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.

A082105 Array A(n, k) = (k*n)^2 + 4*(k*n) + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 13, 13, 1, 1, 22, 33, 22, 1, 1, 33, 61, 61, 33, 1, 1, 46, 97, 118, 97, 46, 1, 1, 61, 141, 193, 193, 141, 61, 1, 1, 78, 193, 286, 321, 286, 193, 78, 1, 1, 97, 253, 397, 481, 481, 397, 253, 97, 1, 1, 118, 321, 526, 673, 726, 673, 526, 321, 118, 1
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Examples

			Array, A(n, k), begins as:
  1,  1,   1,   1,   1,   1, ... A000012;
  1,  6,  13,  22,  33,  46, ... A028872;
  1, 13,  33,  61,  97, 141, ... A082109;
  1, 22,  61, 118, 193, 286, ... ;
  1, 33,  97, 193, 321, 481, ... ;
  1, 46, 141, 286, 481, 726, ... ;
Triangle, T(n, k), begins as:
  1;
  1,  1;
  1,  6,   1;
  1, 13,  13,   1;
  1, 22,  33,  22,   1;
  1, 33,  61,  61,  33,   1;
  1, 46,  97, 118,  97,  46,   1;
  1, 61, 141, 193, 193, 141,  61,  1;
  1, 78, 193, 286, 321, 286, 193, 78,  1;

Links

Crossrefs

Cf. A016897, A028872, A047673, A082043, A082046, A082106, A082107, A082109.

Programs

  • Magma
    [(k*(n-k))^2 + 4*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022
  • Mathematica
    T[n_, k_]:= (k*(n-k))^2 + 4*(k*(n-k)) + 1;
Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)
  • SageMath
    def A082105(n,k): return (k*(n-k))^2 + 4*(k*(n-k)) + 1
flatten([[A082105(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022

Formula

A(n, k) = (k*n)^2 + 4*(k*n) + 1 (square array).
A(n, n) = T(2*n, n) = A082106(n) (main diagonal).
T(n, k) = A(n-k, k) (number triangle).
Sum_{k=0..n} T(n, k) = A082107(n) (diagonal sums).
T(n, n-1) = A028872(n-1), n >= 1.
T(n, n-2) = A082109(n-2), n >= 2.
From G. C. Greubel, Dec 22 2022: (Start)
Sum_{k=0..n} (-1)^k * T(n, k) = ((1+(-1)^n)/2)*A016897(n-1).
T(2*n+1, n+1) = A047673(n+1), n >= 0.
T(n, n-k) = T(n, k). (End)

A082047 Diagonal sums of number array A082046.

Original entry on oeis.org

1, 2, 7, 24, 69, 170, 371, 736, 1353, 2338, 3839, 6040, 9165, 13482, 19307, 27008, 37009, 49794, 65911, 85976, 110677, 140778, 177123, 220640, 272345, 333346, 404847, 488152, 584669, 695914, 823515, 969216, 1134881, 1322498, 1534183
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Links

Crossrefs

Cf. A082045, A082046, A082107.

Programs

  • Magma
    [(n+1)*(n*(n-1)*(n^2+16)+30)/30: n in [0..40]]; // G. C. Greubel, Dec 24 2022
  • Mathematica
    Table[(n+1)*(n*(n-1)*(n^2+16)+30)/30, {n,0,40}] (* G. C. Greubel, Dec 24 2022 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,7,24,69,170},40] (* Harvey P. Dale, Jan 25 2024 *)
  • PARI
    a(n) = (n^5+15*n^3+14*n+30)/30; \\ Michel Marcus, Jan 22 2016
  • SageMath
    [(n+1)*(n*(n-1)*(n^2+16)+30)/30 for n in range(41)] # G. C. Greubel, Dec 24 2022

Formula

a(n) = (n^5+15*n^3+14*n+30)/30 = (n+1)*(n^4-n^3+16*n^2-16*n+30)/30.
From G. C. Greubel, Dec 24 2022: (Start)
G.f.: (1 - 4*x + 10*x^2 - 8*x^3 + 5*x^4)/(1-x)^6.
E.g.f.: (1/30)*(30 +30*x +60*x^2 +40*x^3 +10*x^4 +x^5)*exp(x). (End)

A082106 Main diagonal of number array A082105.

Original entry on oeis.org

1, 6, 33, 118, 321, 726, 1441, 2598, 4353, 6886, 10401, 15126, 21313, 29238, 39201, 51526, 66561, 84678, 106273, 131766, 161601, 196246, 236193, 281958, 334081, 393126, 459681, 534358, 617793, 710646, 813601, 927366, 1052673, 1190278, 1340961, 1505526, 1684801
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Comments

4*a(n) can be written as (n^2 + 2*n + 1)^2 + (n^2 - 2*n + 1)^2 + (n^2 - 2*n - 1)^2 + (n^2 + 2*n - 1)^2. - Bruno Berselli, Jun 20 2014

Links

Crossrefs

Cf. A082047, A082044, A082105, A082107, A082109.

Programs

  • Magma
    [(n^2+2)^2 -3: n in [0..40]]; // G. C. Greubel, Dec 22 2022
  • Mathematica
    Table[n^4+4n^2+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,6,33,118,321},40] (* Harvey P. Dale, Dec 06 2012 *)
  • SageMath
    [(n^2+2)^2 -3 for n in range(41)] # G. C. Greubel, Dec 22 2022

Formula

a(n) = n^4 + 4*n^2 + 1.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Dec 06 2012
G.f.: (1 + x + 13*x^2 + 3*x^3 + 6*x^4)/(1 - x)^5. - Bruno Berselli, Jun 20 2014
E.g.f.: (1 + 5*x + 11*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 22 2022
Sum_{n>=0} 1/a(n) = 1/2 + (Pi/4)*((1/sqrt(2)+1/sqrt(6))*coth(sqrt(2-sqrt(3))*Pi) - (1/sqrt(2)-1/sqrt(6))*coth(sqrt(2+sqrt(3))*Pi)). - Amiram Eldar, Jan 08 2023

A082114 Diagonal sums of number array A082110.

Original entry on oeis.org

1, 2, 9, 32, 89, 210, 441, 848, 1521, 2578, 4169, 6480, 9737, 14210, 20217, 28128, 38369, 51426, 67849, 88256, 113337, 143858, 180665, 224688, 276945, 338546, 410697, 494704, 591977, 704034, 832505, 979136, 1145793, 1334466, 1547273
Offset: 0

Views

Author

Paul Barry, Apr 04 2003

Keywords

Links

Crossrefs

Cf. A082045, A082107, A082110.

Programs

  • Magma
    [(n+1)*(n^4-n^3+26*n^2-26*n+30)/30: n in [0..50]]; // G. C. Greubel, Dec 22 2022
  • Mathematica
    LinearRecurrence[{6,-15,20,-15,6,-1}, {1,2,9,32,89,210}, 51] (* G. C. Greubel, Dec 22 2022 *)
  • SageMath
    [(n+1)*(n^4-n^3+26*n^2-26*n+30)/30 for n in range(51)] # G. C. Greubel, Dec 22 2022

Formula

a(n) = (n+1)*(n^4 - n^3 + 26*n^2 - 26*n + 30)/30.
From G. C. Greubel, Dec 22 2022: (Start)
G.f.: (1 - 4*x + 12*x^2 - 12*x^3 + 7*x^4)/(1-x)^6.
E.g.f.: (1/30)*(30 + 30*x + 90*x^2 + 50*x^3 + 10*x^4 + x^5)*exp(x). (End)
Showing 1-4 of 4 results.

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