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.

A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11
Offset: 0

Views

Author

Omar E. Pol, Apr 27 2008

Keywords

Comments

A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.
The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011

Examples

			The square array of nonnegatives together with polygonal numbers begins:
=========================================================
....................... A   A   .   .   A    A    A    A
....................... 0   0   .   .   0    0    1    1
....................... 0   0   .   .   1    1    3    3
....................... 0   0   .   .   6    7    9    9
....................... 0   0   .   .   9    3    6    6
....................... 0   1   .   .   5    2    0    0
....................... 4   2   .   .   7    9    6    7
=========================================================
Nonnegatives . A001477: 0,  1,  2,  3,  4,   5,   6,   7, ...
Triangulars .. A000217: 0,  1,  3,  6, 10,  15,  21,  28, ...
Squares ...... A000290: 0,  1,  4,  9, 16,  25,  36,  49, ...
Pentagonals .. A000326: 0,  1,  5, 12, 22,  35,  51,  70, ...
Hexagonals ... A000384: 0,  1,  6, 15, 28,  45,  66,  91, ...
Heptagonals .. A000566: 0,  1,  7, 18, 34,  55,  81, 112, ...
Octagonals ... A000567: 0,  1,  8, 21, 40,  65,  96, 133, ...
9-gonals ..... A001106: 0,  1,  9, 24, 46,  75, 111, 154, ...
10-gonals .... A001107: 0,  1, 10, 27, 52,  85, 126, 175, ...
11-gonals .... A051682: 0,  1, 11, 30, 58,  95, 141, 196, ...
12-gonals .... A051624: 0,  1, 12, 33, 64, 105, 156, 217, ...
...
=========================================================
The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.

Links

Crossrefs

Cf. A001477, A057145, A139601, A139617, A139618, A139620, A180662.
A formal extension negative n is in A326728.
Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011
Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).

Programs

  • Magma
    T:= func< n,k | k*(n*(k-1)+2)/2 >;
A139600:= func< n,k | T(n-k, k) >;
[A139600(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024
  • Maple
    T:= (n, k)-> n*(k-1)*k/2+k:
seq(seq(T(d-k, k), k=0..d), d=0..14);  # Alois P. Heinz, Oct 14 2018
  • Mathematica
    T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)
  • Python
    def A139600Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield x
        x, y = x + y + n, y + n
for n in range(8):
    R = A139600Row(n)
    print([next(R) for  in range(11)]) # _Peter Luschny, Aug 04 2019
  • SageMath
    def T(n,k): return k*(n*(k-1)+2)/2
def A139600(n,k): return T(n-k, k)
flatten([[A139600(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

Formula

T(n,k) = n*(k-1)*k/2+k.
T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016
From Stefano Spezia, Apr 12 2024: (Start)
G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)

Extensions

Edited by Omar E. Pol, Jan 05 2009

A014628 Number of segments (and sides) created by diagonals of an n-gon in general position.

Original entry on oeis.org

3, 8, 20, 45, 91, 168, 288, 465, 715, 1056, 1508, 2093, 2835, 3760, 4896, 6273, 7923, 9880, 12180, 14861, 17963, 21528, 25600, 30225, 35451, 41328, 47908, 55245, 63395, 72416, 82368, 93313, 105315, 118440, 132756, 148333, 165243, 183560
Offset: 3

Views

Author

Mohammad K. Azarian

Keywords

Comments

There is a connection to A014626: number of intersection points of diagonals of n-gon, plus number of vertices, b(n) = n*(n+1)*(n^2-7*n+18)/24 and A006522: number of regions created by sides and diagonals of n-gon, c(n) = (n-1)*(n-2)*(n^2-3*n+12)/24. These are related by the Euler-formula: b(n) + c(n) - a(n) = 1. - Georg Wengler, Mar 31 2005

Links

Crossrefs

Cf. A014626, A006522, A134393.

Programs

  • Mathematica
    Table[Binomial[n,2]+2Binomial[n,4],{n,3,50}] (* Harvey P. Dale, Oct 03 2020 *)

Formula

a(n) = (n^4-6*n^3+17*n^2-24*n)/12 + n; or equally n*(n-1)*(n^2-5*n+12)/12.
G.f.: x^3*(3-7*x+10*x^2-5*x^3+x^4)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = C(n,2) + 2*C(n,4). - Gary Detlefs, Jun 06 2010

Extensions

G.f. proposed by Maksym Voznyy, checked and corrected by R. J. Mathar, Sep 16 2009
More terms from Erich Friedman
Offset corrected by Mohammad K. Azarian, Nov 19 2008
Offset corrected by Eric Rowland, Aug 15 2017

A134392 A077028 * A000012, that is Rascal's triangle (as matrix) multiplied by a lower triangular matrix of ones (main diagonal of ones included).

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 15, 14, 10, 5, 1, 26, 25, 20, 13, 6, 1, 42, 41, 35, 26, 16, 7, 1, 64, 63, 56, 45, 32, 19, 8, 1, 93, 92, 84, 71, 55, 38, 22, 9, 1, 130, 129, 120, 105, 86, 65, 44, 25, 10, 1, 176, 175, 165, 148, 126, 101, 75, 50, 28, 11, 1
Offset: 1

Views

Author

Gary W. Adamson, Oct 23 2007

Keywords

Comments

Left border = A000125.
Row sums = A134393.

Examples

			First few rows of the triangle:
   1;
   2,  1;
   4,  3,  1;
   8,  7,  4,  1;
  15, 14, 10,  5,  1;
  26, 25, 20, 13,  6,  1;
  42, 41, 35, 26, 16,  7,  1;
  ...

Crossrefs

Cf. A077028, A000125, A134393.

Programs

  • Mathematica
    rows = 11;
R[n_, k_] /; k <= n := k (n - k) + 1; R[0, 0] = 1; R[, ] = 0;
MR = Table[R[n, k], {n, 0, rows-1}, {k, 0, rows-1}];
MB = Table[Boole[0 <= k <= n], {n, 0, rows-1}, {k, 0, rows -1}];
T = MR.MB;
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 01 2020 *)

Formula

A077028 * A000012 as infinite lower triangular matrices.
Triangle read by rows, partial sums starting from the right of A077028.

Extensions

Typos corrected by Jean-François Alcover, Apr 01 2020
Showing 1-3 of 3 results.

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