A086270 -id:A086270 - cp's OEIS frontend

A144257 Weight array of A086270.

1, 2, 0, 3, 1, 0, 4, 2, 1, 0, 5, 3, 2, 1, 0, 6, 4, 3, 2, 1, 0, 7, 5, 4, 3, 2, 1, 0, 8, 6, 5, 4, 3, 2, 1, 0, 9, 7, 6, 5, 4, 3, 2, 1, 0, 10, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 12, 11, 10, 9, 8, 7, 6, 5

Offset: 1

Clark Kimberling, Sep 16 2008

For the definition of weight array, see A144112.
From Gary W. Adamson, Feb 18 2010: (Start)
Identical to an infinite lower triangular matrix with (1,2,3,...) in every column but the leftmost column shifted one row upwards, giving:
  1;
  2, 0;
  3, 1, 0;
  4, 2, 1, 0;
  5, 3, 2, 1, 0;
  ...
Let the triangle = M. Row sums = A000124; M * [1,2,3,...] = A050407 starting with offset 3: (1, 2, 5, 11, 21, 36, ...); and lim_{n->inf} M^n = the odd-indexed Fibonacci numbers, A001519: (1, 2, 5, 13, ...). (End)

			Northwest corner:
  1 2 3 4 5 6 7 8 9
  0 1 2 3 4 5 6 7 8
  0 1 2 3 4 5 6 7 8
  0 1 2 3 4 5 6 7 8
  0 1 2 3 4 5 6 7 8

Cf. A086270.

Cf. A000124, A050407, A001519. - Gary W. Adamson, Feb 18 2010

Row 1 = A000027. All subsequent rows are 0 followed by A000027.

A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 10, 9, 3, 20, 24, 15, 4, 35, 50, 42, 22, 5, 56, 90, 90, 64, 30, 6, 84, 147, 165, 140, 90, 39, 7, 120, 224, 273, 260, 200, 120, 49, 8, 165, 324, 420, 434, 375, 270, 154, 60, 9, 220, 450, 612, 672, 630, 510, 350, 192, 72, 10, 286, 605, 855, 984, 980, 861, 665, 440, 234, 85, 11, 364, 792, 1155, 1380, 1440, 1344, 1127, 840, 540, 280, 99, 12, 455, 1014, 1518, 1870, 2025, 1980, 1764, 1428, 1035, 650, 330, 114, 13, 560, 1274, 1950, 2464, 2750, 2790, 2604, 2240

Offset: 1

Clark Kimberling, Feb 01 2011

This is the (first) accumulation array of A086270; the second is A185733. See A144112 for the definition of accumulation array.

			Northwest corner:
1....4....10...20...35
2....9....24...50...90
3....15...42...90...165
4....22...64...140..260
5....30...90...200..375

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows 1 to 5: A000292, A006002, A059270, A177814, 5*A002411.
Columns 1 to 4: A000027, A055999, A067728, 10*A000096.
Cf. A144112, A086270, A185732.

f[n_,k_]:=k+n*k(k-1)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A086270 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A086270 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185732 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* acc. arr. A185732 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185732 *)

T(n,k) = k*(k+1)*n*(n+1)*(k*n-n+k+5)/12.

A185733 Second accumulation array of the polygonal number array (A086270), by antidiagonals.

Original entry on oeis.org

1, 5, 3, 15, 16, 6, 35, 50, 34, 10, 70, 120, 110, 60, 15, 126, 245, 270, 200, 95, 21, 210, 448, 560, 500, 325, 140, 28, 330, 756, 1036, 1050, 825, 490, 196, 36, 495, 1200, 1764, 1960, 1750, 1260, 700, 264, 45, 715, 1815, 2820, 3360, 3290, 2695, 1820, 960, 345, 55, 1001, 2640, 4290, 5400, 5670, 5096, 3920, 2520, 1275, 440, 66, 1365, 3718, 6270, 8250, 9150, 8820, 7448, 5460, 3375, 1650, 550, 78

Offset: 1

Clark Kimberling, Feb 02 2011

This is the accumulation array of the accumulation array of A086270. The accumulation array of A185733 is A185734, so that A184733 is the weight array of A185734. Thus, the arrays are members of a two-way infinite chain; see A144112 for definitions.

			Northwest corner:
1....5....15....35....70
3....16...50....120...245
6....34...110...270...560
10...60...200...500..1050

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows 1 to 2: A000332, A004320.
Columns 1 to 3: A000219, A096941, 5*A007603.
Cf. A144112, A086270, A185732, A184734.

f[n_,k_]:=k*(1+k)*(2+k)*n*(1+n)*(10+2*k-n+k*n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k*(k+1)*(k+2)*n*(n+1)*(k*n-n+2*k+10)/144.

A185734 Third accumulation array of the polygonal number array (A086270), by antidiagonals.

Original entry on oeis.org

1, 6, 4, 21, 25, 10, 56, 90, 65, 20, 126, 245, 240, 135, 35, 252, 560, 665, 510, 245, 56, 462, 1134, 1540, 1435, 945, 406, 84, 792, 2100, 3150, 3360, 2695, 1596, 630, 120, 1287, 3630, 5880, 6930, 6370, 4606, 2520, 930, 165, 2002, 5940, 10230

Offset: 1

Clark Kimberling, Feb 02 2011

The chain of accumulation arrays is A144257->A086270->A185732->A185733->A184734.

			Northwest corner:
1....6......21.....56....126
4....25.....90....245....560
10...65....240....665...1540
20...135...510...1435...3360

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A144257, A086270, A185732, A185733.

f[n_,k_]:= k*(1+k)*(2+k)*n*(1+n)*(10+2*k-n+k*n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* array A185733 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A185733 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* the formula for A185734 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* array A185734 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A185734 *)

T(n,k) = C(k+3,4)*C(n+2,3)*(k*n-n+3*k+17)/20, k>=1, n>=1.

T(n,k) = Sum_{j=1..n} Sum_{l=1..k} A185733(j,l), by definition.

A376276 Table T(n, k) n > 0, k > 2 read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. The length of the row number n in column k is equal to the n-th k-gonal number A086270.

Original entry on oeis.org

1, 3, 1, 4, 4, 1, 2, 3, 4, 1, 8, 5, 5, 5, 1, 7, 2, 3, 4, 5, 1, 9, 10, 6, 6, 6, 6, 1, 6, 11, 2, 3, 4, 5, 6, 1, 10, 9, 13, 7, 7, 7, 7, 7, 1, 5, 12, 12, 2, 3, 4, 5, 6, 7, 1, 16, 8, 14, 15, 8, 8, 8, 8, 8, 8, 1, 15, 13, 11, 16, 2, 3, 4, 5, 6, 7, 8, 1, 17, 7, 15, 14, 18, 9, 9, 9, 9, 9, 9, 9, 1, 14, 14, 10, 17, 17, 2, 3, 4, 5, 6, 7, 8, 9, 1, 18, 6, 16, 13, 19, 20, 10, 10, 10

Offset: 1

Boris Putievskiy, Sep 18 2024

A209278 presents an algorithm for generating permutations.

The sequence is an intra-block permutation of integer positive numbers.

			Table begins:
  k =      3   4   5   6   7   8
--------------------------------------
  n = 1:   1,  1,  1,  1,  1,  1, ...
  n = 2:   3,  4,  4,  5,  5,  6, ...
  n = 3:   4,  3,  5,  4,  6,  5, ...
  n = 4:   2,  5,  3,  6,  4,  7, ...
  n = 5:   8,  2,  6,  3,  7,  4, ...
  n = 6:   7, 10,  2,  7,  3,  8, ...
  n = 7:   9, 11, 13,  2,  8,  3, ...
  n = 8:   6,  9, 12, 15,  2,  9, ...
  n = 9:  10, 12, 14, 16, 18,  2, ...
  n =10:   5,  8, 11, 14, 17, 20, ...
  n =11:  16, 13, 15, 17, 19, 21, ...
  n =12:  15, 7,  10, 13, 16, 19, ...
  n =13:  17, 14, 16, 18, 20, 22, ...
  n =14:  14,  6,  9, 12, 15, 18, ...
  n =15:  18, 23, 17, 19, 21, 23, ...
  n =16:  13, 22,  8, 11, 14, 17, ...
  n =17:  19, 24, 18, 20, 22, 24, ...
  n =18:  12, 21,  7, 10, 13, 16, ...
  n =19:  20, 25, 30, 21, 23, 25, ...
  n =20:  11, 20, 29,  9, 12, 15, ...
          ... .
For k = 3 the first 4 blocks have lengths 1,3,6 and 10.
For k = 4 the first 3 blocks have lengths 1,4, and 9.
For k = 5 the first 3 blocks have lengths 1,5, and 12.
Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
  1;
  3, 1;
  4, 4, 1;
  2, 3, 4, 1;
  8, 5, 5, 5, 1;
  7, 2, 3, 4, 5, 1;

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 45.

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Polygonal Number.
Index entries for sequences that are permutations of the natural numbers.
Index to sequences related to polygonal numbers

Cf. A000012, A004526, A028242, A074279, A084964, A086270, A209278, A360010, A375725, A375797

T[n_,k_]:=Module[{L,R,P,Res,result},L=Ceiling[Max[x/.NSolve[x^3*(k-2)+3*x^2-x*(k-5)-6*n==0,x,Reals]]];
R=n-(((L-1)^3)*(k-2)+3*(L-1)^2-(L-1)*(k-5))/6;P=Which[OddQ[R]&&OddQ[k*L*(L-1)/2-L^2+2*L],((k*L*(L-1)/2-L^2+2*L+1-R)+1)/2,OddQ[R]&&EvenQ[k*L*(L-1)/2-L^2+2*L],(R+k*L*(L-1)/2-L^2+2*L+1)/2,EvenQ[R]&&OddQ[k*L*(L-1)/2-L^2+2*L],Ceiling[(k*L*(L-1)/2-L^2+2*L+1)/2]+R/2,EvenQ[R]&&EvenQ[k*L*(L-1)/2-L^2+2*L],Ceiling[(k*L*(L-1)/2-L^2+2*L+1)/2]-R/2];
Res=P+((L-1)^3*(k-2)+3*(L-1)^2-(L-1)*(k-5))/6;result=Res;result]
Nmax=6;Table[T[n,k],{n,1,Nmax},{k,3,Nmax+2}]

T(n,k) = P(n,k) + ((L(n,k)-1)^3*(k-2)+3*(L(n,k)-1)^2-(L(n,k)-1)*(k-5))/6, where L(n,k) = ceiling(x(n,k)), x(n,k) is largest real root of the equation x^3*(k - 2) + 3*x^2 - x*(k - 5) - 6*n = 0. R(n,k) = n - ((L(n,k) - 1)^3*(k-2)+3*(L(n,k)-1)^2-(L(n,k)-1)*(k-5))/6. P(n,k) = ((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 2 - R(n,k)) / 2 if R is odd and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is odd, P(n,k) = (R(n,k) + (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2 if R is odd and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is even, P(n,k) = ceiling(((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2) + (R(n,k) / 2) if R is even and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is odd, P(n,k) = ceiling(((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2) - (R(n,k) / 2) if R is even and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is even.

T(1,n) = A000012(n). T(2,n) = A004526(n+7). T(3,n) = A028242(n+6). T(4,n) = A084964(n+5). T(n-2,n) = A000027(n) for n > 3. L(n,3) = A360010(n). L(n,4) = A074279(n).

A057145 Square array of polygonal numbers T(n,k) = ((n-2)k^2 - (n-4)k)/2, n >= 2, k >= 1, read by antidiagonals upwards.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 9, 10, 5, 1, 6, 12, 16, 15, 6, 1, 7, 15, 22, 25, 21, 7, 1, 8, 18, 28, 35, 36, 28, 8, 1, 9, 21, 34, 45, 51, 49, 36, 9, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11, 1, 12, 30, 52, 75, 96, 112

Offset: 2

N. J. A. Sloane, Sep 12 2000

The set of the "nontrivial" entries T(n>=3,k>=3) is in A090466. - R. J. Mathar, Jul 28 2016

T(n,k) is the smallest number that can be expressed as the sum of k consecutive positive integers that differ by n - 2. In other words: T(n,k) is the sum of k terms of the arithmetic progression with common difference n - 2 and 1st term 1, (see the example). - Omar E. Pol, Apr 29 2020

			Array T(n k) (n >= 2, k >= 1) begins:
  1,  2,  3,  4,   5,   6,   7,   8,   9,  10,  11, ...
  1,  3,  6, 10,  15,  21,  28,  36,  45,  55,  66, ...
  1,  4,  9, 16,  25,  36,  49,  64,  81, 100, 121, ...
  1,  5, 12, 22,  35,  51,  70,  92, 117, 145, 176, ...
  1,  6, 15, 28,  45,  66,  91, 120, 153, 190, 231, ...
  1,  7, 18, 34,  55,  81, 112, 148, 189, 235, 286, ...
  1,  8, 21, 40,  65,  96, 133, 176, 225, 280, 341, ...
  1,  9, 24, 46,  75, 111, 154, 204, 261, 325, 396, ...
  1, 10, 27, 52,  85, 126, 175, 232, 297, 370, 451, ...
  1, 11, 30, 58,  95, 141, 196, 260, 333, 415, 506, ...
  1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, ...
  1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, ...
  1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, ...
-------------------------------------------------------
From _Wolfdieter Lang_, Nov 04 2014: (Start)
The triangle a(k, m) begins:
  k\m 1  2  3  4  5   6   7   8   9  10  11  12 13 14 ...
  2:  1
  3:  1  2
  4:  1  3  3
  5:  1  4  6  4
  6:  1  5  9 10  5
  7:  1  6 12 16 15   6
  8:  1  7 15 22 25  21   7
  9:  1  8 18 28 35  36  28   8
  10: 1  9 21 34 45  51  49  36   9
  11: 1 10 24 40 55  66  70  64  45  10
  12: 1 11 27 46 65  81  91  92  81  55  11
  13: 1 12 30 52 75  96 112 120 117 100  66  12
  14: 1 13 33 58 85 111 133 148 153 145 121  78 13
  15: 1 14 36 64 95 126 154 176 189 190 176 144 91 14
  ...
-------------------------------------------------------
a(2,1) = T(2,1), a(6, 3) = T(4, 3). (End)
.
From _Omar E. Pol_, May 03 2020: (Start)
Illustration of the corner of the square array:
.
  1       2         3           4
  O     O O     O O O     O O O O
.
  1       3         6          10
  O     O O     O O O     O O O O
          O       O O       O O O
                    O         O O
                                O
.
  1       4         9          16
  O     O O     O O O     O O O O
          O       O O       O O O
          O       O O       O O O
                    O         O O
                    O         O O
                                O
                                O
.
  1       5        12          22
  O     O O     O O O     O O O O
          O       O O       O O O
          O       O O       O O O
          O       O O       O O O
                    O         O O
                    O         O O
                    O         O O
                                O
                                O
                                O
(End)

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 189, 1966.
J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag (Copernicus), p. 38, 1996.

T. D. Noe, Rows n = 2..100, flattened
Lukas Andritsch, Boundary algebra of a GL_m-dimer, arXiv:1804.07243 [math.RT], 2018.
Index to sequences related to polygonal numbers

Many rows and columns of this array are in the database.
Cf. A055795 (antidiagonal sums), A064808 (main diagonal).
Cf. A063212, A077028, A086270, A090466, A139600.

/* As square array: */ t:=func; [[t(s,n): s in [1..11]]: n in [2..14]]; // Bruno Berselli, Jun 24 2013

A057145 := proc(n,k)
    ((n-2)*k^2-(n-4)*k)/2 ;
end proc:
seq(seq(A057145(d-k,k),k=1..d-2),d=3..12); # R. J. Mathar, Jul 28 2016

nn = 12; Flatten[Table[k (3 - k^2 - n + k*n)/2, {n, 2, nn}, {k, n - 1}]] (* T. D. Noe, Oct 10 2012 *)

T(2n+4,n) = n^3. - Stuart M. Ellerstein (ellerstein(AT)aol.com), Aug 28 2000
T(n, k) = T(n-1, k) + k*(k-1)/2 [with T(2, k)=k] = T(n, k-1) + 1 + (n-2)*(k-1) [with T(n, 0)=0] = k + (n-2)k(k-1)/2 = k + A063212(n-2, k-1). - Henry Bottomley, Jul 11 2001
G.f. for row n: x*(1+(n-3)*x)/(1-x)^3, n>=2. - Paul Barry, Feb 21 2003
From Wolfdieter Lang, Nov 05 2014: (Start)
The triangle is a(n, m) = T(n-m+1, m) = (1/2)*m*(n*(m-1) + 3 - m^2) for n >= 2, m = 1, 2, ..., n-1 and zero elsewhere.
O.g.f. for column m (without leading zeros): (x*binomial(m,2) + (1+2*m-m^2)*(m/2)*(1-x))/(x^(m-1)*(1-x)^2). (End)
T(n,k) = A139600(n-2,k) = A086270(n-2,k). - R. J. Mathar, Jul 28 2016
Row sums of A077028: T(n+2,k+1) = Sum_{j=0..k} A077028(n,j), where A077028(n,k) = 1+n*k is the square array interpretation of A077028 (the 1D polygonal numbers). - R. J. Mathar, Jul 30 2016
G.f.: x^2*y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3). - Stefano Spezia, Apr 12 2024

a(50)=49 corrected to a(50)=40 by Jean-François Alcover, Jul 22 2011

Edited: Name shortened, offset in Paul Barry's g.f. corrected and Conway-Guy reference added. - Wolfdieter Lang, Nov 04 2014

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)

A177025 Number of ways to represent n as a polygonal number.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 2, 4, 1, 1, 2, 2, 1, 2, 1, 1, 4, 2, 1, 2, 2, 1, 3, 2, 1, 2, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 4, 2, 1, 2, 2, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 3, 1, 1, 2, 3

Offset: 3

Vladimir Shevelev, May 01 2010

nonn

Frequency of n in the array A139601 or A086270 of polygonal numbers.
Since n is always n-gonal number, a(n) >= 1.
Conjecture: Every positive integer appears in the sequence.
Records of 2, 3, 4, 5, ... are reached at n = 6, 15, 36, 225, 561, 1225, ... see A063778. [R. J. Mathar, Aug 15 2010]

J. J. Tattersall, Elementary Number Theory in Nine chapters, 2nd ed (2005), Cambridge Univ. Press, page 22 Problem 26, citing Wertheim (1897)

T. D. Noe, Table of n, a(n) for n = 3..10000
E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.

Cf. A129654, A139601, A090428, A176949, A176948, A176774, A176744, A176747, A176775, A175873, A176874.

A177025 := proc(p)
    local ii,a,n,s,m ;
    ii := 2*p ;
    a := 0 ;
    for n in numtheory[divisors](ii) do
        if n > 2 then
            s := ii/n ;
            if (s-2) mod (n-1) = 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Jan 10 2013

nn = 100; t = Table[0, {nn}]; Do[k = 2; While[p = k*((n - 2) k - (n - 4))/2; p <= nn, t[[p]]++; k++], {n, 3, nn}]; t (* T. D. Noe, Apr 13 2011 *)
Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]] - 1, {n, 3, 100}] (* Jonathan Sondow, May 09 2014 *)

a(n) = sum(i=3, n, ispolygonal(n, i)); \\ Michel Marcus, Jul 08 2014

from sympy import divisors
def a(n):
    i=2*n
    x=0
    for d in divisors(i):
        if d>2:
            s=i/d
            if (s - 2)%(d - 1)==0: x+=1
    return x # Indranil Ghosh, Apr 28 2017, translated from Maple code by R. J. Mathar

a(n) = A129654(n) - 1.

G.f.: x * Sum_{k>=2} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 09 2020

Extended by R. J. Mathar, Aug 15 2010

A086271 Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 9, 10, 1, 6, 12, 16, 15, 1, 7, 15, 22, 25, 21, 1, 8, 18, 28, 35, 36, 28, 1, 9, 21, 34, 45, 51, 49, 36, 1, 10, 24, 40, 55, 66, 70, 64, 45, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66, 1, 13, 33, 58, 85, 111, 133, 148, 153, 145, 121, 78

Offset: 1

Clark Kimberling, Jul 14 2003

The transpose of the array in A086270; diagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.

			Columns 1,2,3 are the triangular, square and pentagonal numbers.
Northwest corner:
       k=1 k=2 k=3 k=4 k=5
  n=1:   1   1   1   1   1 ...
  n=2:   3   4   5   6   7 ...
  n=3:   6   9  12  15  18 ...
  n=4:  10  16  22  28  34 ...
  n=5:  15  25  35  45  55 ...
  ...

G. C. Greubel, Table of n, a(n) for the first 50 diagonals, flattened
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A006522, A086270, A086272, A086273.

Main diagonal gives A006000(n-1).

T[n_, k_] := PolygonalNumber[k+2, n]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 04 2016 *)

T(n, k) = k*C(n,2) + n.
From Stefano Spezia, Sep 02 2022: (Start)
G.f.: x*y*(1 - y + x*y)/((1 - x)^3*(1 - y)^2).
G.f. of n-th row: n*(1 + n - 2*y)*y/(2*(1 - y)^2). (End)

A086272 Rectangular array T(n,k) of central polygonal numbers, by antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 13, 10, 5, 1, 21, 19, 13, 6, 1, 31, 31, 25, 16, 7, 1, 43, 46, 41, 31, 19, 8, 1, 57, 64, 61, 51, 37, 22, 9, 1, 73, 85, 85, 76, 61, 43, 25, 10, 1, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 111, 136, 145, 141, 127, 106, 81, 55, 31, 12, 1, 133, 166, 181, 181, 169

Offset: 1

Clark Kimberling, Jul 14 2003

In the standard notation, the offset is different: the first row are the 2-gonal, the second row the 3-gonal numbers, etc. - R. J. Mathar, Oct 07 2011

			First rows:
1,3,7,13,21,31,43,57,73,91,111,..   A002061
1,4,10,19,31,46,64,85,109,136,166,...  A005448
1,5,13,25,41,61,85,113,145,181,221,..   A001844
1,6,16,31,51,76,106,141,181,226,276,...  A005891
1,7,19,37,61,91,127,169,217,271,331,...   A003215
1,8,22,43,71,106,148,197,253,316,386,...    A069099
1,9,25,49,81,121,169,225,289,361,441,...    A016754
1,10,28,55,91,136,190,253,325,406,496,...    A060544

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7 (2004) # 04.1.6

Cf. A086270, A086271, A086273.

T(k, n)=(k+1)*binomial(n, 2)+1.

cp's OEIS Frontend

A144257 Weight array of A086270.

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A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

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A185733 Second accumulation array of the polygonal number array (A086270), by antidiagonals.

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A185734 Third accumulation array of the polygonal number array (A086270), by antidiagonals.

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A376276 Table T(n, k) n > 0, k > 2 read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. The length of the row number n in column k is equal to the n-th k-gonal number A086270.

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A057145 Square array of polygonal numbers T(n,k) = ((n-2)*k^2 - (n-4)*k)/2, n >= 2, k >= 1, 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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A057145 Square array of polygonal numbers T(n,k) = ((n-2)k^2 - (n-4)k)/2, n >= 2, k >= 1, read by antidiagonals upwards.

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