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 1-3 of 3 results.

A028313 Elements in the 5-Pascal triangle (by row).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 6, 6, 1, 1, 7, 12, 7, 1, 1, 8, 19, 19, 8, 1, 1, 9, 27, 38, 27, 9, 1, 1, 10, 36, 65, 65, 36, 10, 1, 1, 11, 46, 101, 130, 101, 46, 11, 1, 1, 12, 57, 147, 231, 231, 147, 57, 12, 1, 1, 13, 69, 204, 378, 462, 378, 204, 69, 13, 1, 1, 14, 82, 273, 582, 840, 840, 582, 273, 82, 14, 1
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  5,  1;
  1,  6,  6,   1;
  1,  7, 12,   7,   1;
  1,  8, 19,  19,   8,   1;
  1,  9, 27,  38,  27,   9,   1;
  1, 10, 36,  65,  65,  36,  10,  1;
  1, 11, 46, 101, 130, 101,  46, 11,  1;
  1, 12, 57, 147, 231, 231, 147, 57, 12,  1;

Links

Crossrefs

Cf. A000007, A005009, A010892, A022112, A028275, A028314, A028315.
Cf. A028316, A028317, A028318, A028319, A028320, A028321, A028322.
Cf. A028323, A028324, A028325, A029653, A051472, A051936, A051937.
Cf. A178915.

Programs

  • Magma
    [n le 1 select 1 else Binomial(n,k) +3*Binomial(n-2,k-1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 05 2024
  • Mathematica
    Table[If[n<2, 1, Binomial[n,k] +3*Binomial[n-2,k-1]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2024 *)
  • SageMath
    def A028313(n,k): return 1 if n<2 else binomial(n,k) + 3*binomial(n-2,k-1)
flatten([[A028313(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 05 2024

Formula

From Ralf Stephan, Jan 31 2005: (Start)
T(n, k) = C(n, k) + 3*C(n-2, k-1), with T(0, k) = T(1, k) = 1.
G.f.: (1 + 3*x^2*y)/(1 - x*(1+y)). (End)
From G. C. Greubel, Jan 05 2024: (Start)
T(n, n-k) = T(n, k).
T(n, n-1) = n + 3*(1 - [n=1]) = A178915(n+3), n >= 1.
T(n, n-2) = A051936(n+2), n >= 2.
T(n, n-3) = A051937(n+1), n >= 3.
T(2*n, n) = A028322(n).
Sum_{k=0..n} T(n, k) = A005009(n-2) - (3/4)*[n=0] - (3/2)*[n=1].
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n) - 3*[n=2].
Sum_{k=0..floor(n/2)} T(n-k, k) = A022112(n-2) + 3*([n=0] - [n=1]).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = 4*A010892(n) - 3*([n=0] + [n=1]). (End)

Extensions

More terms from Sam Alexander (pink2001x(AT)hotmail.com)

A028314 Elements in the 5-Pascal triangle A028313 that are not 1.

Original entry on oeis.org

5, 6, 6, 7, 12, 7, 8, 19, 19, 8, 9, 27, 38, 27, 9, 10, 36, 65, 65, 36, 10, 11, 46, 101, 130, 101, 46, 11, 12, 57, 147, 231, 231, 147, 57, 12, 13, 69, 204, 378, 462, 378, 204, 69, 13, 14, 82, 273, 582, 840, 840, 582, 273, 82, 14, 15, 96, 355, 855, 1422, 1680, 1422, 855, 355, 96, 15
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Triangle begins as:
   5;
   6,  6;
   7, 12,   7;
   8, 19,  19,   8;
   9, 27,  38,  27,   9;
  10, 36,  65,  65,  36,  10;
  11, 46, 101, 130, 101,  46,  11;
  12, 57, 147, 231, 231, 147,  57,  12;
  13, 69, 204, 378, 462, 378, 204,  69,  13;

Links

Crossrefs

Cf. A000027, A010892, A022112, A028313, A028322,
Cf. A051936, A051937, A121262, A176448.

Programs

  • Magma
    A028314:= func< n,k | Binomial(n+2,k+1) + 3*Binomial(n,k) >;
[A028314(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 06 2024
  • Mathematica
    A028314[n_, k_]:= Binomial[n+2,k+1] + 3*Binomial[n,k];
Table[A028314[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 06 2024 *)
  • SageMath
    def A028314(n,k): return binomial(n+2,k+1) + 3*binomial(n,k)
flatten([[A028314(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 06 2024

Formula

From G. C. Greubel, Jan 06 2024: (Start)
T(n, k) = binomial(n+2, k+1) + 3*binomial(n, k).
T(n, n-k) = T(n, k).
T(n, 0) = T(n, n) = A000027(n+5).
T(n, 1) = T(n, n-1) = A051936(n+4).
T(n, 2) = T(n, n-2) = A051937(n+3).T(2*n, n) = A028322(n+1).
Sum_{k=0..n} T(n, k) = A176448(n).
Sum_{k=0..n} (-1)^k * T(n, k) = 1 + (-1)^n + 3*[n=0].
Sum_{k=0..n} T(n-k, k) = A022112(n+1) - (3-(-1)^n)/2.
Sum_{k=0..n} (-1)^k * T(n-k, k) = 4*A010892(n) - 2*A121262(n+1) - (3 - (-1)^n)/2. (End)
G.f.: (5 - 4*x - 4*x*y + 3*x^2*y)/((1 - x)*(1 - x*y)*(1 - x - x*y)). - Stefano Spezia, Dec 06 2024

Extensions

More terms from James Sellers

A103145 a(n) = (1/6)*(n^3 + 21*n^2 + 74*n + 18).

Original entry on oeis.org

3, 19, 43, 76, 119, 173, 239, 318, 411, 519, 643, 784, 943, 1121, 1319, 1538, 1779, 2043, 2331, 2644, 2983, 3349, 3743, 4166, 4619, 5103, 5619, 6168, 6751, 7369, 8023, 8714, 9443, 10211, 11019, 11868, 12759, 13693, 14671, 15694, 16763
Offset: 0

Views

Author

Creighton Dement, Mar 17 2005

Keywords

Comments

A floretion-generated sequence relating truncated triangle and pyramidal numbers. The following reasoning suggests that (a(n)) may not be the result of some "arbitrary" addition of these sequences--it may possess some geometric meaning of its own: The FAMP identity: "jesrightfor + jesleftfor = jesfor" holds and was used to find the relation a(n) = 2*A051936(n+4)_4 + A051937(n+4)_4 . In the above case, "jesfor" returns the truncated triangular numbers (times -1); "jesrightfor" returns the truncated pyramidal numbers; and (a(n)) is given by "jesleftfor" (times -1). All sequences result from a Force transform of the sequence c(n) = n + 5 (c was not chosen arbitrarily, for details see program code). Specifically, the sequence (a(n)) is the (ForType 1A) jesleftfor-transform of the sequence c(n) = n + 5 with respect to the floretion given in the program code.
Floretion Algebra Multiplication Program, FAMP Code: 1jesleftfor[A*B] with A = .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e and B = + 'i + .5j' + .5k' + .5'ij' + .5'ik'; 1vesfor[A*B](n) = n + 5. ForType: 1A Alternative description: 1jesleftfor[A*B], ForType: 1A, LoopType: tes (first iteration after transforming the zero-sequence A000004).

Links

Crossrefs

Cf. A051936, A051937.

Programs

  • Magma
    I:=[3, 19, 43, 76]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jun 26 2012
  • Mathematica
    CoefficientList[Series[(3-2*x)*(1+3*x-3*x^2)/(1-x)^4,{x,0,40}],x] (* or *) LinearRecurrence[ {4,-6,4,-1},{3,19,43,76},50] (* Vincenzo Librandi, Jun 26 2012 *)
Table[(n^3+21n^2+74n+18)/6,{n,0,50}] (* Harvey P. Dale, Jun 18 2024 *)
  • PARI
    a(n) = (n^3+21*n^2+74*n+18)/6; \\ Altug Alkan, Sep 23 2018
  • Python
    def A103145(n): return (n*(n*(n+21)+74)+18)//6 # Chai Wah Wu, Mar 07 2024

Formula

a(n) = 2*A051936(n+4)_4 + A051937(n+4)_4 (for n = 0, 1, 2, 3) or a(m) = (1/6)*(m^3 + 9m^2 - 46m - 6) = 2*A051936(m) + A051937(m) (for m = 4, 5, 6).
G.f.: (3-2*x)*(1 + 3*x - 3*x^2)/(1-x)^4. - Colin Barker, Apr 30 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 26 2012
Showing 1-3 of 3 results.

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

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