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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A174814 a(n) = n*(n+1)*(5*n+1)/3.

Original entry on oeis.org

0, 4, 22, 64, 140, 260, 434, 672, 984, 1380, 1870, 2464, 3172, 4004, 4970, 6080, 7344, 8772, 10374, 12160, 14140, 16324, 18722, 21344, 24200, 27300, 30654, 34272, 38164, 42340, 46810, 51584, 56672, 62084, 67830, 73920, 80364, 87172, 94354, 101920, 109880
Offset: 0

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Author

Bruno Berselli, Dec 01 2010 - Dec 02 2010

Keywords

Comments

Also zero followed by bisection (even part) of A088003.
Numbers ending in 0, 2 or 4 (cf. 2*A053796(n)). Therefore we can easily see that a(m)^(2*k+1)==-1 (mod 5) only for m in A047219, while a(m)^(2*k)==-1 (mod 5) only for m in A016873 and k odd.

Links

Programs

Formula

G.f.: 2*x*(2+3*x)/(1-x)^4.
a(n) = 2*A033994(n) for n>0.
a(n) = n*A147875(n+1)-sum(k=1..n, A147875(k)) for n>0.
a(-n) = -A144945(n).

A165662 Period 5: repeat 4,4,8,6,8.

Original entry on oeis.org

4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8, 4, 4, 8, 6, 8
Offset: 0

Views

Author

Vincenzo Librandi, Sep 24 2009

Keywords

Comments

This is also the post-period decimal digit of ((n+2)^2-2)/5.
Serves also as the decimal expansion of 1495600/33333 and as the continued fraction representation of (33397+sqrt(12952802))/1649.

Links

Crossrefs

Cf. A008865, A053796, A147973.

Programs

  • Magma
    [(2*n^2+8*n+4) mod 10 : n in [0..100]]; // Wesley Ivan Hurt, Sep 06 2014
  • Maple
    A165662:=n->2*n^2+8*n+4 mod 10: seq(A165662(n), n=0..100); # Wesley Ivan Hurt, Sep 06 2014
  • Mathematica
    Table[Mod[2 n^2 + 8 n + 4, 10], {n, 0, 100}] (* Wesley Ivan Hurt, Sep 06 2014 *)
CoefficientList[Series[2 (2 + 2 x + 4 x^2 + 3 x^3 + 4 x^4)/(1 - x^5), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 06 2014 *)
  • PARI
    a(n)=[4,4,8,6,8][n%5+1] \\ Edward Jiang, Sep 06 2014

Formula

a(n) = (2*n^2 + 8*n + 4) mod 10.
From Wesley Ivan Hurt, Sep 06 2014: (Start)
G.f.: 2*(2 + 2*x + 4*x^2 + 3*x^3 + 4*x^4)/(1-x^5). [corrected by Georg Fischer, May 11 2019]
Recurrence: a(n) = a(n-5).
a(n) = (2*A008865(n+1)) mod 10.
a(n) = (-A147973(n+4)) mod 10.
a(n+1) = 2*A053796(n) + 4. (End)

Extensions

Definition simplified, offset corrected by R. J. Mathar, Sep 25 2009
Name and offset changed by Wesley Ivan Hurt, Sep 06 2014
Showing 1-2 of 2 results.

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