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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A318054 a(n) = n*(n + 1)*(n^2 + n + 22)/24.

Original entry on oeis.org

0, 2, 7, 17, 35, 65, 112, 182, 282, 420, 605, 847, 1157, 1547, 2030, 2620, 3332, 4182, 5187, 6365, 7735, 9317, 11132, 13202, 15550, 18200, 21177, 24507, 28217, 32335, 36890, 41912, 47432, 53482, 60095, 67305, 75147, 83657, 92872, 102830, 113570, 125132, 137557
Offset: 0

Views

Author

Luce ETIENNE, Aug 14 2018

Keywords

Examples

			a(1) = 2; a(2)= 5+2 = 7; a(3) = 10+5+2 = 17; a(4) = 18+10+5+2 = 35; a(5) = 30+18+10+5+2 = 65; a(6) = 47+30+18+10+5+2 = 112.

Links

Crossrefs

Partial sums of A177787.
Cf. A089071, A022856, A152948.

Programs

  • GAP
    List([0..30],n->n*(n+1)*(n^2+n+22)/24); # Muniru A Asiru, Aug 15 2018
  • Maple
    seq(coeff(series(x*(2*x^2-3*x+2)/(1-x)^5, x,n+1),x,n),n=0..30); # Muniru A Asiru, Aug 15 2018
  • PARI
    a(n) = n*(n+1)*(n^2+n+22)/24; \\ Michel Marcus, Aug 17 2018

Formula

G.f.: x*(2*x^2-3*x+2)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = (1/6)*Sum_{i=1..n} (n-i)*((n-i)^2+11), for n >= 1.

A173248 a(0)=1, a(n) = (-1)^n*(n^3-15*n^2-12+2*n)/6, n>0.

Original entry on oeis.org

-1, 4, -10, 19, -30, 42, -54, 65, -74, 80, -82, 79, -70, 54, -30, -3, 46, -100, 166, -245, 338, -446, 570, -711, 870, -1048, 1246, -1465, 1706, -1970, 2258, -2571, 2910, -3276, 3670, -4093, 4546, -5030, 5546, -6095, 6678, -7296, 7950, -8641, 9370
Offset: 0

Views

Author

Roger L. Bagula, Feb 13 2010

Keywords

Comments

Limiting ratio a(n+1)/a(n) is near -1.071806167400881 as n->infinity.

Links

Crossrefs

Cf. A089071, A105811

Programs

  • Mathematica
    p[x_] = (x^4 - x^3 - 1)/(x + 1)^4;
a = Table[SeriesCoefficient[ Series[p[x], {x, 0, 50}], n], {n, 0, 50}]
LinearRecurrence[{-4,-6,-4,-1},{-1,4,-10,19,-30},50] (* Harvey P. Dale, Nov 21 2019 *)

Formula

G.f.: (x^4 - x^3 - 1)/(x + 1)^4.
a(n)= -4*a(n-1) -6*a(n-2) -4*a(n-3) -a(n-4).

Extensions

Definition simplified by the Assoc. Editors of the OEIS, Feb 21 2010

A308414 Minimal number of moves after which a group of stones of one color with exactly n liberties is possible on an infinite board in Go.

Original entry on oeis.org

7, 5, 3, 2, 1, 3, 3, 5, 5
Offset: 0

Views

Author

Felix Fröhlich, May 25 2019

Keywords

Examples

			For n = 0: A white stone with 4 black stones adjacent to it has 0 liberties, which is possible after 7 moves, so a(0) = 7.
For n = 1: A white stone with 3 black stones adjacent to it has 1 liberty, which is possible after 5 moves (3 of black and 2 of white), so a(1) = 5.
For n = 2: A white stone with 2 black stones adjacent to it has 2 liberties, which is possible after 3 moves, so a(2) = 3.
For n = 3: A black stone with a single white stone adjacent to it has 3 liberties, which is possible after 2 moves, so a(3) = 2.
For n = 4: A single stone has 4 liberties, with that configuration possible after 1 move of black, so a(4) = 1.
For n = 5: A group of 2 black stones adjacent to a single white stone has 5 liberties, which is possible after 3 moves, so a(5) = 3.
For n = 6: A group of 2 black stones has 6 liberties, which is possible after 3 moves, so a(6) = 3.
For n = 7: A group of 3 black stones in a straight line with a single adjacent white stone or a group of 3 black stones in an L-shape have 7 liberties, both of which are possible after 5 moves, so a(7) = 5.
For n = 8: A group of 3 black stones in a straight line has 8 liberties, which is possible after 5 moves, so a(8) = 5.

Links

Crossrefs

Cf. A089071, A275346, A320666.
Previous Showing 11-13 of 13 results.

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