A051872 -id:A051872 - cp's OEIS frontend

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

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

Omar E. Pol, Apr 27 2008

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

			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.

Alois P. Heinz, Rows n = 0..140, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.
Index to sequences related to polygonal numbers

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).

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

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

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 *)

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

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

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)

Edited by Omar E. Pol, Jan 05 2009

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66

Offset: 0

Omar E. Pol, Apr 27 2008

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. - Omar E. Pol, Dec 21 2008

			The square array of polygonal numbers begins:
========================================================
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,
And so on ..............................................
========================================================

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

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).
Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.
Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.
Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.
Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.
Cf. A139619, A139620.

T:= func< n,k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n,k | T(n-k, k) >;
[A139601(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def T(n,k): return k*((n+1)*(k-1)+2)/2
def A139601(n,k): return T(n-k, k)
flatten([[A139601(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

A287326 Triangle read by rows: T(n, k) = 6k(n-k) + 1; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 25, 19, 1, 1, 25, 37, 37, 25, 1, 1, 31, 49, 55, 49, 31, 1, 1, 37, 61, 73, 73, 61, 37, 1, 1, 43, 73, 91, 97, 91, 73, 43, 1, 1, 49, 85, 109, 121, 121, 109, 85, 49, 1, 1, 55, 97, 127, 145, 151, 145, 127, 97, 55, 1, 1, 61, 109, 145, 169, 181, 181, 169, 145, 109, 61, 1

Offset: 0

Kolosov Petro, Aug 31 2017

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(1, n, k), i.e T(n, k) is partial case of L(m, n, k) for m = 1.
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
  ----------------------------------------
  k=    0   1   2   3   4   5   6   7   8
  ----------------------------------------
  n=0:  1;
  n=1:  1,  1;
  n=2:  1,  7,  1;
  n=3:  1, 13, 13,  1;
  n=4:  1, 19, 25, 19,  1;
  n=5:  1, 25, 37, 37, 25,  1;
  n=6:  1, 31, 49, 55, 49, 31,  1;
  n=7:  1, 37, 61, 73, 73, 61, 37,  1;
  n=8:  1, 43, 73, 91, 97, 91, 73, 43,  1;

Georg Fischer, Table of n, a(n) for n = 0..495 [rows 0..10 and 12..30 from Kolosov Petro]
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Columns k=0..6 give A000012, A016921, A017533, A161705, A103214, A128470, A158065.
Column sums k=0..4 give A000027, A000567, A051866, A051872, A255185.
Row sums give A001093.
Various cases of L(m, n, k): This sequence (m=1), A300656(m=2), A300785(m=3). See comments for L(m, n, k).
Differences of cubes n^3 are T(A000124(n), 1).
Cf. A000124, A000578, A003154, A003215, A007318, A008458, A016969.
Cf. A038593, A055012, A068601, A077028, A094053, A166873, A227776.
Cf. A294317, A302971, A304042.

Flat(List([0..11],n->List([0..n],k->6*k*(n-k)+1))); # Muniru A Asiru, Oct 09 2018

/* As triangle */ [[6*k*(n-k) + 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 26 2018

T := (n, k) -> 6*k*(n-k) + 1:
seq(seq(T(n, k), k=0..n), n=0..11); # Muniru A Asiru, Oct 09 2018

T[n_, k_] := 6 k (n - k) + 1; Column[Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jun 02 2019 *)

t(n, k) = 6*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */
trianglerows(9) \\ Felix Fröhlich, Jan 09 2018

def A287326(n,k): return 6*k*(n-k) + 1
flatten([[A287326(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 25 2024

T(n, k) = 6*k*(n-k) + 1.
G.f. of column k: n^k*(1+(6*k-1)*n)/(1-n)^2.
G.f.: (1 - x - x*y + 7*x^2*y)/((1 - x)^2*(1 - x*y)^2). - Stefano Spezia, Oct 09 2018 [Adapted by Stefano Spezia, Sep 25 2024]
From Kolosov Petro, Jun 05 2019: (Start)
T(n, k) = 1/2 * T(A294317(n, k), k) + 1/2.
T(n+1, k) = 2*T(n, k) - T(n-1, k), for n >= k.
T(n, k) = 6*A077028(n, k) - 5.
T(2n, n) = A227776(n).
T(2n+1, n) = A003154(n+1).
T(2n+3, n) = A166873(n+1).
Sum_{k=0..n-1} T(n, k) = Sum_{k=1..n} T(n, k) = A000578(n).
Sum_{k=1..n-1} T(n, k) = A068601(n).
(n+1)^3 - n^3 = T(A000124(n), 1). (End)
Sum_{k=0..n} (-1)^k*T(n, k) = (-1/2)*(1 + (-1)^n)*A016969(floor(n/2) - 1). - G. C. Greubel, Sep 25 2024

A188896 Numbers n such that there is no square n-gonal number greater than 1.

Original entry on oeis.org

10, 20, 52, 164, 340, 580, 884, 1252, 1684, 2180, 2740, 4052, 4804, 5620, 6500, 7444, 8452, 9524, 10660, 11860, 13124, 14452, 15844, 17300, 18820, 20404, 22052, 25540, 27380, 29284, 31252, 33284, 35380, 37540, 39764, 42052, 44404, 46820, 49300, 51844

Offset: 1

T. D. Noe, Apr 13 2011

nonn

It is easy to find squares that are triangular, pentagonal, hexagonal, etc. So it is somewhat surprising that there are no square 10-gonal numbers other than 0 and 1. For these n, the equation 2*x^2 = (n-2)*y^2 - (n-4)*y has no integer solutions x>1 and y>1.
Chu shows how to transform the equation into a generalized Pell equation. When n has the form 2k^2+2 (A005893), then the Pell equation has only a finite number of solutions and it is simple to select the n that produce no integer solutions greater than 1.
The general case is in A188950.

Muniru A Asiru, Table of n, a(n) for n = 1..244
Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781-802.

Cf. A001107 (10-gonal numbers), A051872 (20-gonal numbers), A188892, A100252, A188950, A005893.

Subsequence of A271624. - Muniru A Asiru, Oct 16 2016

P[n_,k_]:=1/2n(n(k-2)+4-k); data1=2#^2+2&/@Range[2,161]; data2=Head[Reduce[m^2==P[n,#] && 1Ant King, Mar 01 2012 *)

A255184 25-gonal numbers: a(n) = n(23n-21)/2.

Original entry on oeis.org

0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, 8722, 9367, 10035, 10726, 11440, 12177, 12937, 13720, 14526, 15355, 16207, 17082, 17980

Offset: 0

Luciano Ancora, Apr 03 2015

If b(n,k) = n*((k-2)*n-(k-4))/2 is n-th k-gonal number, then b(n,k) = A000217(n) + (k-3)* A000217(n-1) (see Deza in References section, page 21, where the formula is attributed to Bachet de Méziriac).
Also, b(n,k) = b(n,k-1) + A000217(n-1) (see Deza and Picutti in References section, page 20 and 137 respectively, where the formula is attributed to Nicomachus). Some examples:
for k=4, A000290(n) = A000217(n) + A000217(n-1);
for k=5, A000326(n) = A000290(n) + A000217(n-1);
for k=6, A000384(n) = A000326(n) + A000217(n-1), etc.
This is the case k=25.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (23rd row of the table).
E. Picutti, Sul numero e la sua storia, Feltrinelli Economica (1977), pages 131-147.

Luciano Ancora, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. k-gonal numbers: A000217 (k=3), A000290 (k=4), A000326 (k=5), A000384 (k=6), A000566 (k=7), A000567 (k=8), A001106 (k=9), A001107 (k=10), A051682 (k=11), A051624 (k=12), A051865 (k=13), A051866 (k=14), A051867 (k=15), A051868 (k=16), A051869 (k=17), A051870 (k=18), A051871 (k=19), A051872 (k=20), A051873 (k=21), A051874 (k=22), A051875 (k=23), A051876 (k=24), this sequence (k=25), A255185 (k=26), A255186 (k=27), A161935 (k=28), A255187 (k=29), A254474 (k=30).

k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // Bruno Berselli, Apr 10 2015

Mathematica
```
Table[n (23 n - 21)/2, {n, 40}]
```

a(n)=n*(23*n-21)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(-1 - 22*x)/(-1 + x)^3.
a(n) = A000217(n) + 22*A000217(n-1) = A051876(n) + A000217(n-1), see comments.
Product_{n>=2} (1 - 1/a(n)) = 23/25. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 23*x^2/2). - Nikolaos Pantelidis, Feb 05 2023

A317302 Square array T(n,k) = (n - 2)(k - 1)k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, -3, 0, 1, 2, 0, -8, 0, 1, 3, 3, -2, -15, 0, 1, 4, 6, 4, -5, -24, 0, 1, 5, 9, 10, 5, -9, -35, 0, 1, 6, 12, 16, 15, 6, -14, -48, 0, 1, 7, 15, 22, 25, 21, 7, -20, -63, 0, 1, 8, 18, 28, 35, 36, 28, 8, -27, -80, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, -35, -99, 0, 1, 10, 24, 40, 55, 66

Offset: 0

Omar E. Pol, Aug 09 2018

Note that the formula gives several kinds of numbers, for example:
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the nonnegative numbers.
For n >= 3, row n gives the n-gonal numbers (see Crossrefs section).

			Array begins:
------------------------------------------------------------------------
n\k  Numbers       Seq. No.   0   1   2   3   4    5    6    7    8
------------------------------------------------------------------------
0    ............ (A258837):  0,  1,  0, -3, -8, -15, -24, -35, -48, ...
1    ............ (A080956):  0,  1,  1,  0, -2,  -5,  -9, -14, -20, ...
2    Nonnegatives  A001477:   0,  1,  2,  3,  4,   5,   6,   7,   8, ...
3    Triangulars   A000217:   0,  1,  3,  6, 10,  15,  21,  28,  36, ...
4    Squares       A000290:   0,  1,  4,  9, 16,  25,  36,  49,  64, ...
5    Pentagonals   A000326:   0,  1,  5, 12, 22,  35,  51,  70,  92, ...
6    Hexagonals    A000384:   0,  1,  6, 15, 28,  45,  66,  91, 120, ...
7    Heptagonals   A000566:   0,  1,  7, 18, 34,  55,  81, 112, 148, ...
8    Octagonals    A000567:   0,  1,  8, 21, 40,  65,  96, 133, 176, ...
9    9-gonals      A001106:   0,  1,  9, 24, 46,  75, 111, 154, 204, ...
10   10-gonals     A001107:   0,  1, 10, 27, 52,  85, 126, 175, 232, ...
11   11-gonals     A051682:   0,  1, 11, 30, 58,  95, 141, 196, 260, ...
12   12-gonals     A051624:   0,  1, 12, 33, 64, 105, 156, 217, 288, ...
13   13-gonals     A051865:   0,  1, 13, 36, 70, 115, 171, 238, 316, ...
14   14-gonals     A051866:   0,  1, 14, 39, 76, 125, 186, 259, 344, ...
15   15-gonals     A051867:   0,  1, 15, 42, 82, 135, 201, 280, 372, ...
...

Omar E. Pol, Polygonal numbers.
The OEIS, Polygonal numbers.
University of Surrey, Dept. of Mathematics, Polygonal Numbers - or Numbers as Shapes.
Eric Weisstein's World of Mathematics, Figurate Number.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Column 0 gives A000004.
Column 1 gives A000012.
Column 2 gives A001477, which coincides with the row numbers.
Main diagonal gives A060354.
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the same as column 2.
For n >= 3, row n gives the n-gonal numbers: A000217 (n=3), A000290 (n=4), A000326 (n=5), A000384 (n=6), A000566 (n=7), A000567 (n=8), A001106 (n=9), A001107 (n=10), A051682 (n=11), A051624 (n=12), A051865 (n=13), A051866 (n=14), A051867 (n=15), A051868 (n=16), A051869 (n=17), A051870 (n=18), A051871 (n=19), A051872 (n=20), A051873 (n=21), A051874 (n=22), A051875 (n=23), A051876 (n=24), A255184 (n=25), A255185 (n=26), A255186 (n=27), A161935 (n=28), A255187 (n=29), A254474 (n=30).
Cf. A139600, A139601.
Cf. A303301 (similar table but with generalized polygonal numbers).

T(n,k) = A139600(n-2,k) if n >= 2.

T(n,k) = A139601(n-3,k) if n >= 3.

A274968 Even numbers n >= 4 which are not m-gonal number for 3 <= m < n.

Original entry on oeis.org

4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 302

Offset: 1

Daniel Forgues, Jul 12 2016

nonn

An m-gonal number, m >= 3, i.e., of the form n = (k/2)*[(m-2)*k - (m-4)], yields a nontrivial factorization of n if and only if of order k >= 3.
Except for a(1) = 4, all a(n) are congruent to 2 (mod 6), although from 8 to 302, the numbers
     92:  5-gonal of order 8,
    176:  5-gonal of order 11, 8-gonal of order 8,
    260: 11-gonal of order 8,
  are not in this sequence.
Even numbers n for which A176948(n) = n.
Since we are looking for solutions of (m-2)*k^2 - (m-4)*k - 2*n = 0, with m >= 3 and k >= 3, the largest order k we need to consider is
    k = {(m-4) + sqrt[(m-4)^2 + 8*(m-2)*n]}/[2*(m-2)] with m = 3, thus
    k <= (1/2)*{-1 + sqrt[1 + 8*n]}.
Or, since we are looking for solutions of 2n = m*k*(k-1) - 2*k*(k-2), with m >= 3 and k >= 3, the largest m we need to consider is
    m = [2n + 2*k*(k-2)]/[k*(k-1)] with k = 3, thus m <= (n+3)/3.
Composite numbers n which are divisible by 3 are m-gonal numbers of order 3, with m = (n + 3)/3. Thus all a(n) are coprime to 3.
a(1) = 4 is the only square number: 4-gonal with order k = 2.
All integers of the form n = 6j + 4, with j >= 1, are m-gonal numbers of order k = 4, with m = j + 2, which means that none are in this sequence. - Daniel Forgues, Aug 01 2016

			20 is in this sequence because 20 is trivially a 20-gonal number of order k = 2 (element of A051872) but not an m-gonal number for 3 <= k <= (1/2)*{-1 + sqrt[1 + 8*20]}.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
OEIS Wiki, Polygonal numbers

Cf. A051872, A176948, A176949, A274967.

lista(nn) = {forstep(n=4, nn, 2, sp = n; forstep(k=n, 3, -1, if (ispolygonal(n, k), sp=k);); if (sp == n, print1(n, ", ")););} \\ Michel Marcus, Sep 06 2016

A274968_list = []
for n in range(4,10**6,2):
    k = 3
    while k*(k+1) <= 2*n:
        if not (2*(k*(k-2)+n)) % (k*(k - 1)):
            break
        k += 1
    else:
        A274968_list.append(n) # Chai Wah Wu, Jul 28 2016

def is_A274968(n):
    if is_odd(n): return False
    for m in (3..(n+3)//3):
        if pari('ispolygonal')(n, m):
            return False
    return True
print([n for n in (3..302) if is_A274968(n)]) # Peter Luschny, Jul 28 2016

A117798 Icosagonal numbers divisible by 20.

Original entry on oeis.org

0, 20, 820, 1200, 3440, 4180, 7860, 8960, 14080, 15540, 22100, 23920, 31920, 34100, 43540, 46080, 56960, 59860, 72180, 75440, 89200, 92820, 108020, 112000, 128640, 132980, 151060, 155760, 175280, 180340, 201300, 206720, 229120, 234900, 258740

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 29 2006

Intersection of A008602 and A051872. - Michel Marcus, Feb 27 2014

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A008602, A051872.

Select[Table[PolygonalNumber[20, n], {n, 0, 200}], Divisible[#, 20] &] (* Amiram Eldar, Mar 22 2021 *)

isok(n) = ispolygonal(n, 20) && !(n % 20); \\ Michel Marcus, Feb 27 2014

A261343 50-gonal numbers: a(n) = 48n(n-1)/2 + n.

Original entry on oeis.org

0, 1, 50, 147, 292, 485, 726, 1015, 1352, 1737, 2170, 2651, 3180, 3757, 4382, 5055, 5776, 6545, 7362, 8227, 9140, 10101, 11110, 12167, 13272, 14425, 15626, 16875, 18172, 19517, 20910, 22351, 23840, 25377, 26962, 28595, 30276, 32005, 33782, 35607, 37480

Offset: 0

Sergey Pavlov, Aug 15 2015

According to the common formula for the polygonal numbers: (s-2)*n*(n-1)/2 + n (here s = 50).

96*a(n) + 23^2 is a square.

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001107, A051872, A254474.

JavaScript
```
function a(n){return 48*n*(n-1)/2+n}
```

[n*(24*n-23): n in [0..40]]; // Vincenzo Librandi, Aug 17 2015

A261343:=n->n*(24*n-23): seq(A261343(n), n=0..40); # Wesley Ivan Hurt, Aug 20 2015

PolygonalNumber[50,Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 11 2019 *)

first(m)=vector(m,n,n--;n*(24*n-23)) \\ Anders Hellström, Aug 15 2015

a(n) = n*(24*n - 23).
G.f.: x*(1+47*x)/(1-x)^3. - Vincenzo Librandi, Aug 17 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Aug 17 2015
E.g.f.: exp(x)*(x + 24*x^2). - Nikolaos Pantelidis, Feb 10 2023

A108667 Triangle read by rows: T(n,k) = 9k*n + 14(n+k) + 20 (0 <= k <= n).

Original entry on oeis.org

20, 34, 57, 48, 80, 112, 62, 103, 144, 185, 76, 126, 176, 226, 276, 90, 149, 208, 267, 326, 385, 104, 172, 240, 308, 376, 444, 512, 118, 195, 272, 349, 426, 503, 580, 657, 132, 218, 304, 390, 476, 562, 648, 734, 820, 146, 241, 336, 431, 526, 621, 716, 811

Offset: 0

Emeric Deutsch, Jun 14 2005

Kekulé numbers for certain benzenoids. T(n,n) = 9n^2 + 28n + 20 = A051872(n+2).

			Triangle begins:
20;
34,57;
48,80,112;
62,103,144,185;

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 102).

Cf. A051872.

T:=proc(n,k) if k<=n then 9*k*n+14*(n+k)+20 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

G.f.: (20 - 6z - 3t*z + t^2*z^2 - 16t*z^2 - 4t^2*z^3)/((1-z)^2*(1-t*z)^3).

cp's OEIS Frontend

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

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A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.

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A287326 Triangle read by rows: T(n, k) = 6*k*(n-k) + 1; n >= 0, 0 <= k <= n.

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A188896 Numbers n such that there is no square n-gonal number greater than 1.

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A255184 25-gonal numbers: a(n) = n*(23*n-21)/2.

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A317302 Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

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A274968 Even numbers n >= 4 which are not m-gonal number for 3 <= m < n.

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

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

A287326 Triangle read by rows: T(n, k) = 6k(n-k) + 1; n >= 0, 0 <= k <= n.

A255184 25-gonal numbers: a(n) = n(23n-21)/2.

A317302 Square array T(n,k) = (n - 2)(k - 1)k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

A261343 50-gonal numbers: a(n) = 48n(n-1)/2 + n.