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.

A144562 Triangle read by rows: T(n, k) = 2*n*k + n + k - 1.

Original entry on oeis.org

3, 6, 11, 9, 16, 23, 12, 21, 30, 39, 15, 26, 37, 48, 59, 18, 31, 44, 57, 70, 83, 21, 36, 51, 66, 81, 96, 111, 24, 41, 58, 75, 92, 109, 126, 143, 27, 46, 65, 84, 103, 122, 141, 160, 179, 30, 51, 72, 93, 114, 135, 156, 177, 198, 219, 33, 56, 79, 102, 125, 148, 171, 194, 217, 240, 263
Offset: 1

Views

Author

Vincenzo Librandi, Jan 06 2009

Keywords

Comments

Rearrangement of A153238, numbers n such that 2*n+3 is not prime (we have 2*T(n,k) + 3 = (2*n+1)*(2*k+1), as 2*n+3 is odd it consists of (at least) two odd factors and all such factors appear by definition).

Examples

			Triangle begins:
   3;
   6, 11;
   9, 16, 23;
  12, 21, 30, 39;
  15, 26, 37, 48,  59;
  18, 31, 44, 57,  70,  83;
  21, 36, 51, 66,  81,  96, 111;
  24, 41, 58, 75,  92, 109, 126, 143;
  27, 46, 65, 84, 103, 122, 141, 160, 179;
  ...

Links

Crossrefs

Cf. A067076, A144640, A153238.
Main diagonal gives A142463.
T(2n,n) gives A180863(n+1).

Programs

  • Magma
    [2*n*k+n+k-1: k in [1..n], n in [1..11]]; /* or, see example: */ [[2*n*k+n+k-1: k in [1..n]]: n in [1..9]]; // Bruno Berselli, Dec 04 2011
  • Maple
    A144562:= (n,k) -> 2*n*k +n +k -1; seq(seq(A144562(n,k), k=1..n), n=1..12); # G. C. Greubel, Mar 01 2021
  • Mathematica
    T[n_, k_]:= 2*n*k +n +k -1; Table[T[n, k], {n, 11}, {k, n}]//Flatten
  • PARI
    T(n,k)=2*n*k+n+k-1 \\ Charles R Greathouse IV, Dec 28 2011
  • Sage
    flatten([[2*n*k+n+n-1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 01 2021

Formula

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n - 1)/2 = A144640(n). - G. C. Greubel, Mar 01 2021
G.f.: x*y*(3 + 2*x*y + 2*x^3*y^2 - x^2*y*(6 + y))/((1 - x)^2*(1 - x*y)^3). - Stefano Spezia, Nov 04 2024

Extensions

Edited by Ray Chandler, Jan 07 2009

A178242 Numerator of n*(5+n)/((n+1)*(n+4)).

Original entry on oeis.org

0, 3, 7, 6, 9, 25, 33, 21, 26, 63, 75, 44, 51, 117, 133, 75, 84, 187, 207, 114, 125, 273, 297, 161, 174, 375, 403, 216, 231, 493, 525, 279, 296, 627, 663, 350, 369, 777, 817, 429, 450, 943, 987, 516, 539, 1125, 1173, 611, 636, 1323, 1375, 714, 741, 1537, 1593
Offset: 0

Views

Author

Paul Curtz, Dec 20 2010

Keywords

Comments

Sequence of differences denominator(n) - numerator(n) = 1,2,2,1... = A014695(n).
Denominator: A160050(n+2).

Examples

			The reduced fractions are 0, 3/5, 7/9, 6/7, 9/10, 25/27, 33/35, 21/22, 26/27, 63/65, 75/77, 44/45, ..

Links

Crossrefs

Cf. A001793, A014695, A028557, A060819, A160050, A176027.

Programs

  • Magma
    [Floor(n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8) : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011
  • Maple
    A178242 := proc(n) n*(5+n)/(n+1)/(n+4) ;  numer(%) ;end proc:
seq(A178242(n),n=0..80) ; # R. J. Mathar, Dec 20 2010
  • Mathematica
    f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 5] &, 50, 0]
Table[Numerator[n*(5+n)/((n+1)*(n+4))], {n,0,50}] (* G. C. Greubel, Sep 21 2018 *)
  • PARI
    vector(50, n, n--; numerator(n*(5+n)/((n+1)*(n+4)))) \\ G. C. Greubel, Sep 21 2018

Formula

a(n) = numerator(A176027(n)/A001793(n+1)).
a(n) = A060819(n)*A060819(n+5).
a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9).
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).
G.f.: x*(-3+2*x-3*x^2-3*x^3+x^7) / ( (x-1)^3*(x^2+1)^3 ).
a(n) = n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8 = n*(n+5)*(3-i^(n*(n+1)))/8, where i=sqrt(-1); also a(n) = a(n-4)*A028557(n)/A028557(n-4) for n>4. - Bruno Berselli, Dec 30 2010
From Peter Bala, Aug 07 2022: (Start)
a(n) = numerator of n*(n+5)/4.
a(n) is quasi-polynomial in n: a(4*n) = n*(4*n+5) = A343560(n+1); a(4*n+1) = (2*n+3)*(4*n+1); a(4*n+2) = (2*n+1)*(4*n+7); a(4*n+3) = (n+2)*(4*n+3) = A180863(n+2). (End)
Sum_{n>=1} 1/a(n) = 112/75 - Pi/10. - Amiram Eldar, Aug 16 2022

A343560 a(n) = (n-1)*(4*n+1).

Original entry on oeis.org

0, 9, 26, 51, 84, 125, 174, 231, 296, 369, 450, 539, 636, 741, 854, 975, 1104, 1241, 1386, 1539, 1700, 1869, 2046, 2231, 2424, 2625, 2834, 3051, 3276, 3509, 3750, 3999, 4256, 4521, 4794, 5075, 5364, 5661, 5966, 6279, 6600, 6929, 7266, 7611, 7964, 8325
Offset: 1

Views

Author

Zachary Dove, Apr 19 2021

Keywords

Comments

A polynomial curve. However, write 0, 1, 2, ... in a square spiral, with 0 at the origin and 1 immediately below it; sequence gives numbers parallel to the negative y-axis (see Example section). This sequence only encounters composite numbers as it expands to infinity.

Examples

			  On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence, not including "0", will lie parallel to the negative y-axis, located within the fourth quadrant, as seen in the example below:
  99  64--65--66--67--68--69--70--71--72
   |   |                               |
  98  63  36--37--38--39--40--41--42  73
   |   |   |                       |   |
  97  62  35  16--17--18--19--20  43  74
   |   |   |   |               |   |   |
  96  61  34  15   4---5---6  21  44  75
   |   |   |   |   |       |   |   |   |
  95  60  33  14   3  *0*  7  22  45  76
   |   |   |   |   |   |   |   |   |   |
  94  59  32  13   2---1   8  23  46  77
   |   |   |   |           |   |   |   |
  93  58  31  12--11--10--*9* 24  47  78
   |   |   |                   |   |   |
  92  57  30--29--28--27-*26*-25  48  79
   |   |                           |   |
  91  56--55--54--53--52-*51*-50--49  80
   |                                   |
  90--89--88--87--86--85-*84*-83--82--81

Links

Crossrefs

Cf. A164754, A001107, A180863.

Programs

  • C
    int a(int n) { return 4*n*n-3*n-1; }
  • Maple
    A343560 := n -> 4*n^2 - 3*n - 1;
seq(A343560(n), n = 1 .. 50);
  • Mathematica
    A343560[n_] := (4*n + 1)*(n - 1); Array[A343560, 100] (* or *)
LinearRecurrence[{3, -3, 1}, {0, 9, 26}, 100] (* Paolo Xausa, Aug 27 2025 *)

Formula

a(n) = A164754(n+1) + 1 = A001107(n+1), for n >= 2.
G.f.: x^2*(-9+x)/(x-1)^3 . - R. J. Mathar, Sep 15 2021
Sum_{n>=2} 1/a(n) = 24/25 -3*log(2)/5 -Pi/10. - R. J. Mathar, May 30 2022
Showing 1-3 of 3 results.

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