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.

A226449 a(n) = n*(5*n^2-8*n+5)/2.

Original entry on oeis.org

0, 1, 9, 39, 106, 225, 411, 679, 1044, 1521, 2125, 2871, 3774, 4849, 6111, 7575, 9256, 11169, 13329, 15751, 18450, 21441, 24739, 28359, 32316, 36625, 41301, 46359, 51814, 57681, 63975, 70711, 77904, 85569, 93721, 102375, 111546, 121249, 131499, 142311, 153700
Offset: 0

Views

Author

Bruno Berselli, Jun 07 2013

Keywords

Comments

Sequences of the type b(m)+m*b(m-1), where b is a polygonal number:
A006003(n) = A000217(n) + n*A000217(n-1) (b = triangular numbers);
A069778(n) = A000290(n+1) + (n+1)*A000290(n) (b = square numbers);
A143690(n) = A000326(n+1) + (n+1)*A000326(n) (b = pentagonal numbers);
A212133(n) = A000384(n) + n*A000384(n-1) (b = hexagonal numbers);
a(n) = A000566(n) + n*A000566(n-1) (b = heptagonal numbers);
A226450(n) = A000567(n) + n*A000567(n-1) (b = octagonal numbers);
A226451(n) = A001106(n) + n*A001106(n-1) (b = nonagonal numbers);
A204674(n) = A001107(n+1) + (n+1)*A001107(n) (b = decagonal numbers).

Links

Crossrefs

Cf. (see the comment) A000566, A006003, A069778, A143690, A204674, A212133, A226450, A226451.

Programs

  • Magma
    [n*(5*n^2-8*n+5)/2: n in [0..40]];
  • Magma
    I:=[0,1,9,39]; [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 (5 n^2 - 8 n + 5)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 5 x + 9 x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,1,9,39},50] (* Harvey P. Dale, May 19 2017 *)
  • PARI
    a(n)=n*(5*n^2-8*n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: x*(1+5*x+9*x^2)/(1-x)^4.
a(n) - a(-n) = A008531(n) for n>0.

A033292 A Connell-like sequence: take 1 number = 1 (mod Q), 2 numbers = 2 (mod Q), 3 numbers = 3 (mod Q), etc., where Q = 3.

Original entry on oeis.org

1, 2, 5, 6, 9, 12, 13, 16, 19, 22, 23, 26, 29, 32, 35, 36, 39, 42, 45, 48, 51, 52, 55, 58, 61, 64, 67, 70, 71, 74, 77, 80, 83, 86, 89, 92, 93, 96, 99, 102, 105, 108, 111, 114, 117, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 177, 180
Offset: 1

Views

Author

Gary E. Stevens

Keywords

Comments

Left border of the triangle (1, 2, 6, 13, 23, 36, ...) = A143689 = A000326(n) - 3n, where A000326 = the pentagonal numbers, right border. - Gary W. Adamson, Aug 29 2008
Row sums = A143690: (1, 7, 27, 70, 145, 261, 427, 652, ...). - Gary W. Adamson, Aug 29 2008
Central terms = A136392. - Reinhard Zumkeller, Jan 18 2012

Links

Programs

  • Haskell
    a033292 n k = a033292_tabl !! (n-1) !! (k-1)
a033292_row n = a033292_tabl !! (n-1)
a033292_tabl = f 1 [1..] where
   f k xs = ys : f (k+1) (dropWhile (<= last ys) xs) where
     ys  = take k $ filter ((== 0) . (`mod` 3) . (subtract k)) xs
-- Reinhard Zumkeller, Jan 18 2012 2011
  • Mathematica
    row[n_] := n*(3*n-1)/2 + Range[1, 3*n+1, 3]; Flatten[ Table[ row[n], {n, 0, 11}]] (* Jean-François Alcover, Aug 03 2012 *)

Extensions

More terms from jeroen.lahousse(AT)icl.com
Offset changed by Reinhard Zumkeller, Jan 18 2012

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
Showing 1-4 of 4 results.

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