cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A163657 Triangle T(m,n) = 2*m*n + m + n + 8 read by rows.

Original entry on oeis.org

12, 15, 20, 18, 25, 32, 21, 30, 39, 48, 24, 35, 46, 57, 68, 27, 40, 53, 66, 79, 92, 30, 45, 60, 75, 90, 105, 120, 33, 50, 67, 84, 101, 118, 135, 152, 36, 55, 74, 93, 112, 131, 150, 169, 188, 39, 60, 81, 102, 123, 144, 165, 186, 207, 228, 42, 65, 88, 111, 134, 157, 180
Offset: 1

Views

Author

Vincenzo Librandi, Aug 02 2009

Keywords

Comments

If p=2*n+1 is a prime number, then T(n,n) = (p^2+15)/2.
First column: 3*A020705; second column: 5*A020705; third column: A017029. - Vincenzo Librandi, Nov 18 2012

Examples

			Triangle begins:
12;
15, 20;
18, 25, 32;
21, 30, 39, 48;
24, 35, 46, 57, 68;
27, 40, 53, 66, 79, 92;
30, 45, 60, 75, 90, 105, 120;
33, 50, 67, 84, 101, 118, 135, 152; etc.
		

Crossrefs

Programs

  • Magma
    [2*n*k+n+k+8: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 18 2012
  • Mathematica
    Flatten[Table[2nm + m + n + 8, {n, 10}, {m, n}]] (* Vincenzo Librandi, Nov 18 2012 *)

Formula

T(n,m) = A163672(n,m)+1.

Extensions

Edited by R. J. Mathar, Oct 12 2009

A153049 Numbers n such that 2*n - 13 is not a prime.

Original entry on oeis.org

7, 11, 14, 17, 19, 20, 23, 24, 26, 29, 31, 32, 34, 35, 38, 39, 41, 44, 45, 47, 49, 50, 52, 53, 54, 56, 59, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 77, 78, 79, 80, 83, 84, 86, 87, 89, 91, 92, 94, 95, 98, 99, 100, 101, 104, 107, 108, 109, 110, 111, 113, 114, 115, 116
Offset: 1

Views

Author

Vincenzo Librandi, Dec 17 2008

Keywords

Comments

All entries of A163672 are in the sequence because 2*A163672(n,m)-13 = (2n+1)(2m+1) is not prime. - R. J. Mathar, Jan 05 2011

Examples

			Distribution of a(n) > 7 in the following triangular array:
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;
41,64,87,110,133,156,179,202,225,248,271;
44,69,94,119,144,169,194,219,244,269,294,319; etc.
which is obtained by (2*h*k + k + h + 7) with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013
		

Crossrefs

Programs

  • Magma
    [n: n in [7..120] | not IsPrime(2*n - 13)]; // Vincenzo Librandi, Oct 15 2012
  • Mathematica
    Select[Range[7, 200],!PrimeQ[2 # - 13] &] (* Vincenzo Librandi, Oct 15 2012 *)

A163673 a(n) = n*(2*n^2 + 5*n + 15)/2.

Original entry on oeis.org

0, 11, 33, 72, 134, 225, 351, 518, 732, 999, 1325, 1716, 2178, 2717, 3339, 4050, 4856, 5763, 6777, 7904, 9150, 10521, 12023, 13662, 15444, 17375, 19461, 21708, 24122, 26709, 29475, 32426, 35568, 38907, 42449, 46200, 50166
Offset: 0

Views

Author

Vincenzo Librandi, Aug 03 2009

Keywords

Crossrefs

Programs

  • Mathematica
    CoefficientList[Series[x*(11-11*x+6*x^2)/(x-1)^4, {x, 0, 40}], x] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 11, 33, 72}, 50] (* Vincenzo Librandi, Mar 06 2012 *)
    Table[n (2n^2+5n+15)/2,{n,0,60}] (* Harvey P. Dale, Apr 01 2023 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(x*(11-11*x+6*x^2)/(x-1)^4)) \\ G. C. Greubel, Aug 02 2017

Formula

Row sums from A163672: a(n) = Sum_{m=1..n} (2*m*n + m + n + 7) = A163661(n) - n.
G.f.: x*(11 - 11*x + 6*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*(22 + 11*x + 2*x^2)*exp(x). - G. C. Greubel, Aug 02 2017

Extensions

Edited by R. J. Mathar, Aug 05 2009
Showing 1-3 of 3 results.