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 1-6 of 6 results.

A082109 Third row of number array A082105.

Original entry on oeis.org

1, 13, 33, 61, 97, 141, 193, 253, 321, 397, 481, 573, 673, 781, 897, 1021, 1153, 1293, 1441, 1597, 1761, 1933, 2113, 2301, 2497, 2701, 2913, 3133, 3361, 3597, 3841, 4093, 4353, 4621, 4897, 5181, 5473, 5773, 6081, 6397, 6721, 7053, 7393, 7741, 8097, 8461
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Links

Crossrefs

Cf. A000217, A082105, A082108, A000217.
Column 2 of array A188646.

Programs

Formula

a(n) = 4*n^2 + 8*n + 1.
a(n) = a(n-1) + 8*n + 4, with a(0)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: (1 + 10*x - 3*x^2)/(1-x)^3. - Bruno Berselli, Apr 18 2011
E.g.f.: (1 + 12*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022
From Amiram Eldar, Jan 18 2023: (Start)
Sum_{n>=0} 1/a(n) = 1/6 - cot(sqrt(3)*Pi/2)*sqrt(3)*Pi/12.
Sum_{n>=0} (-1)^n/a(n) = cosec(sqrt(3)*Pi/2)*sqrt(3)*Pi/12 - 1/6. (End)

A082107 Diagonal sums of number array A082105.

Original entry on oeis.org

1, 2, 8, 28, 79, 190, 406, 792, 1437, 2458, 4004, 6260, 9451, 13846, 19762, 27568, 37689, 50610, 66880, 87116, 112007, 142318, 178894, 222664, 274645, 335946, 407772, 491428, 588323, 699974, 828010, 974176, 1140337, 1328482, 1540728
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Links

Crossrefs

Cf. A082043, A082045, A082046, A082105, A082106, A082109, A082110.

Programs

  • Magma
    [(n^5+20*n^3+9*n+30)/30: n in [0..50]]; // G. C. Greubel, Dec 22 2022
  • Mathematica
    LinearRecurrence[{6,-15,20,-15,6,-1}, {1,2,8,28,79,190}, 51] (* G. C. Greubel, Dec 22 2022 *)
  • SageMath
    [(n^5+20*n^3+9*n+30)/30 for n in range(51)] # G. C. Greubel, Dec 22 2022

Formula

a(n) = (n^5 + 20*n^3 + 9*n + 30)/30.
G.f.: (1-4*x+11*x^2-10*x^3+6*x^4)/(1-x)^6 . - R. J. Mathar, Mar 27 2019
E.g.f.: (1/30)*(30 +30*x +75*x^2 +45*x^3 +10*x^4 +x^5)*exp(x). - G. C. Greubel, Dec 22 2022

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

A082043 Square array, A(n, k) = (k*n)^2 + 2*k*n + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 25, 16, 1, 1, 25, 49, 49, 25, 1, 1, 36, 81, 100, 81, 36, 1, 1, 49, 121, 169, 169, 121, 49, 1, 1, 64, 169, 256, 289, 256, 169, 64, 1, 1, 81, 225, 361, 441, 441, 361, 225, 81, 1, 1, 100, 289, 484, 625, 676, 625, 484, 289, 100, 1
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Examples

			Array, A(n, k), begins as:
  1,   1,   1,    1,    1,    1,    1,    1,     1, ... A000012;
  1,   4,   9,   16,   25,   36,   49,   64,    81, ... A000290;
  1,   9,  25,   49,   81,  121,  169,  225,   289, ... A016754;
  1,  16,  49,  100,  169,  256,  361,  484,   625, ... A016778;
  1,  25,  81,  169,  289,  441,  625,  841,  1089, ... A016814;
  1,  36, 121,  256,  441,  676,  961, 1296,  1681, ... A016862;
  1,  49, 169,  361,  625,  961, 1369, 1849,  2401, ... A016922;
  1,  64, 225,  484,  841, 1296, 1849, 2500,  3249, ... A016994;
  1,  81, 289,  625, 1089, 1681, 2401, 3249,  4225, ... A017078;
  1, 100, 361,  784, 1369, 2116, 3025, 4096,  5329, ... A017174;
  1, 121, 441,  961, 1681, 2601, 3721, 5041,  6561, ... A017282;
  1, 144, 529, 1156, 2025, 3136, 4489, 6084,  7921, ... A017402;
  1, 169, 625, 1369, 2401, 3721, 5329, 7225,  9409, ... A017534;
  1, 196, 729, 1600, 2809, 4356, 6241, 8464, 11025, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,   1;
  1,   4,   1;
  1,   9,   9,   1;
  1,  16,  25,  16,   1;
  1,  25,  49,  49,  25,   1;
  1,  36,  81, 100,  81,  36,   1;
  1,  49, 121, 169, 169, 121,  49,   1;
  1,  64, 169, 256, 289, 256, 169,  64,   1;
  1,  81, 225, 361, 441, 441, 361, 225,  81,   1;
  1, 100, 289, 484, 625, 676, 625, 484, 289, 100,  1;

Links

Crossrefs

Rows include A000290, A016754, A016778, A016814, A016862, A016922, A016994, A017078, A017174, A017282, A017402, A017534.
Diagonals include A000583, A058031, A062938, A082044 (main diagonal).
Diagonal sums (row sums if viewed as number triangle) are A082045.
Cf. A082039, A082045, A082046, A082105.

Programs

  • Magma
    A082043:= func< n,k | (k*(n-k))^2 +2*k*(n-k) +1 >;
[A082043(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 24 2022
  • Mathematica
    T[n_, k_]:= (k*(n-k))^2 +2*k*(n-k) +1;
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 24 2022 *)
  • SageMath
    def A082043(n,k): return (k*(n-k))^2 +2*k*(n-k) +1
flatten([[A082043(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Dec 24 2022

Formula

A(n, k) = (k*n)^2 + 2*k*n + 1 (square array).
T(n, k) = (k*(n-k))^2 + 2*k*(n-k) + 1 (number triangle).
A(k, n) = A(n, k).
T(n, n-k) = T(n, k).
A(n, n) = T(2*n, n) = A082044(n).
A(n, n-1) = T(2*n+1, n-1) = A058031(n), n >= 1.
A(n, n-2) = T(2*(n-1), n) = A000583(n-1), n >= 2.
A(n, n-3) = T(2*n-3, n) = A062938(n-3), n >= 3.
Sum_{k=0..n} T(n, k) = A082045(n) (diagonal sums).
Sum_{k=0..n} (-1)^k * T(n, k) = (1/4)*(1+(-1)^n)*(2 - 3*n). - G. C. Greubel, Dec 24 2022

A082046 Square array, A(n, k) = (k*n)^2 + 3*k*n + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 29, 19, 1, 1, 29, 55, 55, 29, 1, 1, 41, 89, 109, 89, 41, 1, 1, 55, 131, 181, 181, 131, 55, 1, 1, 71, 181, 271, 305, 271, 181, 71, 1, 1, 89, 239, 379, 461, 461, 379, 239, 89, 1, 1, 109, 305, 505, 649, 701, 649, 505, 305, 109, 1
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Examples

			Array, A(n, k), begins as:
  1,  1,   1,   1,   1,    1,    1,    1, ... A000012;
  1,  5,  11,  19,  29,   41,   55,   71, ... A028387;
  1, 11,  29,  55,  89,  131,  181,  239, ... A082108;
  1, 19,  55, 109, 181,  271,  379,  505, ... A069131;
  1, 29,  89, 181, 305,  461,  649,  869, ... ;
  1, 41, 131, 271, 461,  701,  991, 1331, ... ;
  1, 55, 181, 379, 649,  991, 1405, 1891, ... ;
  1, 71, 239, 505, 869, 1331, 1891, 2549, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,  1;
  1,  5,   1;
  1, 11,  11,   1;
  1, 19,  29,  19,   1;
  1, 29,  55,  55,  29,   1;
  1, 41,  89, 109,  89,  41,   1;
  1, 55, 131, 181, 181, 131,  55,  1;
  1, 71, 181, 271, 305, 271, 181, 71,  1;

Links

Crossrefs

Cf. A028387, A057721, A069131, A072025, A082039, A082043, A082047, A082105, A082108.

Programs

  • Magma
    [(k*(n-k))^2 + 3*(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 + 3*(k*(n-k)) + 1;
Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)
  • SageMath
    def A082046(n,k): return (k*(n-k))^2 + 3*(k*(n-k)) + 1
flatten([[A082046(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 + 3*k*n + 1 (square array).
A(k, n) = A(n, k).
A(n, n) = T(2*n, n) = A057721(n).
A(n, n+1) = A072025(n).
T(n, k) = (k*(n-k))^2 + 3*k*(n-k) + 1 (antidiagonals).
Sum_{k=0..n} T(n, k) = A082047(n) (antidiagonal sums).
From G. C. Greubel, Dec 22 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*(1 - 2*n).
T(2*n+1, n-1) = T(2*n-1, n-1) = A072025(n-1). (End)

A082110 Array A(n,k) = (k*n)^2 + 5*(k*n) + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 15, 15, 1, 1, 25, 37, 25, 1, 1, 37, 67, 67, 37, 1, 1, 51, 105, 127, 105, 51, 1, 1, 67, 151, 205, 205, 151, 67, 1, 1, 85, 205, 301, 337, 301, 205, 85, 1, 1, 105, 267, 415, 501, 501, 415, 267, 105, 1, 1, 127, 337, 547, 697, 751, 697, 547, 337, 127, 1
Offset: 0

Views

Author

Paul Barry, Apr 04 2003

Keywords

Examples

			Square array, A(n, k), begins as:
  1,   1,   1,   1,    1,    1,    1,    1,    1, ... A000012;
  1,   7,  15,  25,   37,   51,   67,   85,  105, ... A082111;
  1,  15,  37,  67,  105,  151,  205,  267,  337, ... A082112;
  1,  25,  67, 127,  205,  301,  415,  547,  697, ...
  1,  37, 105, 205,  337,  501,  697,  925, 1185, ...
  1,  51, 151, 301,  501,  751, 1051, 1401, 1801, ...
  1,  67, 205, 415,  697, 1051, 1477, 1975, 2545, ...
  1,  85, 267, 547,  925, 1401, 1975, 2647, 3417, ...
  1, 105, 337, 697, 1185, 1801, 2545, 3417, 4417, ...
Antidiagonals, T(n, k), begins as:
  1;
  1,  1;
  1,  7,   1;
  1, 15,  15,   1;
  1, 25,  37,  25,   1;
  1, 37,  67,  67,  37,   1;
  1, 51, 105, 127, 105,  51,   1;
  1, 67, 151, 205, 205, 151,  67,  1;
  1, 85, 205, 301, 337, 301, 205, 85,  1;

Links

Crossrefs

Cf. A082039, A082043, A082046, A082105, A082111, A082112, A082113, A082114.

Programs

  • Magma
    [(k*(n-k))^2 + 5*(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 + 5*(k*(n-k)) + 1;
Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)
  • SageMath
    def A082110(n,k): return (k*(n-k))^2 + 5*(k*(n-k)) + 1
flatten([[A082110(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 + 5*(k*n) + 1 (Square array).
A(k, n) = A(n, k).
A(2, k) = A082111(k).
A(3, k) = A082112(k).
A(n, n) = T(2*n, n) = A082113(n) (main diagonal).
T(n, k) = (k*(n-k))^2 + 5*k*(n-k) + 1 (number triangle).
Sum_{k=0..n} T(n, k) = A082114(n) (diagonal sums of the array).
From G. C. Greubel, Dec 22 2022: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} (-1)^k*T(n, k) = (1 - 3*n)*(1 + (-1)^n)/2. (End)
Showing 1-6 of 6 results.

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