A112414 -id:A112414 - cp's OEIS frontend

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

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.

A086822 a(n) = floor((6n^0+5n^1+4n^2+3n^3) / (1n^0+1n^1+1*n^2)).

Original entry on oeis.org

6, 8, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175

Offset: 1

Wang Dan (wangdan01(AT)lycos.com), Aug 07 2003

			a(3) = floor((6*3^0+5*3^1+4*3^2+3*3^3)/(1*3^0+1*3^1+1*3^2)) = floor(138/13) = floor(10.615) = 10.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A086790, A086814.

Table[Floor[(6+5n+4n^2+3n^3)/(1+n+n^2)],{n,60}] (* Harvey P. Dale, Apr 22 2018 *)

Vec(x*(x^3-4*x+6)/(x-1)^2 + O(x^100)) \\ Colin Barker, May 23 2015

a(n) = floor(3n+1+(5+n)/(1+n+n^2)) = 3n+1 = A112414(n-2) for n>2. - R. J. Mathar, Dec 13 2008
a(n) = 2*a(n-1)-a(n-2) for n>4. - Colin Barker, May 23 2015
G.f.: x*(x^3-4*x+6) / (x-1)^2. - Colin Barker, May 23 2015

A163674 Triangle T(n,m) = 2mn + m + n + 9 read by rows.

Original entry on oeis.org

13, 16, 21, 19, 26, 33, 22, 31, 40, 49, 25, 36, 47, 58, 69, 28, 41, 54, 67, 80, 93, 31, 46, 61, 76, 91, 106, 121, 34, 51, 68, 85, 102, 119, 136, 153, 37, 56, 75, 94, 113, 132, 151, 170, 189, 40, 61, 82, 103, 124, 145, 166, 187, 208, 229, 43, 66, 89, 112, 135, 158, 181

Offset: 1

Vincenzo Librandi, Aug 03 2009

2*T(m,n) - 17 =(2*n+1)*(2*m+1) and 2*T(n,n) - 17 is a square. Also:
first column:  A112414;
second column: A016861;
third column:  A017041;
fourth column: A017209. [Vincenzo Librandi, Nov 20 2012]

			Triangle begins:
  13;
  16,  21;
  19,  26,  33;
  22,  31,  40,  49;
  25,  36,  47,  58,  69;
  28,  41,  54,  67,  80,  93;
  31,  46,  61,  76,  91, 106, 121;
  34,  51,  68,  85, 102, 119, 136, 153;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A016861, A017041, A017209, A112414, A153051, A163657.

[2*n*k + n + k + 9: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k + 9; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

for(n=1,10, for(m=1,n, print1(2*m*n + m + n + 9, ", "))) \\ G. C. Greubel, Aug 02 2017

T(n,m) = A163657(n,m) + 1.

Edited by R. J. Mathar, Oct 12 2009

A163652 Triangle read by rows where T(n,m)=2mn + m + n + 6.

Original entry on oeis.org

10, 13, 18, 16, 23, 30, 19, 28, 37, 46, 22, 33, 44, 55, 66, 25, 38, 51, 64, 77, 90, 28, 43, 58, 73, 88, 103, 118, 31, 48, 65, 82, 99, 116, 133, 150, 34, 53, 72, 91, 110, 129, 148, 167, 186, 37, 58, 79, 100, 121, 142, 163, 184, 205, 226, 40, 63, 86, 109, 132, 155, 178

Offset: 1

Vincenzo Librandi, Aug 02 2009

The numbers 2*T(n,m)-11 = (2*n+1)*(2*m+1) are not prime, and 2*T(n,n) = (2n+1)^2.

First column: A112414, second column: A016885, third column: A017005, fourth column: A017173. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
  10;
  13, 18;
  16, 23, 30;
  19, 28, 37, 46;
  22, 33, 44, 55,  66;
  25, 38, 51, 64,  77,  90;
  28, 43, 58, 73,  88,  103, 118;
  31, 48, 65, 82,  99,  116, 133, 150;
  34, 53, 72, 91,  110, 129, 148, 167, 186;
  37, 58, 79, 100, 121, 142, 163, 184, 205, 226;
  40, 63, 86, 109, 132, 155, 178, 201, 224, 247, 270;
  etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153045, A154685, A163657, A112414, A016885, A017005, A017173.

[2*n*k + n + k + 6: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k +  6; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

T(n,m) = A154685(n,m)+2 = A163657(n,m)-2. [R. J. Mathar, Oct 22 2009]

Comment clarified by R. J. Mathar, Oct 22 2009

A154614 Triangle read by rows where T(m,n) = m*n + m + n - 1, 1<=n<=m.

Original entry on oeis.org

2, 4, 7, 6, 10, 14, 8, 13, 18, 23, 10, 16, 22, 28, 34, 12, 19, 26, 33, 40, 47, 14, 22, 30, 38, 46, 54, 62, 16, 25, 34, 43, 52, 61, 70, 79, 18, 28, 38, 48, 58, 68, 78, 88, 98, 20, 31, 42, 53, 64, 75, 86, 97, 108, 119, 22, 34, 46, 58, 70, 82, 94, 106, 118, 130, 142

Offset: 1

Vincenzo Librandi, Jan 16 2009

T(m,n)+2 = (n+1)*(m+1) is not prime.
T(m,m)+2 = (m+1)^2.
First column: A005843; second column: A112414; third column: 2*A020742; fourth column: A016885. - Vincenzo Librandi, Nov 17 2012

			Triangle begins:
2;
4, 7;
6, 10, 14;
8, 13, 18, 23;
10, 16, 22, 28, 34;
12, 19, 26, 33, 40, 47;
14, 22, 30, 38, 46, 54, 62;
16, 25, 34, 43, 52, 61, 70, 79;
18, 28, 38, 48, 58, 68, 78, 88, 98;
20, 31, 42, 53, 64, 75, 86, 97, 108, 119; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A040976, A005843, A112414, A020742, A016885.

[(n*k + n + k - 1): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 17 2012

t[n_,k_]:=n*k + n + k - 1; Table[t[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 17 2012 *)

A154681 Triangle read by rows where T(m,n) = 2mn + m + n +3.

Original entry on oeis.org

7, 10, 15, 13, 20, 27, 16, 25, 34, 43, 19, 30, 41, 52, 63, 22, 35, 48, 61, 74, 87, 25, 40, 55, 70, 85, 100, 115, 28, 45, 62, 79, 96, 113, 130, 147, 31, 50, 69, 88, 107, 126, 145, 164, 183, 34, 55, 76, 97, 118, 139, 160, 181, 202, 223, 37, 60, 83, 106, 129, 152

Offset: 1

Vincenzo Librandi, Jan 18 2009

2*T(m,n) - 5 = (2*n+1)*(2*m+1) is not prime.

First column: A112414; second column: A008587.

			Triangle begins:
  7;
  10, 15;
  13, 20, 27;
  16, 25, 34, 43;
  19, 30, 41, 52,  63;
  22, 35, 48, 61,  74,  87;
  25, 40, 55, 70,  85, 100, 115;
  28, 45, 62, 79,  96, 113, 130, 147;
  31, 50, 69, 88, 107, 126, 145, 164, 183;
  34, 55, 76, 97, 118, 139, 160, 181, 202, 223; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153040, A008587, A089253, A112414.

[(2*n*k + n + k + 3): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 17 2012

t[n_,k_]:=2 n*k + n + k + 3; Table[t[n, k], {n, 10}, {k, n}]//Flatten (* vincenzo Librandi, Nov 17 2012 *)

cp's OEIS Frontend

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

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A086822 a(n) = floor((6*n^0+5*n^1+4*n^2+3*n^3) / (1*n^0+1*n^1+1*n^2)).

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A163674 Triangle T(n,m) = 2*m*n + m + n + 9 read by rows.

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A163652 Triangle read by rows where T(n,m)=2*m*n + m + n + 6.

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A154614 Triangle read by rows where T(m,n) = m*n + m + n - 1, 1<=n<=m.

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Magma

Mathematica

A154681 Triangle read by rows where T(m,n) = 2*m*n + m + n +3.

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Programs

Magma

Mathematica

A086822 a(n) = floor((6n^0+5n^1+4n^2+3n^3) / (1n^0+1n^1+1*n^2)).

A163674 Triangle T(n,m) = 2mn + m + n + 9 read by rows.

A163652 Triangle read by rows where T(n,m)=2mn + m + n + 6.

A154681 Triangle read by rows where T(m,n) = 2mn + m + n +3.