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.

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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.

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 1-2 of 2 results.

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