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.

A168326 a(n) = (6*n - 3*(-1)^n - 1)/2.

Original entry on oeis.org

4, 4, 10, 10, 16, 16, 22, 22, 28, 28, 34, 34, 40, 40, 46, 46, 52, 52, 58, 58, 64, 64, 70, 70, 76, 76, 82, 82, 88, 88, 94, 94, 100, 100, 106, 106, 112, 112, 118, 118, 124, 124, 130, 130, 136, 136, 142, 142, 148, 148, 154, 154, 160, 160, 166, 166, 172, 172, 178, 178
Offset: 1

Views

Author

Vincenzo Librandi, Nov 23 2009

Keywords

Links

Crossrefs

Cf. A016957, A168236.

Programs

  • Magma
    [n eq 1 select n+3 else 6*n-Self(n-1)-4: n in [1..70]]; // Vincenzo Librandi, Sep 17 2013
  • Mathematica
    With[{c = 6 Range[0, 30] + 4}, Riffle[c, c]] (* or *) RecurrenceTable[ {a[1] == 4, a[n] == 6 n - a[n-1] - 4}, a, {n, 60}] (* Harvey P. Dale, Jun 12 2012 *)
Table[3 n - 3 (-1)^n/2 - 1/2, {n, 70}] (* Bruno Berselli, Sep 17 2013 *)
CoefficientList[Series[(4 + 2 x^2) / ((1 + x) (1 - x)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 17 2013 *)

Formula

a(n) = 6*n - a(n-1) - 4, with n>1, a(1)=4.
From Vincenzo Librandi, Sep 17 2013: (Start)
G.f.: 2*x*(2 + x^2)/((1+x)*(1-x)^2).
a(n) = 2*A168236(n) = A168300(n) - 1 = A168329(n) + 1 = A168301(n+1) - 3.
a(n) = a(n-1) +a(n-2) -a(n-3). (End)
E.g.f.: (1/2)*(-3 + 4*exp(x) + (6*x - 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 18 2016

Extensions

New definition by Bruno Berselli, Sep 17 2013

A168414 a(n) = (18*n - 9*(-1)^n - 3)/4.

Original entry on oeis.org

6, 6, 15, 15, 24, 24, 33, 33, 42, 42, 51, 51, 60, 60, 69, 69, 78, 78, 87, 87, 96, 96, 105, 105, 114, 114, 123, 123, 132, 132, 141, 141, 150, 150, 159, 159, 168, 168, 177, 177, 186, 186, 195, 195, 204, 204, 213, 213, 222, 222, 231, 231, 240, 240, 249, 249, 258
Offset: 1

Views

Author

Vincenzo Librandi, Nov 25 2009

Keywords

Links

Programs

  • Magma
    [6+9*Floor((n-1)/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013
  • Mathematica
    Table[6 + 9 Floor[(n - 1)/2], {n, 70}] (* or *) CoefficientList[Series[3 (2 + x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)
LinearRecurrence[{1,1,-1},{6,6,15},60] (* Harvey P. Dale, May 17 2017 *)

Formula

a(n) = 9*n - a(n-1) - 6, n>1.
From R. J. Mathar, Jul 10 2011: (Start)
a(n) = 3*A168236(n).
G.f.: 3*x*(2 + x^2) / ( (1+x)*(x-1)^2 ). (End)
a(n) = 6 + 9*Floor((n-1)/2). - Vincenzo Librandi, Sep 19 2013
From G. C. Greubel, Jul 22 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
E.g.f.: (3/4)*(-3 + 4*exp(x) +(6*x - 1)*exp(2*x))*exp(-x). (End)

Extensions

Definition replaced by Lava formula of Nov 2009. - R. J. Mathar, Jul 10 2011

A319128 Interleave n*(3*n - 2), 3*n^2 + n - 1, n=0,0,1,1, ... .

Original entry on oeis.org

0, -1, 1, 3, 8, 13, 21, 29, 40, 51, 65, 79, 96, 113, 133, 153, 176, 199, 225, 251, 280, 309, 341, 373, 408, 443, 481, 519, 560, 601, 645, 689, 736, 783, 833, 883, 936, 989, 1045, 1101, 1160, 1219, 1281, 1343, 1408, 1473, 1541, 1609, 1680, 1751
Offset: 0

Views

Author

Paul Curtz, Sep 11 2018

Keywords

Comments

A144391(n) = -1, 3, 13, 29, 51, ... is in the hexagonal spiral begining with -1 (like from 0 in A000567):
.
55--54--53--52--51
/ \
56 32--31--30--29 50
/ / \ \
57 33 15--14--13 28 49
/ / / \ \ \
58 34 16 4---3 12 27 48
/ / / / \ \ \ \
59 35 17 5 -1 2 11 26 47
/ / / / / / / /
36 18 6 0---1 10 25 46
\ \ \ / / /
37 19 7---8---9 24 45
\ \ / /
38 20--21--22--23 44
\ /
39--40--41--42--43
.
A000567(n) = 0, 1, 8, 21, 40, ... is in the first hexagonal spiral.
The bisections 0, 1, 8, 21, ... and -1, 3, 13, 29, ... are on the respective main antidiagonals.
a(-n) = 0, 1, 5, 9, 16, 23, ... . The bisections n*(3*n + 2) and 3*n^2 - n - 1 are in both spirals on main diagonals.
The bisections of a(n) are in the second spiral: ... 29, 13, 3, -1, 0, 1, 8, 21, ... .
The bisections of a(-n) are in the first and in the second spiral: ... 33, 16, 5, 0, 1, 9, 23, ... .

Links

Crossrefs

Main diagonal of A318958.
Cf. A000567, A144391, A168236, A045944, A144390.

Programs

  • GAP
    Flat(List([0..30],n->[3*n^2-2*n,3*n^2+n-1])); # Muniru A Asiru, Sep 19 2018
  • Maple
    seq(op([3*n^2-2*n,3*n^2+n-1]),n=0..30); # Muniru A Asiru, Sep 19 2018
  • Mathematica
    LinearRecurrence[{2, 0, -2, 1}, {0, -1, 1, 3 }, 40] (* Stefano Spezia, Sep 16 2018 *)
  • PARI
    concat(0, Vec(-x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ Colin Barker, Sep 14 2018

Formula

a(n+1) = a(n) + (6*n^2 - 3*(-1)^n - 1)/4, n=0,1,2, ... , a(0) = 0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>3.
From Colin Barker, Sep 14 2018: (Start)
G.f.: -x*(1 - 3*x - x^2) / ((1 - x)^3*(1 + x)).
a(n) = (-4*n + 3*n^2) / 4 for n even.
a(n) = (-3 - 4*n + 3*n^2) / 4 for n odd.
(End)
a(n) = (-3 + 3*(-1)^n - 8*n + 6*n^2)/8. - Colin Barker, Sep 14 2018
E.g.f.: (x*(3*x - 1)*cosh(x) + (3*x^2 - x - 3)*sinh(x))/4. - Stefano Spezia, Mar 15 2020
Showing 1-3 of 3 results.

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