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 1-4 of 4 results.

A222182 Numbers m such that 2*m + 11 is a square.

Original entry on oeis.org

-5, -1, 7, 19, 35, 55, 79, 107, 139, 175, 215, 259, 307, 359, 415, 475, 539, 607, 679, 755, 835, 919, 1007, 1099, 1195, 1295, 1399, 1507, 1619, 1735, 1855, 1979, 2107, 2239, 2375, 2515, 2659, 2807, 2959, 3115, 3275, 3439, 3607, 3779, 3955, 4135, 4319, 4507, 4699
Offset: 1

Views

Author

Bruno Berselli, Mar 01 2013

Keywords

Comments

Except the first term, main diagonal of A155546. - Vincenzo Librandi, Mar 04 2013

Crossrefs

Cf. numbers n such that 2*n + 2*k + 1 is a square: A046092 (k=0), A142463 (k=1), A090288 (k=2), A059993 (k=3), A139570 (k=4), this sequence (k=5), A181510 (k=6).
Cf. A005408 (square roots of 2*a(n)+11), A155546.
After a(2), subsequence of A168489.

Programs

  • Magma
    [m: m in [-5..5000] | IsSquare(2*m+11)];
    
  • Magma
    I:=[-5,-1,7]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Mar 04 2013
    
  • Mathematica
    Table[2 n^2 - 2 n - 5, {n, 50}]
  • Maxima
    makelist(coeff(taylor(-(5-14*x+5*x^2)/(1-x)^3, x, 0, n), x, n), n, 0, 50);
    
  • PARI
    a(n)=2*n^2-2*n-5 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: -x*(5 - 14*x + 5*x^2)/(1-x)^3.
a(n) = a(-n+1) = 2*n^2 - 2*n - 5.
a(n) = A046092(n-1) - 5.
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(11)*Pi/2)/(2*sqrt(11)). - Amiram Eldar, Dec 23 2022
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: exp(x)*(2*x^2 - 5) + 5.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A153083 Numbers such that 2*n + 11 is not prime.

Original entry on oeis.org

2, 5, 7, 8, 11, 12, 14, 17, 19, 20, 22, 23, 26, 27, 29, 32, 33, 35, 37, 38, 40, 41, 42, 44, 47, 50, 52, 53, 54, 55, 56, 57, 59, 61, 62, 65, 66, 67, 68, 71, 72, 74, 75, 77, 79, 80, 82, 83, 86, 87, 88, 89, 92, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 107, 110, 112
Offset: 1

Views

Author

Vincenzo Librandi, Dec 18 2008

Keywords

Comments

One less than the associated entry in A155723. [ R. J. Mathar, Jan 05 2011]

Examples

			Distribution of the terms in the following triangular array:
*
2,7;
5,12,19;
8,17,26,35;
11,22,33,44,55;
14,27,40,53,66,79;
17,32,47,62,77,92,107;
20,37,54,71,88,105,122,139;
23,42,61,80,99,118,137,156,175;
26,47,68,89,110,131,152,173,194,215;
29,52,75,98,121,144,167,190,213,236,259;
32,57,82,107,132,157,182,207,232,257,282,307;
where * marks the decimal values of (2*h*k + k + h - 5) with h >= k >= 1. - _Vincenzo Librandi_, Jan 16 2013
		

Crossrefs

Programs

  • Magma
    [n: n in [1..120] | not IsPrime(2*n + 11)]; // Vincenzo Librandi, Nov 21 2012
  • Mathematica
    Select[Range[100],!PrimeQ[2#+11]&] (* Harvey P. Dale, Jul 18 2011 *)

A162258 a(n) = (2*n^3 + 5*n^2 - 9*n)/2.

Original entry on oeis.org

-1, 9, 36, 86, 165, 279, 434, 636, 891, 1205, 1584, 2034, 2561, 3171, 3870, 4664, 5559, 6561, 7676, 8910, 10269, 11759, 13386, 15156, 17075, 19149, 21384, 23786, 26361, 29115, 32054, 35184, 38511, 42041, 45780, 49734, 53909, 58311, 62946, 67820
Offset: 1

Views

Author

Vincenzo Librandi, Jun 29 2009

Keywords

Crossrefs

Cf. A155546.

Programs

  • Mathematica
    LinearRecurrence[{4, -6, 4, -1}, {-1, 9, 36, 86}, 50] (* or *) CoefficientList[Series[(2+7*x-3*x^2)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 04 2012 *)

Formula

Row sums from A155546: a(n) = Sum_{m=1..n} (2*m*n + m + n - 5).
From Vincenzo Librandi, Mar 04 2012: (Start)
G.f.: x*(-1 + 13*x - 6*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

Extensions

New name from Vincenzo Librandi, Mar 04 2012

A324937 Triangle read by rows: T(n, k) = 2*n*k + n + k - 8.

Original entry on oeis.org

-4, -1, 4, 2, 9, 16, 5, 14, 23, 32, 8, 19, 30, 41, 52, 11, 24, 37, 50, 63, 76, 14, 29, 44, 59, 74, 89, 104, 17, 34, 51, 68, 85, 102, 119, 136, 20, 39, 58, 77, 96, 115, 134, 153, 172, 23, 44, 65, 86, 107, 128, 149, 170, 191, 212, 26, 49, 72, 95, 118, 141, 164, 187, 210, 233, 256
Offset: 1

Views

Author

Vincenzo Librandi, Mar 25 2019

Keywords

Examples

			Triangle begins:
  -4;
  -1, 4;
   2, 9,  16;
   5, 14, 23, 32;
   8, 19, 30, 41, 52;
  11, 24, 37, 50, 63, 76;
  14, 29, 44, 59, 74, 89,  104;
  17, 34, 51, 68, 85, 102, 119, 136;
  20, 39, 58, 77, 96, 115, 134, 153, 172;  etc.
		

Crossrefs

Similar sequence T(n,k) = 2*n*k+n+k-h: A144562 (h=1); A154680 (h=2); A154684 (h=3); A155724 (h=4); A155546 (h=5); A155550 (h=6); A144670 (h=7); this sequence (h=8); A155551 (h=9).

Programs

  • Magma
    [2*n*k+n+k-8: k in [1..n], n in [1..11]]; /* As triangle */ [[2*n*k+n+k-8: k in [1..n]]: n in [1.. 15]];
  • Mathematica
    t[n_, k_]:=2 n k + n + k - 8; Table[t[n, k], {n, 11}, {k, n}]//Flatten

Formula

G.f.: x*y*(9*x^3*y^2 - 4*x^2*y*(5 + 2*y) + x*(7 + 16*y) - 4)/((1 - x)^2*(1 - x*y)^3). - Stefano Spezia, Jul 29 2025
Showing 1-4 of 4 results.