A086272 -id:A086272 - cp's OEIS frontend

A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals.

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

Offset: 1

Clark Kimberling, Jul 14 2003

The antidiagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.
This is the accumulation array (cf. A144112) of A144257 (which is the weight array of this sequence). - Clark Kimberling, Sep 16 2008
By rows, the sequence beginning (1, N, ...) is the binomial transform of (1, (N-1), (N-2), 0, 0, 0, ...); and is the second partial sum of (1, (N-2), (N-2), (N-2), ...). Example: The sequence (1, 4, 9, 16, 25, ...) is the binomial transform of (1, 3, 2, 0, 0, 0, ...) and the second partial sum of (1, 2, 2, 2, ...). - Gary W. Adamson, Aug 23 2015

			First 6 rows:
=========================================
n\k|  1   2    3    4    5    6     7
---|-------------------------------------
1  |  1   3    6   10   15   21    28 ... (A000217, triangular numbers)
2  |  1   4    9   16   25   36    49 ... (A000290, squares)
3  |  1   5   12   22   35   51    70 ... (A000326, pentagonal numbers)
4  |  1   6   15   28   45   66    91 ... (A000384, hexagonal numbers)
5  |  1   7   18   34   55   81   112 ... (A000566, heptagonal numbers)
6  |  1   8   21   40   65   96   133 ... (A000567, octagonal numbers)
...
The array formed by the complements: A183225.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 76 at p. 189.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Wikipedia, Polygonal number: Table of values.

Cf. A006522, A057145, A086271, A086272, A086273, A139601, A183225.
Cf. A000217, A000290, A000326, A000384, A000566, A000567.
Cf. A114112, A144257 .

T:=func; [T(k,n-k+1): k in [1..n], n in [1..12]]; // Bruno Berselli, Dec 19 2014

t[n_, k_] := n*Binomial[k, 2] + k; Table[ t[k, n - k + 1], {n, 12}, {k, n}] // Flatten

T(n, k) = n*binomial(k, 2) + k = A057145(n+2,k).
2*T(n, k) = T(n+r, k) + T(n-r, k), where r = 0, 1, 2, 3, ..., n-1 (see table in Example field). - Bruno Berselli, Dec 19 2014
From Stefano Spezia, Sep 02 2022: (Start)
G.f.: x*y*(1 - x + x*y)/((1 - x)^2*(1 - y)^3).
G.f. of k-th column: k*(1 + k - 2*x)*x/(2*(1 - x)^2). (End)

Extended by Clark Kimberling, Jan 01 2011

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)

A086273 Rectangular array T(n,k) of central polygonal numbers, by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 7, 1, 5, 10, 13, 1, 6, 13, 19, 21, 1, 7, 16, 25, 31, 31, 1, 8, 19, 31, 41, 46, 43, 1, 9, 22, 37, 51, 61, 64, 57, 1, 10, 25, 43, 61, 76, 85, 85, 73, 1, 11, 28, 49, 71, 91, 106, 113, 109, 91, 1, 12, 31, 55, 81, 106, 127, 141, 145, 136, 111, 1, 13, 34, 61, 91, 121, 148

Offset: 1

Clark Kimberling, Jul 14 2003

Transpose of the array at A086272.

			Northwest corner:
  1    1    1    1    1    1    1    1    1    1
  3    4    5    6    7    8    9   10   11   12 A000027
  7   10   13   16   19   22   25   28   31   34 A112414, A016777
  13   19   25   31   37   43   49   55   61   67 A016921
  21   31   41   51   61   71   81   91  101  111 A017281
  31   46   61   76   91  106  121  136  151  166
  43   64   85  106  127  148  169  190  211  232
  57   85  113  141  169  197  225  253  281  309
  73  109  145  181  217  253  289  325  361  397
  91  136  181  226  271  316  361  406  451  496
111  166  221  276  331  386  441  496  551  606
133  199  265  331  397  463  529  595  661  727
157  235  313  391  469  547  625  703  781  859
183  274  365  456  547  638  729  820  911 1002
211  316  421  526  631  736  841  946 1051 1156
241  361  481  601  721  841  961 1081 1201 1321

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

Cf. A086270, A086271, A086272, A086274.

A086273 := proc(n,k) (k+1)*n*(n-1)/2+1 ; end proc: # R. J. Mathar, Jun 05 2011

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

A125764 Array of partial sums of rows of array in A086271, read by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 7, 3, 10, 15, 12, 4, 15, 26, 27, 18, 5, 21, 40, 48, 42, 25, 6, 28, 57, 75, 76, 60, 33, 7, 36, 77, 108, 120, 110, 81, 42, 8, 45, 100, 147, 174, 175, 150, 105, 52, 9, 55, 126, 192, 238, 255, 240, 196, 132, 63, 10, 66, 155, 243, 312, 350, 351, 315, 248, 162, 75, 11

Offset: 1

Jonathan Vos Post and Joshua Zucker, Feb 03 2007

Row 3 is = 3rd triangular number + 3rd square + 3rd pentagonal number + 3rd hexagonal number + ... + 3rd k-gonal number. First column is triangular numbers. A086271 Rectangular array T(n,k) of polygonal numbers, by diagonals.

			Partial row sum array begins:
1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... n.
2 | 3, 7, 12, 18, 25, 33, 42, 52, (n(n+1)/2)-3.
3 | 6, 15, 27, 42, 60, 81, 105, ... (3/2)n^2 + (9/2) n.
4 | 10, 26, 48, 76, 110, 150, ... 3n^2 + 7n.
5 | 15, 40, 75, ... 5n^2 + 10n.
6 | 21, 57, 108, ... (15/2)n^2 + (27/2)n.

Eric Weisstein's World of Mathematics, Polygonal Number. See equation (4), our partial sums are on this as array element values.

Cf. A000096, A000217, A000290, A000326, A000384, A086270, A086271, A086272, A086273.

A086271 := proc(n,k) k*binomial(n,2)+n ; end: A125764 := proc(n,k) add(A086271(n,i),i=1..k) ; end: for d from 1 to 15 do for k from 1 to d do printf("%d, ",A125764(d-k+1,k)) ; od: od: # R. J. Mathar, Nov 02 2007

a(k,n) = (k*(k-1)/2)n^2 + (k*(k+3)/4)n. a(k,n) = row k of array of partial sums = k-th triangular number + k-th square + k-th pentagonal number + k-th hexagonal number + ... = A000217(k) + A000290(k) + A000326(k) + A000384(k) + ... a(1,n) = n. a(2,n) = (n(n+1)/2)-3 = A000217(n) - 3. a(3,n) = 3*n(n+3)/2 = A000096 with offset 3.

More terms from R. J. Mathar, Nov 02 2007

Keyword tabl added by Michel Marcus, Apr 08 2013

A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

Original entry on oeis.org

1, 4, 11, 26, 55, 105, 184, 301, 466, 690, 985, 1364, 1841, 2431, 3150, 4015, 5044, 6256, 7671, 9310, 11195, 13349, 15796, 18561, 21670, 25150, 29029, 33336, 38101, 43355, 49130, 55459, 62376, 69916, 78115, 87010, 96639, 107041, 118256, 130325

Offset: 1

Dan Graybill (clopen(AT)comcast.net), Aug 12 2008

Consider the array of triangular, square and centered polygonal numbers (irregular variant of A086272 and A086273):
 3   6  10  15  21  28  36  45  55  A000217
 4   9  16  25  36  49  64  81 100  A000290
 6  16  31  51  76 106 141 181 226  A005891
 7  19  37  61  91 127 169 217 271  A003215
 8  22  43  71 106 148 197 253 316  A069099
 9  25  49  81 121 169 225 289 361  A016754
10  28  55  91 136 190 253 325 406  A060544
11  31  61 101 151 211 281 361 451  A062786
12  34  67 111 166 232 309 397 496  A069125
13  37  73 121 181 253 337 433 541  A003154
14  40  79 131 196 274 365 469 586  A069126
15  43  85 141 211 295 393 505 631  A069127
etc. The sequence contains the antidiagonal sums of this array. - R. J. Mathar, Jun 05 2011
For comparison, the antidiagonal sums of A086270 are essentially A006522 starting at the 4th term. - R. J. Mathar, Sep 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217.

a(n) = (n-1)*(n^3+11*n^2-38*n+120)/24, n>1. - R. J. Mathar, Sep 12 2008

G.f.: x*(1-x+x^2+x^3-x^5)/(1-x)^5. - Alexander R. Povolotsky, Jun 06 2011

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

A194127 Listed by antidiagonals, array A(k,n) = digital root of n-th nonzero centered k-gonal number.

Original entry on oeis.org

1, 1, 4, 1, 5, 1, 1, 6, 4, 1, 1, 7, 7, 7, 4, 1, 8, 1, 4, 5, 1, 1, 9, 4, 1, 6, 7, 1, 1, 1, 7, 7, 7, 4, 4, 4, 1, 2, 1, 4, 8, 1, 7, 5, 1, 1, 3, 4, 1, 9, 7, 1, 6, 1, 1, 1, 4, 7, 7, 1, 4, 4, 7, 1, 1, 4, 1, 5, 1, 4, 2, 1, 7, 8, 1, 1, 5, 1, 1, 6, 4, 1, 3, 7, 1, 9, 1, 1, 6, 4, 1

Offset: 1

Jonathan Vos Post, Aug 15 2011

This is to centered k-gonal numbers as A193043 is to k-gonal numbers. The main diagonal begins: 1, 5, 7, 1, 8, 4, 1, 2.

			The array begins:
=====================================
....|n=1|n=2|n=3|n=4|n=5|n=6|n=7|n=8|
====|===|===|===|===|===|===|===|===|
k=3.|.1.|.4.|.1.|.1.|.4.|.1.|.1.|.4.|
k=4.|.1.|.5.|.4.|.7.|.5.|.7.|.4.|.5.|
k=5.|.1.|.6.|.7.|.4.|.6.|.4.|.7.|.6.|
k=6.|.1.|.7.|.1.|.1.|.7.|.1.|.1.|.7.|
k=7.|.1.|.8.|.4.|.7.|.8.|.7.|.4.|.8.|
k=8.|.1.|.9.|.7.|.4.|.9.|.4.|.7.|.9.|
k=9.|.1.|.1.|.1.|.1.|.1.|.1.|.1.|.1.|
k=10|.1.|.2.|.4.|.7.|.2.|.7.|.4.|.2.|
=====================================

Eric W. Weisstein, Digital Root
Eric W. Weisstein, Centered Polygonal Number

Cf. A086272, A086273, A010888, A193043.

dr[n_]:=1+Mod[n-1, 9]; A[k_,n_]:=dr[k*n*(n-1)/2+1];
Flatten[Table[A[d-i+3, i], {d, 13}, {i, d}]] (* Ray Chandler, Aug 16 2011 *)

Extended by Ray Chandler, Aug 16 2011

cp's OEIS Frontend

A086274 Antidiagonal sums of A086272 (and of A086273).

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A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals.

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

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A086273 Rectangular array T(n,k) of central polygonal numbers, by antidiagonals.

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A125764 Array of partial sums of rows of array in A086271, read by antidiagonals.

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A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

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A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

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A194127 Listed by antidiagonals, array A(k,n) = digital root of n-th nonzero centered k-gonal number.

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