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.

Previous Showing 11-18 of 18 results.

A226450 a(n) = n*(3*n^2 - 5*n + 3).

Original entry on oeis.org

0, 1, 10, 45, 124, 265, 486, 805, 1240, 1809, 2530, 3421, 4500, 5785, 7294, 9045, 11056, 13345, 15930, 18829, 22060, 25641, 29590, 33925, 38664, 43825, 49426, 55485, 62020, 69049, 76590, 84661, 93280, 102465, 112234, 122605, 133596, 145225, 157510, 170469
Offset: 0

Views

Author

Bruno Berselli, Jun 07 2013

Keywords

Comments

See the comment in A226449.
For n >= 3, also the detour index of the n-barbell graph. - Eric W. Weisstein, Dec 20 2017

Links

Crossrefs

Cf. A000567.
Similar sequences of the type b(m)+m*b(m-1), where b is a polygonal number: A006003, A069778, A143690, A204674, A212133, A226449, A226451.

Programs

  • Magma
    [n*(3*n^2-5*n+3): n in [0..40]];
  • Magma
    I:=[0,1,10,45]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013
  • Mathematica
    Table[n (3 n^2 - 5 n + 3), {n, 0, 40}]
CoefficientList[Series[x (1 + 6 x + 11 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 10, 45, 124}, {0, 20}] (* Eric W. Weisstein, Dec 20 2017 *)
  • PARI
    a(n) = n*(3*n^2 - 5*n + 3); \\ Altug Alkan, Dec 20 2017

Formula

G.f.: x*(1+6*x+11*x^2)/(1-x)^4.
a(n) = A000567(n) + n*A000567(n-1).

A226451 a(n) = n*(7*n^2-12*n+7)/2.

Original entry on oeis.org

0, 1, 11, 51, 142, 305, 561, 931, 1436, 2097, 2935, 3971, 5226, 6721, 8477, 10515, 12856, 15521, 18531, 21907, 25670, 29841, 34441, 39491, 45012, 51025, 57551, 64611, 72226, 80417, 89205, 98611, 108656, 119361, 130747, 142835, 155646, 169201, 183521
Offset: 0

Views

Author

Bruno Berselli, Jun 07 2013

Keywords

Comments

See the comment in A226449.

Links

Crossrefs

Cf. A001106.
Similar sequences of the type b(m)+m*b(m-1), where b is a polygonal number: A006003, A069778, A143690, A204674, A212133, A226449, A226450.

Programs

  • Magma
    [n*(7*n^2-12*n+7)/2: n in [0..40]];
  • Magma
    I:=[0,1,11,51]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 18 2013
  • Mathematica
    Table[n (7 n^2 - 12 n + 7)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 7 x + 13 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: x*(1+7*x+13*x^2)/(1-x)^4.
a(n) = A001106(n) + n*A001106(n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Oct 15 2023

A121682 Triangle read by rows: T(i,j) = (T(i-1,j) + i)*i.

Original entry on oeis.org

1, 6, 4, 27, 21, 9, 124, 100, 52, 16, 645, 525, 285, 105, 25, 3906, 3186, 1746, 666, 186, 36, 27391, 22351, 12271, 4711, 1351, 301, 49, 219192, 178872, 98232, 37752, 10872, 2472, 456, 64, 1972809, 1609929, 884169, 339849, 97929, 22329, 4185, 657, 81, 19728190, 16099390, 8841790, 3398590, 979390, 223390, 41950, 6670, 910, 100
Offset: 1

Views

Author

Thomas Wieder, Aug 15 2006

Keywords

Comments

The first column is A030297 = a(n) = n*(n+a(n-1)). The main diagonal are the squares A000290 = n^2. The first lower diagonal (6,21,52,...) is A069778 = q-factorial numbers 3!_q. See also A121662.

Examples

			Triangle begins:
      1
      6     4
     27    21     9
    124   100    52   16
    645   525   285  105  25
   3906  3186  1746  666  186  36
  27391 22351 12271 4711 1351 301 49
  ...

References

  • T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Crossrefs

Cf. A030297, A000290, A069778, A121662.
Row sums give A337001.

Programs

  • Maple
    T:= proc(i, j) option remember;
      `if`(j<1 or j>i, 0, (T(i-1, j)+i)*i)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 22 2022
  • Mathematica
    T[n_, k_] /; 1 <= k <= n := T[n, k] = (T[n-1, k]+n)*n;
T[, ] = 0;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2022 *)
  • Python
    def T(i, j): return (T(i-1, j)+i)*i if 1 <= j <= i else 0
print([T(r, c) for r in range(1, 11) for c in range(1, r+1)]) # Michael S. Branicky, Jun 22 2022

Extensions

Edited by N. J. A. Sloane, Sep 15 2006
Formula in name corrected by Alois P. Heinz, Jun 22 2022

A153258 n^3 - (n+2)^2.

Original entry on oeis.org

-4, -8, -8, 2, 28, 76, 152, 262, 412, 608, 856, 1162, 1532, 1972, 2488, 3086, 3772, 4552, 5432, 6418, 7516, 8732, 10072, 11542, 13148, 14896, 16792, 18842, 21052, 23428, 25976, 28702, 31612, 34712, 38008, 41506, 45212, 49132, 53272, 57638
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

Keywords

Links

Crossrefs

Cf. A045991, A081437, A069778, A153257

Programs

  • Mathematica
    a[n_] := n^3-(n+2)^2; lst={}; Do[AppendTo[lst, a[n]], {n, 0, 5!}]; lst

Formula

G.f.: 2*x*(x^3+4*x-2)/(x-1)^4. [Colin Barker, Oct 08 2012]

A153259 a(n)=n^3-(3*(n+3))^2.

Original entry on oeis.org

-81, -143, -217, -297, -377, -451, -513, -557, -577, -567, -521, -433, -297, -107, 143, 459, 847, 1313, 1863, 2503, 3239, 4077, 5023, 6083, 7263, 8569, 10007, 11583, 13303, 15173, 17199, 19387, 21743, 24273, 26983, 29879, 32967, 36253, 39743
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

Keywords

Links

Crossrefs

Cf. A045991, A081437, A069778, A153257, A153258.

Programs

  • Mathematica
    a[n_]:=n^3-(3*(n+3))^2;lst={};Do[AppendTo[lst,a[n]],{n,0,5!}];lst
Table[n^3-(3(n+3))^2,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{-81,-143,-217,-297},40] (* Harvey P. Dale, Jul 10 2013 *)
  • PARI
    a(n)=n^3-(3*n+9)^2 \\ Charles R Greathouse IV, Oct 18 2022

Formula

a(1)=-81, a(2)=-143, a(3)=-217, a(4)=-297, a(n)=4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 10 2013

A153260 a(n) = n^3 - 3*(n+3)^2.

Original entry on oeis.org

-27, -47, -67, -81, -83, -67, -27, 43, 149, 297, 493, 743, 1053, 1429, 1877, 2403, 3013, 3713, 4509, 5407, 6413, 7533, 8773, 10139, 11637, 13273, 15053, 16983, 19069, 21317, 23733, 26323, 29093, 32049, 35197, 38543, 42093, 45853, 49829, 54027
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

Keywords

Links

Crossrefs

Cf. A045991, A081437, A069778, A153257, A153258, A153259.

Programs

  • Magma
    [n^3-3*(n+3)^2: n in [0..40] ]; // Vincenzo Librandi, Aug 25 2011
  • Mathematica
    a[n_]:=n^3-3*(n+3)^2; a/@ Range[0, 50]
Table[n^3-3(n+3)^2,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{-27,-47,-67,-81},51] (* Harvey P. Dale, Aug 24 2011 *)
  • PARI
    vector(40, n, n--; n^3-3*(n+3)^2) \\ G. C. Greubel, Nov 10 2018

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=-27, a(1)=-47, a(2)=-67, a(3)=-81. - Harvey P. Dale, Aug 24 2011
G.f.: (x*(x*(13*x - 41) + 61) - 27)/(x-1)^4. - Harvey P. Dale, Aug 24 2011
E.g.f.: (-27 - 20*x + x^3)*exp(x). - G. C. Greubel, Nov 10 2018

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Aug 25 2011

A318765 a(n) = (n + 2)*(n^2 + n - 1).

Original entry on oeis.org

-2, 3, 20, 55, 114, 203, 328, 495, 710, 979, 1308, 1703, 2170, 2715, 3344, 4063, 4878, 5795, 6820, 7959, 9218, 10603, 12120, 13775, 15574, 17523, 19628, 21895, 24330, 26939, 29728, 32703, 35870, 39235, 42804, 46583, 50578, 54795, 59240, 63919, 68838, 74003, 79420, 85095
Offset: 0

Views

Author

Bruno Berselli, Sep 04 2018

Keywords

Comments

First differences are in A004538.
a(n) is divisible by 11 for n = 3, 7, 9, 14, 18, 20, 25, 29, 31, 36, 40, ... with formula (1/3)*(11*m + (1 + (m mod 3))*(-1)^((m-1) mod 3) + 8), m >= 0.

Links

Crossrefs

Cf. A004538.
Subsequence of A047216.
Similar sequences (see Table in Links section): A011379, A027444, A033445, A034262, A045991, A069778.

Programs

  • GAP
    List([0..50], n -> (n+2)*(n^2+n-1));
  • Julia
    [(n+2)*(n^2+n-1) for n in 0:50] |> println
  • Magma
    [(n+2)*(n^2+n-1): n in [0..50]];
  • Maple
    seq((n+2)*(n^2+n-1),n=0..43); # Paolo P. Lava, Sep 04 2018
  • Mathematica
    Table[(n + 2) (n^2 + n - 1), {n, 0, 50}]
  • Maxima
    makelist((n+2)*(n^2+n-1), n, 0, 50);
  • PARI
    vector(50, n, n--; (n+2)*(n^2+n-1))
  • Python
    [(n+2)*(n**2+n-1) for n in range(50)]
  • Sage
    [(n+2)*(n^2+n-1) for n in (0..50)]

Formula

O.g.f.: (-2 + 11*x - 4*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-2 + 5*x + 6*x^2 + x^3)*exp(x).
a(n) = -A033445(-n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 5. - Wesley Ivan Hurt, Dec 18 2020

A228398 The number of permutations of length n sortable by 3 prefix reversals (in the pancake sorting sense).

Original entry on oeis.org

1, 2, 6, 21, 52, 105, 186, 301, 456, 657, 910, 1221, 1596, 2041, 2562, 3165, 3856, 4641, 5526, 6517, 7620, 8841, 10186, 11661, 13272, 15025, 16926, 18981, 21196, 23577, 26130, 28861, 31776, 34881, 38182, 41685, 45396, 49321, 53466, 57837, 62440, 67281, 72366
Offset: 1

Views

Author

Vincent Vatter, Aug 21 2013

Keywords

Comments

Essentially the same as A069778.

Examples

			There are only 3 permutations of length 4 which cannot be sorted by 3 pancake reversals.

Links

Crossrefs

Cf. A000027, A002522.

Programs

  • Mathematica
    CoefficientList[Series[(1/x) (-1 + (x^6 - 3 x^5 + 6 x^4 + 4 x^2 - 3 x + 1)/(x - 1)^4), {x, 0, 50}], x] (* Bruno Berselli, Aug 22 2013 *)

Formula

G.f.: -1 + (x^6 - 3*x^5 + 6*x^4 + 4*x^2 - 3*x + 1)/(x - 1)^4.
a(n) = (n-1)*(n^2-3*n+3) for n>2, a(1)=1, a(2)=2. [Bruno Berselli, Aug 22 2013]
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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