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

A189375 Expansion of 1/((1-x)^5*(x^3+x^2+x+1)^3).

Original entry on oeis.org

1, 2, 3, 4, 8, 12, 16, 20, 30, 40, 50, 60, 80, 100, 120, 140, 175, 210, 245, 280, 336, 392, 448, 504, 588, 672, 756, 840, 960, 1080, 1200, 1320, 1485, 1650, 1815, 1980, 2200, 2420, 2640, 2860, 3146, 3432, 3718, 4004, 4368
Offset: 0

Views

Author

Johannes W. Meijer, Apr 29 2011

Keywords

Comments

The Gi1 triangle sums of A139600 lead to the sequence given above, see the formulas. For the definitions of the Gi1 and other triangle sums see A180662.

Links

Crossrefs

Cf. A139600, A189374, A189376.

Programs

  • Maple
    a:= n-> coeff(series(1/((1-x)^5*(x^3+x^2+x+1)^3), x, n+1), x, n):
seq(a(n), n=0..50);
  • Mathematica
    CoefficientList[Series[1/((1-x)^5(x^3+x^2+x+1)^3),{x,0,50}],x] (* or *) LinearRecurrence[{2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1},{1,2,3,4,8,12,16,20,30,40,50,60,80,100},50] (* Harvey P. Dale, Dec 05 2014 *)

Formula

a(n) = sum(A056594(n-k)*A115269(k), k=0..n).
Gi1(n) = A189375(n-4) - A189375(n-5) - A189375(n-8) + 2*A189375(n-9) with A189375(n)=0 for n <= -1.
a(n) = (2*n^4+56*n^3+538*n^2+2044*n+2469+3*((2*n^2+28*n+89)*(-1)^n+(4*(-1)^((2*n-1+(-1)^n)/4)*(n^2+16*n+57-(n^2+12*n+29)*(-1)^n))))/3072. - Luce ETIENNE, Jun 25 2015

A189376 Expansion of 1/((1-x)^5*(x^3+x^2+x+1)^2).

Original entry on oeis.org

1, 3, 6, 10, 17, 27, 40, 56, 78, 106, 140, 180, 230, 290, 360, 440, 535, 645, 770, 910, 1071, 1253, 1456, 1680, 1932, 2212, 2520, 2856, 3228, 3636, 4080, 4560, 5085, 5655, 6270, 6930, 7645, 8415, 9240, 10120, 11066, 12078
Offset: 0

Views

Author

Johannes W. Meijer, Apr 29 2011

Keywords

Comments

The Gi2 triangle sums of A139600 lead to the sequence given above, see the formulas. For the definitions of the Gi2 and other triangle sums see A180662.

Links

Crossrefs

Cf. A139600, A189374, A189375.

Programs

  • Maple
    a:= n-> coeff (series (1/((1-x)^5*(x^3+x^2+x+1)^2), x, n+1), x, n):
seq (a(n), n=0..50);
  • Mathematica
    CoefficientList[Series[1/((1-x)^5(x^3+x^2+x+1)^2),{x,0,50}],x] (* or *) LinearRecurrence[{3,-3,1,2,-6,6,-2,-1,3,-3,1},{1,3,6,10,17,27,40,56,78,106,140},50] (* Harvey P. Dale, Apr 12 2015 *)

Formula

a(n) = sum(A144678(n-k), k=0..n).
Gi2(n) = A189376(n-1) - A189376(n-2) - A189376(n-5) + 2*A189376(n-6) with A189376(n)=0 for n <= -1.
a(0)=1, a(1)=3, a(2)=6, a(3)=10, a(4)=17, a(5)=27, a(6)=40, a(7)=56, a(8)=78, a(9)=106, a(10)=140, a(n)=3*a(n-1)-3*a(n-2)+a(n-3)+ 2*a(n-4)- 6*a(n-5)+6*a(n-6)-2*a(n-7)-a(n-8)+3*a(n-9)-3*a(n-10)+a(n-11). - Harvey P. Dale, Apr 12 2015
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