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 1-8 of 8 results.

A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 21, 13, 1, 1, 17, 37, 37, 17, 1, 1, 21, 57, 77, 57, 21, 1, 1, 25, 81, 137, 137, 81, 25, 1, 1, 29, 109, 221, 277, 221, 109, 29, 1, 1, 33, 141, 333, 501, 501, 333, 141, 33, 1, 1, 37, 177, 477, 837, 1005, 837, 477, 177, 37, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 13 2007

Keywords

Comments

Row sums = A131062: (1, 2, 7, 20, 49, 110, 235, ...); the binomial transform of (1, 1, 4, 4, 4, ...).
Triangle equals 4*A007318 - 3*A000012 as infinite lower triangular matrices. - Emeric Deutsch, Jun 21 2007

Examples

			First few rows of the triangle are
  1;
  1,  1;
  1,  5,  1;
  1,  9,  9,  1;
  1, 13, 21, 13,  1;
  1, 17, 37, 37, 17,  1;
  1, 21, 57, 77, 57, 21, 1;
  ...

Links

Crossrefs

Cf. A109128, A123203, A131060, A131063, A131064, A131065, A131066, A131067, A131068.

Programs

  • Magma
    [4*Binomial(n, k) -3: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
  • Maple
    T := proc (n, k) if k <= n then 4*binomial(n, k)-3 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 21 2007
  • Mathematica
    Table[4*Binomial[n, k] -3, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
  • Sage
    [[4*binomial(n, k) -3 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

Formula

G.f.:(1 - z - t*z + 4*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 21 2007

Extensions

More terms from Emeric Deutsch, Jun 21 2007

A131065 Triangle read by rows: T(n,k) = 6*binomial(n,k) - 5 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 31, 19, 1, 1, 25, 55, 55, 25, 1, 1, 31, 85, 115, 85, 31, 1, 1, 37, 121, 205, 205, 121, 37, 1, 1, 43, 163, 331, 415, 331, 163, 43, 1, 1, 49, 211, 499, 751, 751, 499, 211, 49, 1, 1, 55, 265, 715, 1255, 1507, 1255, 715, 265, 55, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 13 2007

Keywords

Comments

Row sums = A131066.
The matrix inverse starts:
1;
-1, 1;
6, -7, 1;
-66, 78, -13, 1;
1086, -1284, 216, -19, 1;
-23826, 28170, -4740, 420, -25, 1;
653406, -772536, 129990, -11520, 690, -31, 1; - R. J. Mathar, Mar 12 2013

Examples

			First few rows of the triangle are:
  1;
  1,  1;
  1,  7,  1;
  1, 13, 13,  1;
  1, 19, 31, 19,  1;
  1, 25, 55, 55, 25, 1;
...

Links

Crossrefs

Cf. A109128, A123203, A131060, A131061, A131063, A131064, A131066, A131067, A131068.

Programs

  • Magma
    [6*Binomial(n,k) -5: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
  • Maple
    T := proc (n, k) if k <= n then 6*binomial(n, k)-5 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007
  • Mathematica
    Table[6*Binomial[n,k]-5,{n,0,15},{k,0,n}]//Flatten (* Harvey P. Dale, May 15 2016 *)
  • Sage
    [[6*binomial(n,k) -5 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

Formula

G.f.: (1-z-t*z+6*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

Extensions

More terms from Emeric Deutsch, Jun 20 2007

A131060 3*A007318 - 2*A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 16, 10, 1, 1, 13, 28, 28, 13, 1, 1, 16, 43, 58, 43, 16, 1, 1, 19, 61, 103, 103, 61, 19, 1, 1, 22, 82, 166, 208, 166, 82, 22, 1, 1, 25, 106, 250, 376, 376, 250, 106, 25, 1, 1, 28, 133, 358, 628, 754, 628, 358, 133, 28, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 13 2007

Keywords

Comments

Row sums = A097813: (1, 2, 6, 16, 38, 84, 178, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  4,  1;
  1,  7,  7,  1;
  1, 10, 16, 10,  1;
  1, 13, 28, 28, 13,  1;
  1, 16, 43, 58, 43, 16,  1;
  ...

Links

Crossrefs

Cf. A109128, A123203, A131061, A131063, A131064, A131065, A131066, A131067, A131068.

Programs

  • Magma
    [3*Binomial(n,k) -2: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
  • Maple
    A131060:= (n,k) -> 3*binomial(n, k)-2; seq(seq(A131060(n, k), k = 0..n), n = 0.. 10); # G. C. Greubel, Mar 12 2020
  • Mathematica
    T[n_, k_] = 3*Binomial[n, k] -2; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 20 2008 *)
  • Sage
    [[3*binomial(n,k) -2 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

Formula

T(n,k) = 3*binomial(n,k) - 2. - Roger L. Bagula, Aug 20 2008

Extensions

More terms from Roger L. Bagula, Aug 20 2008

A131063 Triangle read by rows: T(n,k) = 5*binomial(n,k) - 4 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 16, 26, 16, 1, 1, 21, 46, 46, 21, 1, 1, 26, 71, 96, 71, 26, 1, 1, 31, 101, 171, 171, 101, 31, 1, 1, 36, 136, 276, 346, 276, 136, 36, 1, 1, 41, 176, 416, 626, 626, 416, 176, 41, 1, 1, 46, 221, 596, 1046, 1256, 1046, 596, 221, 46, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 13 2007

Keywords

Comments

Row sums = A131064: (1, 2, 8, 24, 60, 136, 292, ...), the binomial transform of (1, 1, 5, 5, 5, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  6,  1;
  1, 11, 11,  1;
  1, 16, 26, 16,  1;
  1, 21, 46, 46, 21,  1;
  1, 26, 71, 96, 71, 26,  1;
  ...

Links

Crossrefs

Cf. A109128, A123203, A131060, A131061, A131064, A131065, A131066, A131067, A131068.

Programs

  • GAP
    Print(Flat(List([0..10],n->List([0..n],k->5*Binomial(n,k)-4)))); # Muniru A Asiru, Feb 21 2019
  • Magma
    [5*Binomial(n, k) -4: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
  • Maple
    T := proc (n, k) if k <= n then 5*binomial(n, k)-4 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007
  • Mathematica
    Table[5*Binomial[n,k] -4, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
  • Sage
    [[5*binomial(n, k) -4 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

Formula

G.f.: (1-z-t*z+5*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

Extensions

More terms from Emeric Deutsch, Jun 20 2007

A131067 Triangle read by rows: T(n,k) = 7*binomial(n,k) - 6 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 22, 36, 22, 1, 1, 29, 64, 64, 29, 1, 1, 36, 99, 134, 99, 36, 1, 1, 43, 141, 239, 239, 141, 43, 1, 1, 50, 190, 386, 484, 386, 190, 50, 1, 1, 57, 246, 582, 876, 876, 582, 246, 57, 1, 1, 64, 309, 834, 1464, 1758, 1464, 834, 309, 64, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 13 2007

Keywords

Comments

Row sums = A131068: (1, 2, 10, 32, 82, 188, 406, ...), the binomial transform of (1, 1, 7, 7, 7, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  8,  1;
  1, 15, 15,  1;
  1, 22, 36, 22,  1;
  1, 29, 64, 64, 29, 1;
  ...

Links

Crossrefs

Cf. A131064, A131066.
Sequence m*binomial(n,k) - (m-1): A007318 (m=1), A109128 (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), this sequence (m=7), A131068 (m=8).

Programs

  • Magma
    [7*Binomial(n, k) -6: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
  • Maple
    T := proc (n, k) if k <= n then 7*binomial(n, k)-6 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007
  • Mathematica
    Table[7*Binomial[n, k] -6, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
  • Sage
    [[7*binomial(n, k) -6 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

Formula

G.f.: G(t,z) = (1-z-t*z+7*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

Extensions

More terms from Emeric Deutsch, Jun 20 2007

A131064 Binomial transform of [1, 1, 5, 5, 5, ...].

Original entry on oeis.org

1, 2, 8, 24, 60, 136, 292, 608, 1244, 2520, 5076, 10192, 20428, 40904, 81860, 163776, 327612, 655288, 1310644, 2621360, 5242796, 10485672, 20971428, 41942944, 83885980, 167772056, 335544212, 671088528, 1342177164, 2684354440
Offset: 0

Views

Author

Gary W. Adamson, Jun 13 2007

Keywords

Comments

Row sums of triangle A131063. - Emeric Deutsch, Jun 20 2007

Examples

			a(3) = 24 = sum of row 4 terms of A131063: (1 + 11 + 11 + 1).
a(3) = 24 = (1, 3, 3, 1) dot (1, 1, 5, 5).

Links

Crossrefs

Cf. A109128, A123203, A131060, A131061, A131063, A131065, A131066, A131067, A131068.

Programs

  • GAP
    Print(List([0..30],n->5*2^n-4*n-4)); # Muniru A Asiru, Feb 21 2019
  • Magma
    I:=[1, 2, 8]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2) + 2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 05 2012
  • Maple
    a := proc (n) options operator, arrow; 5*2^n-4*n-4 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007
  • Mathematica
    CoefficientList[Series[(1-2x+5x^2)/((1-2x)(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 05 2012 *)
LinearRecurrence[{4,-5,2},{1,2,8},30] (* Harvey P. Dale, Dec 29 2014 *)
  • Sage
    [5*2^n -4*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

Formula

From Emeric Deutsch, Jun 20 2007: (Start)
a(n) = 5*2^n - 4*(n + 1).
G.f.: (1-2*x+5*x^2)/((1-2*x)*(1-x)^2). (End)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Vincenzo Librandi, Jul 05 2012
E.g.f.: 5*exp(2*x) - 4*(1+x)*exp(x). - G. C. Greubel, Mar 12 2020

Extensions

Corrected and extended by Emeric Deutsch, Jun 20 2007

A131066 Binomial transform of [1, 1, 6, 6, 6, ...].

Original entry on oeis.org

1, 2, 9, 28, 71, 162, 349, 728, 1491, 3022, 6089, 12228, 24511, 49082, 98229, 196528, 393131, 786342, 1572769, 3145628, 6291351, 12582802, 25165709, 50331528, 100663171, 201326462, 402653049, 805306228, 1610612591, 3221225322
Offset: 0

Views

Author

Gary W. Adamson, Jun 13 2007

Keywords

Comments

Row sums of triangle A131065. - Emeric Deutsch, Jun 20 2007

Examples

			a(3) = 28 = sum of row 4 of triangle A131065: (1 + 13 + 13 + 1).
a(3) = 28 = (1, 3, 3, 1) dot (1, 1, 6, 6) = (1 + 3 + 18 + 6).

Links

Crossrefs

Cf. A109128, A123203, A131060, A131061, A131063, A131064, A131065, A131067, A131068.

Programs

  • GAP
    Print(List([0..30],n->6*2^n-5*n-5)); # Muniru A Asiru, Feb 21 2019
  • Magma
    [6*2^n -5*(n+1): n in [0..30]]; // G. C. Greubel, Mar 12 2020
  • Maple
    a := proc (n) options operator, arrow; 6*2^n-5*n-5 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007
  • Mathematica
    Table[6*2^n -5*(n+1), {n,0,30}] (* G. C. Greubel, Mar 12 2020 *)
  • Sage
    [6*2^n -5*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

Formula

From Emeric Deutsch, Jun 20 2007: (Start)
a(n) = 6*2^n - 5*(n + 1).
G.f.: (1 - 2*x + 6*x^2)/((1-2*x)*(1-x)^2). (End)
E.g.f.: 6*exp(2*x) - 5*(1 + x)*exp(x). - G. C. Greubel, Mar 12 2020
a(n) = 2*a(n - 1) + 5*n - 5. - Kritsada Moomuang, Jul 03 2020
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Jul 10 2020

Extensions

Corrected and extended by Emeric Deutsch, Jun 20 2007

A097810 a(n) = 7*2^n - 3*n - 6.

Original entry on oeis.org

1, 5, 16, 41, 94, 203, 424, 869, 1762, 3551, 7132, 14297, 28630, 57299, 114640, 229325, 458698, 917447, 1834948, 3669953, 7339966, 14679995, 29360056, 58720181, 117440434, 234880943, 469761964, 939524009, 1879048102, 3758096291
Offset: 0

Views

Author

Paul Barry, Aug 25 2004

Keywords

Links

Crossrefs

Cf. A079583, A097809, A131068.

Programs

  • Mathematica
    s=1;lst={s};Do[s+=(s+=n)+n++;AppendTo[lst, s], {n, 1, 5!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Table[7*2^n-3n-6,{n,0,30}] (* or *) LinearRecurrence[{4,-5,2},{1,5,16},30] (* Harvey P. Dale, Nov 15 2011 *)

Formula

G.f.: (1 + x + x^2)/((1-x)^2*(1-2*x)).
a(n) = 2*a(n-1) + 3*n, n > 0, a(0)=1.
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
From Elmo R. Oliveira, Mar 06 2025: (Start)
E.g.f.: exp(x)*(7*exp(x) - 3*(x + 2)).
a(n) = A131068(n+1)/2. (End)
Showing 1-8 of 8 results.

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