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 1-4 of 4 results.

A109906 A triangle based on A000045 and Pascal's triangle: T(n,m) = Fibonacci(n-m+1) * Fibonacci(m+1) * binomial(n,m).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 6, 6, 3, 5, 12, 24, 12, 5, 8, 25, 60, 60, 25, 8, 13, 48, 150, 180, 150, 48, 13, 21, 91, 336, 525, 525, 336, 91, 21, 34, 168, 728, 1344, 1750, 1344, 728, 168, 34, 55, 306, 1512, 3276, 5040, 5040, 3276, 1512, 306, 55, 89, 550, 3060, 7560, 13650, 16128, 13650, 7560, 3060, 550, 89
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Keywords

Comments

Row sums give A081057.

Examples

			Triangle T(n,k) begins:
   1;
   1,   1;
   2,   2,    2;
   3,   6,    6,    3;
   5,  12,   24,   12,     5;
   8,  25,   60,   60,    25,     8;
  13,  48,  150,  180,   150,    48,    13;
  21,  91,  336,  525,   525,   336,    91,   21;
  34, 168,  728, 1344,  1750,  1344,   728,  168,   34;
  55, 306, 1512, 3276,  5040,  5040,  3276, 1512,  306,  55;
  89, 550, 3060, 7560, 13650, 16128, 13650, 7560, 3060, 550, 89;
  ...

Links

Crossrefs

Cf. A141611, A141617, A000045, A081057.
Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A111006, A114197, A162741, A228074.

Programs

  • Haskell
    a109906 n k = a109906_tabl !! n !! k
a109906_row n = a109906_tabl !! n
a109906_tabl = zipWith (zipWith (*)) a058071_tabl a007318_tabl
-- Reinhard Zumkeller, Aug 15 2013
  • Maple
    f:= n-> combinat[fibonacci](n+1):
T:= (n, k)-> binomial(n, k)*f(k)*f(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 26 2023
  • Mathematica
    Clear[t, n, m] t[n_, m_] := Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Formula

T(n,m) = Fibonacci(n-m+1)*Fibonacci(m+1)*binomial(n,m).
T(n,k) = A058071(n,k) * A007318(n,k). - Reinhard Zumkeller, Aug 15 2013

Extensions

Offset changed by Reinhard Zumkeller, Aug 15 2013

A110023 A triangle of coefficients based on A000931 and Pascal's triangle: a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)*a(m + 1)*Binomial[n, m].

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 6, 2, 3, 8, 24, 8, 3, 4, 15, 40, 40, 15, 4, 5, 24, 90, 80, 90, 24, 5, 7, 35, 168, 210, 210, 168, 35, 7, 9, 56, 280, 448, 630, 448, 280, 56, 9, 12, 81, 504, 840, 1512, 1512, 840, 504, 81, 12, 16, 120, 810, 1680, 3150, 4032, 3150, 1680, 810, 120, 16
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Keywords

Comments

Row sums are:
{1, 2, 6, 16, 46, 118, 318, 840, 2216, 5898, 15584}

Examples

			{1},
{1, 1},
{2, 2, 2},
{2, 6, 6, 2},
{3, 8, 24, 8, 3},
{4, 15, 40, 40, 15, 4},
{5, 24, 90, 80, 90, 24, 5},
{7, 35, 168, 210, 210, 168, 35, 7},
{9, 56, 280, 448, 630, 448, 280, 56, 9},
{12, 81, 504, 840, 1512, 1512, 840, 504, 81, 12},
{16, 120, 810, 1680, 3150, 4032, 3150, 1680, 810, 120, 16}

Crossrefs

Cf. A141611, A141617, A000931.

Programs

  • Mathematica
    Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}] Flatten[%]

Formula

a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)*a(m + 1)*Binomial[n, m].

A110102 A triangle of coefficients based on A000931: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 4, 3, 4, 4, 3, 4, 5, 4, 6, 4, 6, 4, 5, 7, 5, 8, 6, 6, 8, 5, 7, 9, 7, 10, 8, 9, 8, 10, 7, 9, 12, 9, 14, 10, 12, 12, 10, 14, 9, 12, 16, 12, 18, 14, 15, 16, 15, 14, 18, 12, 16
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Keywords

Comments

Row sums are:
{1, 2, 5, 8, 14, 22, 34, 52, 77, 114, 166}

Examples

			{1},
{1, 1},
{2, 1, 2},
{2, 2, 2, 2},
{3, 2, 4, 2, 3},
{4, 3, 4, 4, 3, 4},
{5, 4, 6, 4, 6, 4, 5},
{7, 5, 8, 6, 6, 8, 5, 7},
{9, 7, 10, 8, 9, 8, 10, 7, 9},
{12, 9, 14, 10, 12, 12, 10, 14, 9, 12},
{16, 12, 18, 14, 15, 16, 15, 14, 18, 12, 16}

Crossrefs

Cf. A141611, A141617, A000931.

Programs

  • Mathematica
    Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Formula

a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1).

A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 6, 4, 4, 6, 15, 6, 16, 6, 15, 32, 15, 24, 24, 15, 32, 65, 32, 60, 36, 60, 32, 65, 147, 65, 128, 90, 90, 128, 65, 147, 306, 147, 260, 192, 225, 192, 260, 147, 306, 660, 306, 588, 390, 480, 480, 390, 588, 306, 660, 1424, 660, 1224, 882, 975, 1024, 975, 882
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Keywords

Comments

Row sums are:
{1, 2, 9, 20, 58, 142, 350, 860, 2035, 4848, 11354}.

Examples

			{1},
{1, 1},
{4, 1, 4},
{6, 4, 4, 6},
{15, 6, 16, 6, 15},
{32, 15, 24, 24, 15, 32},
{65, 32, 60, 36, 60, 32, 65},
{147, 65, 128, 90, 90, 128, 65, 147},
{306, 147, 260, 192, 225, 192, 260, 147, 306},
{660, 306, 588, 390, 480, 480, 390, 588, 306, 660},
{1424, 660, 1224, 882, 975, 1024, 975, 882, 1224, 660, 1424}

Crossrefs

Cf. A141611, A141617, A000931, A000045, A058071.

Programs

  • Mathematica
    Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Formula

a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].
Showing 1-4 of 4 results.

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