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 1-5 of 5 results.

A082044 Main diagonal of A082043: a(n) = n^4 + 2*n^2 + 1.

Original entry on oeis.org

1, 4, 25, 100, 289, 676, 1369, 2500, 4225, 6724, 10201, 14884, 21025, 28900, 38809, 51076, 66049, 84100, 105625, 131044, 160801, 195364, 235225, 280900, 332929, 391876, 458329, 532900, 616225, 708964, 811801, 925444, 1050625, 1188100
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Comments

a(n) = longest side b of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (n^2+2, n^4+2*n^2+1, n^4+3*n^2+1).

Examples

			G.f. = 1 + 4*x + 25*x^2 + 100*x^3 + 289*x^4 + 676*x^5 + 1369*x^6 + ...

Links

Crossrefs

Cf. A000217, A002061 A002522, A059826, A082043, A082047.
See A120062 for sequences related to integer-sided triangles with integer inradius n.

Programs

  • Magma
    [(n^2+1)^2: n in [0..40]]; // G. C. Greubel, Dec 24 2022
  • Maple
    seq(fibonacci(3,n)^2,n=0..33); # Zerinvary Lajos, Apr 09 2008
  • Mathematica
    Fibonacci[3,Range[0,40]]^2 (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,4,25,100,289},40] (* Harvey P. Dale, Feb 27 2015 *)
  • PARI
    a(n) = n^4+2*n^2+1; \\ Michel Marcus, Jan 22 2016
  • SageMath
    [(n^2+1)^2 for n in range(41)] # G. C. Greubel, Dec 24 2022

Formula

a(n) = n^4 + 2*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, Feb 27 2015
a(n) = (4*A000217(n-1)^2 + 2*A002061(n))^2 / a(n-1). - Bruce J. Nicholson, Apr 17 2017
a(n) = A002522(n)^2 = (n^2 + 1)^2 = a(-n) for all n in Z. - Michael Somos, Apr 17 2017
G.f.: (1 - x + 15*x^2 + 5*x^3 + 4*x^4) / (1 - x)^5. - Michael Somos, Apr 17 2017
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=0} 1/a(n) = Pi^2*csch(Pi)^2/4 + Pi*coth(Pi)/4 + 1/2.
Sum_{n>=0} (-1)^n/a(n) = Pi^2*csch(Pi)*coth(Pi)/4 + Pi*csch(Pi)/4 + 1/2. (End)
E.g.f.: (1 + 3*x + 9*x^2 + 6*x^3 + x^4)*exp(x). - 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)

A082045 Diagonal sums of number array A082043.

Original entry on oeis.org

1, 2, 6, 20, 59, 150, 336, 680, 1269, 2218, 3674, 5820, 8879, 13118, 18852, 26448, 36329, 48978, 64942, 84836, 109347, 139238, 175352, 218616, 270045, 330746, 401922, 484876, 581015, 691854, 819020, 964256, 1129425, 1316514, 1527638
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Links

Crossrefs

Cf. A082043, A082047.

Programs

  • Magma
    [(n^5+10*n^3+19*n+30)/30: n in [0..40]]; // G. C. Greubel, Dec 24 2022
  • Mathematica
    LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,6,20,59,150},41] (* Harvey P. Dale, Mar 18 2018 *)
  • SageMath
    [(n^5+10*n^3+19*n+30)/30 for n in range(41)] # G. C. Greubel, Dec 24 2022

Formula

a(n) = (n^5+10*n^3+19*n+30)/30.
G.f.: (1-4*x+9*x^2-6*x^3+4*x^4)/(1-x)^6. - R. J. Mathar, Oct 29 2014
E.g.f.: (1/30)*(30 +30*x +45*x^2 +35*x^3 +10*x^4 +x^5)*exp(x). - G. C. Greubel, Dec 24 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

A082113 a(n) = n^4 + 5*n^2 + 1.

Original entry on oeis.org

1, 7, 37, 127, 337, 751, 1477, 2647, 4417, 6967, 10501, 15247, 21457, 29407, 39397, 51751, 66817, 84967, 106597, 132127, 162001, 196687, 236677, 282487, 334657, 393751, 460357, 535087, 618577, 711487, 814501, 928327, 1053697, 1191367
Offset: 0

Views

Author

Paul Barry, Apr 04 2003

Keywords

Comments

Main diagonal of number array A082110.

Links

Crossrefs

Cf. A082044, A082047, A082106.

Programs

  • Magma
    [n^4+5*n^2+1: n in [0..40]]; // G. C. Greubel, Dec 22 2022
  • Mathematica
    Table[n^4+5n^2+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,7,37,127,337},40] (* Harvey P. Dale, May 16 2019 *)
  • SageMath
    [n^4+5*n^2+1 for n in range(41)] # G. C. Greubel, Dec 22 2022

Formula

a(n) = n^4 + 5*n^2 + 1.
G.f.: (1+2*x+12*x^2+2*x^3+7*x^4) / (1-x)^5. - R. J. Mathar, Dec 03 2014
E.g.f.: (1 + 6*x + 12*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 22 2022
Showing 1-5 of 5 results.

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

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