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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A211436 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w+3x+4y=0.

Original entry on oeis.org

1, 1, 7, 11, 21, 27, 43, 51, 73, 83, 109, 123, 155, 169, 207, 225, 267, 287, 335, 357, 411, 435, 493, 521, 585, 613, 683, 715, 789, 823, 903, 939, 1025, 1063, 1153, 1195, 1291, 1333, 1435, 1481, 1587, 1635, 1747, 1797, 1915, 1967, 2089, 2145
Offset: 0

Views

Author

Clark Kimberling, Apr 11 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[2 w + 3 x + 4 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211436 *)
(t - 1)/2                    (* integers *)

Formula

Empirical g.f.: (1 + x + 6*x^2 + 9*x^3 + 12*x^4 + 9*x^5 + 6*x^6 + x^7 + x^8) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)). - Colin Barker, May 15 2017

A211437 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w*x*y=n.

Original entry on oeis.org

1, 4, 12, 12, 24, 12, 36, 12, 40, 24, 36, 12, 72, 12, 36, 36, 60, 12, 72, 12, 72, 36, 36, 12, 120, 24, 36, 40, 72, 12, 108, 12, 84, 36, 36, 36, 144, 12, 36, 36, 120, 12, 108, 12, 72, 72, 36, 12, 180, 24, 72, 36, 72, 12, 120, 36, 120, 36, 36, 12, 216, 12, 36, 72
Offset: 0

Views

Author

Clark Kimberling, Apr 12 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422, A007425.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w*x*y - n, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211437 *)
t/4                  (* A007425 for n>0 *)

A211438 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+2y=0.

Original entry on oeis.org

1, 3, 13, 19, 39, 49, 79, 93, 133, 151, 201, 223, 283, 309, 379, 409, 489, 523, 613, 651, 751, 793, 903, 949, 1069, 1119, 1249, 1303, 1443, 1501, 1651, 1713, 1873, 1939, 2109, 2179, 2359, 2433, 2623, 2701, 2901, 2983, 3193, 3279, 3499, 3589
Offset: 0

Views

Author

Clark Kimberling, Apr 11 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w + 2 x + 2 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211438 *)
(t - 1)/2                    (* A118277  *)

Formula

Conjecture: a(n) = (5+3*(-1)^n+2*(7+3*(-1)^n)*n+14*n^2)/8. G.f.: (1+2*x+8*x^2+2*x^3+x^4)/((1-x)^3*(1+x)^2). [Colin Barker, Apr 18 2012]

A211439 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+3x+3y=0.

Original entry on oeis.org

1, 3, 5, 19, 25, 31, 59, 69, 79, 121, 135, 149, 205, 223, 241, 311, 333, 355, 439, 465, 491, 589, 619, 649, 761, 795, 829, 955, 993, 1031, 1171, 1213, 1255, 1409, 1455, 1501, 1669, 1719, 1769, 1951, 2005, 2059, 2255, 2313, 2371, 2581, 2643, 2705
Offset: 0

Views

Author

Clark Kimberling, Apr 11 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w + 3 x + 3 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211439 *)
(t - 1)/2                    (* integers *)

Formula

Conjectures from Colin Barker, May 15 2017: (Start)
G.f.: (1 + 2*x + 2*x^2 + 12*x^3 + 2*x^4 + 2*x^5 + x^6) / ((1 - x)^3*(1 + x + x^2)^2).
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>6.
(End)

A211440 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and 2w+3x+3y=0.

Original entry on oeis.org

1, 3, 5, 17, 23, 29, 53, 63, 73, 109, 123, 137, 185, 203, 221, 281, 303, 325, 397, 423, 449, 533, 563, 593, 689, 723, 757, 865, 903, 941, 1061, 1103, 1145, 1277, 1323, 1369, 1513, 1563, 1613, 1769, 1823, 1877, 2045, 2103, 2161, 2341, 2403, 2465
Offset: 0

Views

Author

Clark Kimberling, Apr 11 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Links

Crossrefs

Cf. A211422.

Programs

  • Maple
    seq(op([10*j^2+6*j+1, 10*j^2 + 10*j + 3, 10*j^2 + 14*j + 5]),j=0..30); # Robert Israel, Apr 03 2019
  • Mathematica
    t[n_] := t[n] = Flatten[Table[2 w + 3 x + 3 y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 30}]  (* A211440 *)
(t - 1)/2                    (* integers *)
LinearRecurrence[{1,0,2,-2,0,-1,1},{1,3,5,17,23,29,53},60] (* Harvey P. Dale, Aug 29 2021 *)

Formula

Conjectures from Colin Barker, May 15 2017: (Start)
G.f.: (1 + 3*x + x^2)*(1 - x + 4*x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^2).
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>6.
(End)
From Robert Israel, Apr 03 2019: (Start)
a(3*j) = 10*j^2+6*j+1.
a(3*j+1) = 10*j^2 + 10*j + 3.
a(3*j+2) = 10*j^2 + 14*j + 5.
This has the conjectured g.f. and recurrence. (End)

A211481 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w+2x+3y=n.

Original entry on oeis.org

1, 3, 8, 16, 26, 38, 54, 71, 91, 114, 139, 166, 197, 229, 264, 302, 342, 384, 430, 477, 527, 580, 635, 692, 753, 815, 880, 948, 1018, 1090, 1166, 1243, 1323, 1406, 1491, 1578, 1669, 1761, 1856, 1954, 2054, 2156, 2262, 2369, 2479, 2592, 2707
Offset: 0

Views

Author

Clark Kimberling, Apr 12 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w + 2 x + 3 y - n, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211481 *)

Formula

Conjecture: a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6) for n>5. - Colin Barker, Nov 08 2014
Empirical g.f.: -(3*x^4+5*x^3+4*x^2+2*x+1) / ((x-1)^3*(x+1)*(x^2+x+1)). - Colin Barker, Nov 08 2014

A211482 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w*x+w*y+x*y=w*x*y.

Original entry on oeis.org

1, 10, 22, 35, 50, 62, 80, 92, 104, 116, 128, 140, 152, 164, 176, 188, 200, 212, 224, 236, 248, 260, 272, 284, 296, 308, 320, 332, 344, 356, 368, 380, 392, 404, 416, 428, 440, 452, 464, 476, 488, 500, 512, 524, 536, 548, 560, 572, 584, 596, 608, 620
Offset: 0

Views

Author

Clark Kimberling, Apr 12 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w*x + w*y + x*y - w*x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211482 *)

Formula

Conjectures from Colin Barker, May 15 2017: (Start)
G.f.: (1 + 8*x + 3*x^2 + x^3 + 2*x^4 - 3*x^5 + 6*x^6 - 6*x^7) / (1 - x)^2.
a(n) = 12*n + 8 for n>5.
a(n) = 2*a(n-1) - a(n-2) for n>7.
(End)

A211483 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and (w+n)^2=x+y.

Original entry on oeis.org

1, 5, 10, 16, 22, 30, 38, 46, 55, 65, 75, 85, 95, 107, 119, 131, 143, 155, 168, 182, 196, 210, 224, 238, 252, 268, 284, 300, 316, 332, 348, 364, 381, 399, 417, 435, 453, 471, 489, 507, 525, 545, 565, 585, 605, 625, 645, 665, 685, 705, 726, 748, 770
Offset: 0

Views

Author

Clark Kimberling, Apr 12 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[(n + w)^2 - x - y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211483 *)

A211507 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=x*y-n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 3, 9, 9, 4, 17, 11, 7, 18, 16, 19, 21, 16, 21, 35, 24, 8, 35, 42, 17, 44, 40, 25, 47, 28, 47, 56, 28, 30, 76, 69, 11, 52, 68, 54, 81, 36, 33, 99, 61, 38, 99, 77, 51, 76, 88, 63, 67, 92, 88, 140, 40, 12, 153, 90, 71, 98, 121, 98, 104, 100, 41
Offset: 0

Views

Author

Clark Kimberling, Apr 14 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w^2 - x*y + n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211507 *)

A211508 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=n-x*y.

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 4, 6, 5, 6, 8, 8, 8, 12, 8, 10, 12, 12, 11, 16, 12, 16, 16, 12, 16, 20, 14, 16, 22, 20, 16, 24, 17, 22, 24, 16, 22, 30, 16, 24, 30, 26, 20, 28, 26, 28, 32, 18, 26, 38, 21, 30, 36, 28, 28, 34, 30, 36, 34, 26, 34, 46, 20, 28, 46, 36, 32, 40, 30, 42, 40
Offset: 0

Views

Author

Clark Kimberling, Apr 14 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w^2 + x*y - n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211508 *)
Previous Showing 41-50 of 105 results. Next

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