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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A211509 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=2n-x*y.

Original entry on oeis.org

0, 1, 0, 2, 3, 4, 4, 4, 8, 7, 6, 10, 10, 8, 16, 10, 11, 16, 14, 8, 22, 12, 18, 24, 18, 13, 26, 18, 20, 24, 24, 10, 36, 22, 20, 30, 29, 20, 42, 20, 22, 30, 36, 24, 44, 24, 28, 40, 38, 15, 48, 26, 34, 48, 38, 18, 50, 38, 42, 38, 44, 18, 66, 42, 35, 50, 38, 32, 68, 24, 36
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 - 2 n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211509 *)

A211510 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2 = x*y - 2n.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 1, 6, 5, 4, 13, 0, 16, 12, 7, 8, 22, 10, 27, 20, 20, 8, 41, 14, 27, 32, 21, 36, 66, 0, 28, 38, 40, 36, 71, 12, 53, 60, 57, 16, 83, 14, 80, 60, 32, 64, 75, 50, 98, 62, 47, 16, 144, 36, 100, 88, 53, 52, 153, 36, 94, 76, 91, 98, 129, 20, 92, 124, 102
Offset: 0

Views

Author

Clark Kimberling, Apr 14 2012

Keywords

Comments

For a guide to related sequences, see A211422.
The original name was "... and w^2 = x*y + 2n", but this would yield 2 instead of 0 for a(3), as observed by Pontus von Brömssen. The corresponding sequence seems not to be in the OEIS yet. - M. F. Hasler, Jan 26 2020

Examples

			From _Bernard Schott_, Jan 26 2020: (Start)
For n = 4, there are 3 ordered solutions with (1,3,3), (2,3,4) and (2,4,3) so a(4) = 3.
For n = 5, there is no solution, hence a(5) = 0.
The only solution for n = 6 is (2,4,4) with 2^2 = 4*4 - 2*6, hence a(6) = 1. (End)

Links

Crossrefs

Cf. A211422.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w^2 - x*y + 2 n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211510 *)
  • PARI
    apply( {A211510(n)=sum(w=1,n-2,my(w2n=(w^2-1)\n+2,s); fordiv(w^2+2*n,x, x>w2n||next; x>n&&break; s++);s)}, [1..100]) \\ M. F. Hasler, Jan 26 2020
  • Python
    import sympy
def A211510(n): return sum(x<=n and x*n>=w**2+2*n for w in range(1,n+1) for x in sympy.divisors(w**2+2*n)) # Pontus von Brömssen, Jan 26 2020

Extensions

Name corrected by Pontus von Brömssen, Jan 26 2020

A211511 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=2x*y.

Original entry on oeis.org

0, 0, 2, 2, 4, 4, 6, 6, 10, 12, 14, 14, 18, 18, 20, 20, 24, 24, 30, 30, 32, 32, 34, 34, 40, 44, 46, 48, 50, 50, 54, 54, 62, 62, 64, 64, 72, 72, 74, 74, 80, 80, 82, 82, 84, 86, 88, 88, 94, 100, 110, 110, 112, 112, 118, 118, 124, 124, 126, 126, 132, 132, 134, 136
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 - 2 x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 60}]  (* A211511 *)
t/2                          (* integers *)

A211512 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=3x*y.

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 6, 6, 6, 8, 8, 8, 14, 14, 14, 16, 18, 18, 22, 22, 22, 24, 24, 24, 30, 32, 32, 38, 38, 38, 44, 44, 46, 48, 48, 48, 56, 56, 56, 58, 58, 58, 62, 62, 62, 66, 66, 66, 76, 80, 82, 84, 84, 84, 92, 92, 92, 94, 94, 94, 104, 104, 104, 108, 112, 112, 116
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 - 3 x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211512 *)
t/2                          (* integers *)

A211513 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2*w^2=x*y.

Original entry on oeis.org

0, 0, 2, 2, 4, 4, 6, 6, 10, 14, 16, 16, 18, 18, 20, 20, 24, 24, 32, 32, 34, 34, 36, 36, 40, 46, 48, 52, 54, 54, 56, 56, 66, 66, 68, 68, 76, 76, 78, 78, 82, 82, 84, 84, 86, 90, 92, 92, 96, 104, 118, 118, 120, 120, 128, 128, 132, 132, 134, 134, 136, 136, 138, 142
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[2 w^2 - x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211513 *)
t/2                          (* integers *)

A211514 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 3*w^2=x*y.

Original entry on oeis.org

0, 0, 0, 2, 4, 4, 6, 6, 8, 10, 10, 10, 16, 16, 16, 18, 22, 22, 24, 24, 26, 28, 28, 28, 34, 38, 38, 48, 50, 50, 52, 52, 56, 58, 58, 58, 64, 64, 64, 66, 68, 68, 70, 70, 72, 74, 74, 74, 86, 94, 98, 100, 102, 102, 112, 112, 114, 116, 116, 116, 122, 122, 122, 124
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[3 w^2 - x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 90}]  (* A211514 *)
t/2                          (* integers *)

A211515 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^3=x*y.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 3, 3, 6, 8, 8, 8, 8, 8, 8, 8, 10, 10, 12, 12, 12, 12, 12, 12, 14, 16, 16, 21, 21, 21, 21, 21, 25, 25, 25, 25, 27, 27, 27, 27, 29, 29, 29, 29, 29, 29, 29, 29, 31, 33, 35, 35, 35, 35, 39, 39, 41, 41, 41, 41, 41, 41, 41, 41, 48, 48, 48, 48, 48, 48, 48
Offset: 0

Views

Author

Clark Kimberling, Apr 14 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Crossrefs

Cf. A211422, A010057.

Programs

  • Mathematica
    t[n_] := t[n] = Flatten[Table[w^3 - x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211515 *)

Formula

a(n) = Sum_{i=1..n} Sum_{j=1..n} A010057(i*j). - Ridouane Oudra, Oct 17 2020

A211516 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=x+y.

Original entry on oeis.org

0, 0, 1, 3, 3, 5, 7, 9, 12, 14, 16, 18, 20, 24, 28, 32, 34, 36, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 96, 102, 108, 114, 118, 122, 126, 130, 134, 140, 146, 152, 158, 164, 170, 176, 182, 186, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245
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, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211516 *)

Formula

Conjecture: a(n) = -(2*n + 5/3)*w(n) + w(n)^2 + (2/3)*w(n)^3 + (2*n + 5/6)*w(2*n) - (1/2)*w(2*n)^2 - (1/3)*w(2*n)^3, where w = A000196. - Velin Yanev, Nov 16 2021

A211517 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^3=(x+y)^2.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 5, 7, 7, 7, 7, 7, 7, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 33, 33, 33, 33, 33, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 98, 100, 102, 104, 106, 108
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^3 - (x + y)^2, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211517 *)

A211519 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w=2x-3y.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 11, 15, 19, 23, 29, 34, 40, 47, 54, 61, 70, 78, 87, 97, 107, 117, 129, 140, 152, 165, 178, 191, 206, 220, 235, 251, 267, 283, 301, 318, 336, 355, 374, 393, 414, 434, 455, 477, 499, 521, 545, 568, 592, 617, 642, 667, 694, 720, 747, 775
Offset: 1

Views

Author

Clark Kimberling, Apr 14 2012

Keywords

Comments

For a guide to related sequences, see A211422.

Examples

			For n = 4, 1 = 2*2-3*1, 2 = 2*4-3*2 and 3 = 2*3-3*1, so (1,2,1), (2,4,2) and (3,3,1) are solutions and a(4) = 3. - _Bernard Schott_, Jan 27 2020

Links

Crossrefs

Cf. A211422.

Programs

  • Magma
    [ #[:w, x, y in [1..n]|w eq 2*x-3*y]: n in [1..56]]; // Marius A. Burtea, Jan 26 2020
  • Magma
    R:=PowerSeriesRing(Integers(), 56); [0] cat Coefficients(R!(x^2*(1+x+x^3) / ((1-x)^3*(1+x)*(1+x+x^2)))); // Marius A. Burtea, Jan 26 2020
  • Mathematica
    t[n_] := t[n] = Flatten[Table[w - 2 x + 3 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 1, 80}]  (* A211519 *)
FindLinearRecurrence[t]
LinearRecurrence[{1,1,0,-1,-1,1},{0,1,2,3,6,8},56] (* Ray Chandler, Aug 02 2015 *)
  • PARI
    concat(0, Vec(x*(1 + x + x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Dec 02 2017
  • PARI
    a(n)=(n-1)^2\4 + (n+1)\3 \\ Charles R Greathouse IV, Jun 12 2020

Formula

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
G.f.: x^2*(1 + x + x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)). - Colin Barker, Dec 02 2017
a(n) = floor(((n-1)^2)/4) + floor((n-2)/3) + 1. - Ridouane Oudra, Jun 12 2020
a(n) = A001399(n-2)+A001399(n-3)+A001399(n-5). - R. J. Mathar, Jun 23 2021~

Extensions

Name and offset corrected by Pontus von Brömssen, Jan 26 2020
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