690 ... 2*|w - x| <= n + |y - z| ........ B
A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div
...
In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.
R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.
A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A
A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A
A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A
A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div
A212744 ... w=range (=max-min) ............... E
A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}
A212746 ... R is in {w,x,y,z} ................ E
A212569 ... R is not in {w,x,y,z} ............ E
A212749 ... w=R or x
A212750 ... w=R or x=R or y
A212751 ... w=R or x=R or y
A212752 ... wR ......... A
A212753 ... wR or z>R ......... D
A212754 ... wR or y>R or z>R ......... D
A002415 ... w = x + R ........................ D
A212755 ... |w - x| = R ...................... D
A212756 ... 2w = x + R
A212757 ... 2w = R
A212758 ... w = floor(R/2)
A002413 ... w = floor((x+y+z/2))
A212759 ... w, x, y are even
A212760 ... w is even and x = y + z .......... E
A212761 ... w is odd and x and y are even .... F . 2 div
A212762 ... w and x are odd y is even ........ F . 2 div
A212763 ... w, x, y are odd .................. F
A212764 ... w, x, y are even and z is odd .... F
A030179 ... w and x are even and y and z odd
A212765 ... w is even and x,y,z are odd ...... F
A212766 ... w is even and x is odd ........... A . 2 div
A212767 ... w and x are even and w+x=y+z ..... E
A212889 ... R is even ........................ A
A212890 ... R is odd ......................... A . 2 div
A212742 ... w-x, x-y, y-z are all even ....... A
A212892 ... w-x, x-y, y-z are all odd ........ A
A212893 ... w-x, x-y, y-z have same parity ... A
A005915 ... min{|w-x|, |x-y|, |y-z|} = 0
A212894 ... min{|w-x|, |x-y|, |y-z|} = 1
A212895 ... min{|w-x|, |x-y|, |y-z|} = 2
A179824 ... min{|w-x|, |x-y|, |y-z|} > 0
A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1
A212897 ... min{|w-x|, |x-y|, |y-z|} > 1
A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2
A212899 ... min{|w-x|, |x-y|, |y-z|} > 2
A212901 ... |w-x| = |x-y| = |y-z|
A212900 ... |w-x|, |x-y|, |y-z| are distinct . G
A212902 ... |w-x| < |x-y| < |y-z| ............ G
A212903 ... |w-x| <= |x-y| <= |y-z| .......... G
A212904 ... |w-x| + |x-y| + |y-z| = n ........ H
A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H
...
Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.
B: 3,-1,-5,5,1,-3,1
C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1
D: 4,-5,0,5,-4,1
E: 1,3,-3,-3,3,1,-1
F: 1,4,-4,-6,6,4,-4,-1,1
G: 2,1,-3,-1,1,3,-1,-2,1
H: 2,1,-4,1,2,-1

Examples

			a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).

References

  • A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
  • P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

Links

Crossrefs

Cf. A000583 (n^4), A210000, A211809, A212959.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},
      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211795 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Formula

a(n) = n^4 - A211809(n).

A209981 Number of singular 2 X 2 matrices having all elements in {-n,...,n}.

Original entry on oeis.org

1, 33, 129, 289, 545, 833, 1313, 1729, 2369, 3041, 3905, 4577, 5857, 6657, 7905, 9345, 10881, 11937, 13953, 15137, 17441, 19521, 21537, 22977, 26177, 28257, 30657, 33249, 36577, 38401, 42721, 44673, 48257, 51617, 54785, 58529, 63905
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Examples

			Among the 33 matrices counted by a(1) are these (in compact notation):
(-1,-1,-1,-1), (0,0,0,0), (1,-1,-1,1), (1,1,1,1).

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = -n; 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]
Table[c[n, 0], {n, 0, z1}]  (* A209981 *)
Table[c[n, 1], {n, 0, z1}]  (* A209982 *)
%/4                         (* A206258 *)
2 %                         (* A209983 *)
Table[c[n, 2], {n, 0, z1}]  (* A209984 *)
%/4                         (* A209985 *)
Table[c[n, 3], {n, 0, z1}]  (* A209986 *)
%/8                         (* A209987 *)
Table[c[n, 4], {n, 0, z1}]  (* A209988 *)
%/4                         (* A209989 *)
Table[c[n, 5], {n, 0, z1}]  (* A209990 *)
%/8                         (* A209997 *)

Formula

a(n) = 8*A134506(n) + (4*n + 1)^2. - Andrew Howroyd, May 04 2020

A209978 a(n) = A196227(n)/2.

Original entry on oeis.org

0, 0, 1, 4, 7, 14, 17, 28, 35, 46, 53, 72, 79, 102, 113, 128, 143, 174, 185, 220, 235, 258, 277, 320, 335, 374, 397, 432, 455, 510, 525, 584, 615, 654, 685, 732, 755, 826, 861, 908, 939, 1018, 1041, 1124, 1163, 1210, 1253, 1344, 1375, 1458, 1497
Offset: 0

Views

Author

Clark Kimberling, Mar 16 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Crossrefs

Cf. A196227, A210000.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)-1 + 2*numtheory[phi](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, May 05 2020
  • Mathematica
    a = 1; b = n; z1 = 50;
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]
Table[c[n, 0], {n, 0, z1}]  (* A134506 *)
Table[c[n, 1], {n, 0, z1}]  (* A196227 *)
%/2                         (* A209978 *)
Table[2 c[n, 1], {n, 0, z1}](* A209979 *)
Table[c[n, 2], {n, 0, z1}]  (* A197168 *)
%/2                         (* A209980 *)
Table[c[n, 3], {n, 0, z1}]  (* A210001 *)
Table[c[n, 4], {n, 0, z1}]  (* A210002 *)
Table[c[n, 5], {n, 0, z1}]  (* A210027 *)

A210369 Number of 2 X 2 matrices with all terms in {0,1,...,n} and even determinant.

Original entry on oeis.org

1, 10, 65, 160, 457, 810, 1681, 2560, 4481, 6250, 9841, 12960, 18985, 24010, 33377, 40960, 54721, 65610, 84961, 100000, 126281, 146410, 181105, 207360, 252097, 285610, 342161, 384160, 454441, 506250, 592321, 655360, 759425, 835210, 959617, 1049760
Offset: 0

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

a(n) is also the number of 2 X 2 matrices with all terms in {0,1,...n} and even permanent.
The determinant will be even if either all entries are odd or if both the leading and trailing diagonals have no more than one odd entry each. - Andrew Howroyd, Apr 28 2020

Links

Crossrefs

Cf. A210000, A210370.

Programs

  • Mathematica
    a = 0; b = n; z1 = 28;
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]
u[n_] := Sum[c[n, 2 k], {k, -n^2, n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -n^2, n^2}]
Table[u[n], {n, 0,  z1}] (* A210369 *)
Table[v[n], {n, 0, z1}]  (* A210370 *)
  • PARI
    a(n) = {((n+1)^2 - ceil(n/2)^2)^2 + ceil(n/2)^4} \\ Andrew Howroyd, Apr 28 2020

Formula

a(n) + A210370(n) = n^4.
From Colin Barker, Nov 28 2014: (Start)
a(n) = (13 + 3*(-1)^n + 4*(13+3*(-1)^n)*n + 2*(37+7*(-1)^n)*n^2 + 4*(11+(-1)^n)*n^3 + 10*n^4)/16.
G.f.: -(x^7+9*x^6+27*x^5+83*x^4+59*x^3+51*x^2+9*x+1) / ((x-1)^5*(x+1)^4).
(End)
a(n) = ((n+1)^2 - ceiling(n/2)^2)^2 + ceiling(n/2)^4. - Andrew Howroyd, Apr 28 2020

Extensions

Terms a(29) and beyond from Andrew Howroyd, Apr 28 2020

A210370 Number of 2 X 2 matrices with all elements in {0,1,...,n} and odd determinant.

Original entry on oeis.org

0, 6, 16, 96, 168, 486, 720, 1536, 2080, 3750, 4800, 7776, 9576, 14406, 17248, 24576, 28800, 39366, 45360, 60000, 68200, 87846, 98736, 124416, 138528, 171366, 189280, 230496, 252840, 303750, 331200, 393216, 426496, 501126, 541008, 629856, 677160, 781926, 837520
Offset: 0

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

a(n) is also the number of 2 X 2 matrices with all elements in {0,1,...n} and odd permanent.

Links

Crossrefs

Cf. A210000, A210369.

Programs

  • Mathematica
    a = 0; b = n; z1 = 28;
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]
u[n_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 0,  z1}] (* A210369 *)
Table[v[n], {n, 0, z1}](* A210370 *)
  • PARI
    a(n)={2*((n+1)^2-ceil(n/2)^2)*ceil(n/2)^2} \\ Andrew Howroyd, Apr 28 2020

Formula

a(n) + A210369(n) = n^4.
From Colin Barker, Nov 28 2014: (Start)
a(n) = (3 - 3*(-1)^n - 12*(-1+(-1)^n)*n + (22-14*(-1)^n)*n^2 - 4*(-5+(-1)^n)*n^3 + 6*n^4)/16.
G.f.: -2*x*(3*x^5+17*x^4+16*x^3+28*x^2+5*x+3) / ((x-1)^5*(x+1)^4).
(End)
a(n) = 2*((n+1)^2 - ceiling(n/2)^2)*ceiling(n/2)^2. - Andrew Howroyd, Apr 28 2020

Extensions

Terms a(29) and beyond from Andrew Howroyd, Apr 28 2020

A196227 Number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is 1.

Original entry on oeis.org

0, 0, 2, 8, 14, 28, 34, 56, 70, 92, 106, 144, 158, 204, 226, 256, 286, 348, 370, 440, 470, 516, 554, 640, 670, 748, 794, 864, 910, 1020, 1050, 1168, 1230, 1308, 1370, 1464, 1510, 1652, 1722, 1816, 1878, 2036, 2082, 2248, 2326, 2420, 2506, 2688, 2750, 2916, 2994
Offset: 0

Views

Author

Aldo González Lorenzo, Sep 29 2011

Keywords

Comments

It is also the number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is -1.

Links

Crossrefs

Cf. A171503 (determinants of matrices that include zero), A209978, A210000.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)-2 + 4*numtheory[phi](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, May 05 2020
  • Mathematica
    Table[cnt = 0; Do[If[a*d-b*c == 1, cnt++], {a, n}, {b, n}, {c, n}, {d, n}]; cnt, {n, 50}] (* T. D. Noe, Oct 11 2011 *)
  • PARI
    a(n) = if(n < 1, 0, 4*sum(k=1, n, eulerphi(k)) - 2*(n + 1)) \\ Andrew Howroyd, May 05 2020

Formula

From Andrew Howroyd, May 05 2020: (Start)
a(n) = A171503(n) - (2*n + 1) for n > 0.
a(n) = -2*(n + 1) + 4*Sum_{k=1..n} phi(k) for n > 0.
a(n) = 2 * A209978(n). (End)

Extensions

a(0)=0 prependend by Andrew Howroyd, May 05 2020

A210378 Number of 2 X 2 matrices with all terms in {0,1,...,n} and even trace.

Original entry on oeis.org

1, 8, 45, 128, 325, 648, 1225, 2048, 3321, 5000, 7381, 10368, 14365, 19208, 25425, 32768, 41905, 52488, 65341, 80000, 97461, 117128, 140185, 165888, 195625, 228488, 266085, 307328, 354061, 405000, 462241, 524288, 593505, 668168, 750925, 839808, 937765, 1042568
Offset: 0

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Examples

			Writing the matrices as 4-letter words, the 8 for n=1 are as follows:
0000, 0100, 0010, 0110, 1001, 1101, 1011, 1111

Links

Crossrefs

Cf. A210000, A210379.
See A210000 for a guide to related sequences.

Programs

  • Mathematica
    a = 0; b = n; z1 = 35;
t[n_] := t[n] = Flatten[Table[w + z, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
u[n_] := Sum[c[n, 2 k], {k, 0, 2*n}]
v[n_] := Sum[c[n, 2 k - 1], {k, 1, 2*n - 1}]
Table[u[n], {n, 0, z1}] (* A210378 *)
Table[v[n], {n, 0, z1}] (* A210379 *)

Formula

a(n) + A210379(n) = (n+1)^4.
From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = (n + 1)^2*((2*n + 1 -(-1)^n)^2 + (2*n + 3 + (-1)^n)^2)/16.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 7.
G.f.: (-x^6 - 6*x^5 - 27*x^4 - 28*x^3 - 27*x^2 - 6*x - 1)/((x - 1)^5*(x + 1)^3). (End)
From Amiram Eldar, Mar 15 2024: (Start)
a(n) = (n+1)^2*floor(((n+1)^2+1)/2).
Sum_{n>=0} 1/a(n) = Pi^4/720 + (Pi-2*tanh(Pi/2))*Pi/4. (End)
E.g.f.: ((2 + 15*x + 26*x^2 + 10*x^3 + x^4)*cosh(x) + (1 + 18*x + 25*x^2 + 10*x^3 + x^4)*sinh(x))/2. - Stefano Spezia, Jul 15 2024

A211059 Number of 2 X 2 matrices having all terms in {1,...,n} and positive determinant.

Original entry on oeis.org

0, 5, 33, 112, 288, 605, 1145, 1968, 3176, 4861, 7161, 10152, 14040, 18917, 24961, 32352, 41312, 51949, 64585, 79320, 96472, 116277, 139025, 164840, 194184, 227261, 264385, 305840, 352096, 403245, 459945, 522312, 590840, 665917
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000, A211056.

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, 0, m}]
Table[c1[n, n^2] - c[n, 0], {n, 1, z1}]   (* A211059 *)
2*%     (* A211056 *)

A211064 Number of 2 X 2 matrices having all terms in {1,...,n} and even determinant.

Original entry on oeis.org

1, 10, 41, 160, 337, 810, 1345, 2560, 3761, 6250, 8521, 12960, 16801, 24010, 30017, 40960, 49825, 65610, 78121, 100000, 117041, 146410, 168961, 207360, 236497, 285610, 322505, 384160, 430081, 506250, 562561, 655360, 723521, 835210
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Links

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]
u[n_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 1, z1}] (* A211064 *)
Table[v[n], {n, 1, z1}] (* A211065 *)

Formula

a(n) + A211065(n) = n^4.
From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = n^4 - (2*n + 1 -(-1)^n)^2*(6*n + 1 -(-1)^n)*(2*n - 1 + (-1)^n)/128.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: x*(-x^7 - 9*x^6 - 51*x^5 - 59*x^4 - 83*x^3 - 27*x^2 - 9*x - 1)/((x - 1)^5*(x + 1)^4). (End)

A209973 Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant 2.

Original entry on oeis.org

0, 0, 12, 20, 36, 52, 76, 100, 132, 156, 204, 244, 276, 324, 396, 428, 492, 556, 628, 700, 764, 812, 932, 1020, 1084, 1164, 1308, 1380, 1476, 1588, 1684, 1804, 1932, 2012, 2204, 2300, 2396, 2540, 2756, 2852, 2980, 3140, 3284, 3452, 3612, 3708
Offset: 0

Views

Author

Clark Kimberling, Mar 16 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A210000.)

Formula

a(n) = A197168(n) + 4*n + 2 for n >= 2. - Chai Wah Wu, Nov 28 2016
Showing 1-10 of 100 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.