A163657 -id:A163657 - cp's OEIS frontend

A163672 Triangle T(n,m) = 2mn + m + n + 7 read by rows.

11, 14, 19, 17, 24, 31, 20, 29, 38, 47, 23, 34, 45, 56, 67, 26, 39, 52, 65, 78, 91, 29, 44, 59, 74, 89, 104, 119, 32, 49, 66, 83, 100, 117, 134, 151, 35, 54, 73, 92, 111, 130, 149, 168, 187, 38, 59, 80, 101, 122, 143, 164, 185, 206, 227, 41, 64, 87, 110, 133, 156, 179

Offset: 1

Vincenzo Librandi, Aug 03 2009

2*T(n,n) - 13 = (2n+1)^2.

The numbers 2*T(m,n)-13 =(2*n+1)*(2*m+1) are not prime. Also: first column: A016789; second column: A016897; third column: A017017; fourth column: A017185. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
  11;
  14,  19;
  17,  24,  31;
  20,  29,  38,  47;
  23,  34,  45,  56,  67;
  26,  39,  52,  65,  78,  91;
  29,  44,  59,  74,  89, 104, 119;
  32,  49,  66,  83, 100, 117, 134, 151;

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

Cf. A016789, A016897, A017017, A017185, A153049, A163674, A163657.

[2*n*k + n + k + 7: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k + 7; 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 + 7, ", "))) \\ G. C. Greubel, Aug 02 2017

T(n,m) = A163674(n,m)-2 = A163657(n,m)-1.

Edited by R. J. Mathar, Oct 12 2009

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

A153047 Numbers n such that 2*n-15 is not a prime.

Original entry on oeis.org

12, 15, 18, 20, 21, 24, 25, 27, 30, 32, 33, 35, 36, 39, 40, 42, 45, 46, 48, 50, 51, 53, 54, 55, 57, 60, 63, 65, 66, 67, 68, 69, 70, 72, 74, 75, 78, 79, 80, 81, 84, 85, 87, 88, 90, 92, 93, 95, 96, 99, 100, 101, 102, 105, 108, 109, 110, 111, 112, 114, 115, 116

Offset: 1

Vincenzo Librandi, Dec 17 2008

One more than the associated entry in A153049. - R. J. Mathar, Jan 05 2011

The terms are the values of 2*h*k + k + h + 8, where h and k are positive integers.- Vincenzo Librandi, Jan 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A097480, A153043, A153045, A153049, A163657.

[n: n in [10..120] | not IsPrime(2*n - 15)]; // Vincenzo Librandi, Oct 11 2012

Select[Range[12,200], !PrimeQ[2*#-15]&] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2012 *)

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

A163661 a(n) = n(2n^2 + 5*n + 17)/2.

Original entry on oeis.org

0, 12, 35, 75, 138, 230, 357, 525, 740, 1008, 1335, 1727, 2190, 2730, 3353, 4065, 4872, 5780, 6795, 7923, 9170, 10542, 12045, 13685, 15468, 17400, 19487, 21735, 24150, 26738, 29505, 32457, 35600, 38940, 42483, 46235, 50202, 54390, 58805, 63453

Offset: 0

Vincenzo Librandi, Aug 02 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A163657.

CoefficientList[Series[x*(12-13*x+7*x^2)/(x-1)^4,{x,0,40}],x] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 12, 35, 75}, 50](* Vincenzo Librandi, Mar 06 2012 *)

x='x+O('x^50); concat([0], Vec(x*(12-13*x+7*x^2)/(x-1)^4)) \\ G. C. Greubel, Aug 01 2017

Row sums from A163657: a(n) = Sum_{m=1..n} (2*m*n + m + n + 8).
a(n) = A163655(n) + 2*n.
G.f.: x*(12 - 13*x + 7*x^2)/(x-1)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: (1/2)*x*(24 + 11*x + 2*x^2)*exp(x). - G. C. Greubel, Aug 01 2017

Edited and a(31) corrected by R. J. Mathar, Aug 05 2009

cp's OEIS Frontend

A163672 Triangle T(n,m) = 2mn + m + n + 7 read by rows.

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

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A153047 Numbers n such that 2*n-15 is not a prime.

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

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A163661 a(n) = n*(2*n^2 + 5*n + 17)/2.

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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.

A163661 a(n) = n(2n^2 + 5*n + 17)/2.