A053439 -id:A053439 - cp's OEIS frontend

A128223 a(n) = if n mod 2 = 0 then n*(n+1)/2 otherwise (n+1)^2/2-1.

0, 1, 3, 7, 10, 17, 21, 31, 36, 49, 55, 71, 78, 97, 105, 127, 136, 161, 171, 199, 210, 241, 253, 287, 300, 337, 351, 391, 406, 449, 465, 511, 528, 577, 595, 647, 666, 721, 741, 799, 820, 881, 903, 967, 990, 1057, 1081, 1151, 1176, 1249, 1275, 1351, 1378, 1457, 1485

Offset: 0

Gary W. Adamson, Feb 19 2007

a(n-1) is the length of the shortest path along the edges of the complete graph with n vertices. - Martin Fuller, Dec 06 2007
From Peter Kagey, Jan 25 2015: (Start)
For an irreflexive, non-transitive, symmetric relation, a(n) is the length of a relation chain required to demonstrate that a != b for all distinct elements a and b in S, where S contains n+1 elements.
For example, for the set {1,2,3} the chain requires a(2) = 3 relations (e.g., 1 != 2 != 3 != 1). For the set {1,2,3,4}, the chain requires a(3) = 7 relations (e.g., 1 != 2 != 3 != 4 != 1 != 3 != 2 != 4 -- noting the redundancy of 2!=3 and 3!=2).  (End)
Given a set of n lots of n distinct items, it is possible to sort the items from fully collated (ABCABCABC) to fully sorted (AAABBBCCC), or vice versa, using a sorting algorithm whereby at each step a portion of the overall string is selected and its contents reversed. The minimum number of steps such an algorithm will take is a(n-1). For example, when n=3, a(n-1)=3: ABCABCABC -> ABBACBACC -> ABBAABCCC -> AAABBBCCC. - Elliott Line, Aug 02 2019

			a(5) = 17 = (5 + 1 + 5 + 1 + 5), row 5 of A128222.

Peter Kagey, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Row sums of A128222.
Cf. A024206, row sums of A128221 = A128174 * A127701.
Cf. A053439, A128221, A024206, A127701, A128174.

a128223 n = if even n then n*(n + 1) `div` 2 else (n+1)^2 `div` 2 - 1 -- Peter Kagey, Jul 14 2015

[(-1+(-1)^n-(-3+(-1)^n)*n+2*n^2)/4: n in [0..60]]; // Vincenzo Librandi, Mar 18 2015

f:=n-> if n mod 2 = 0 then n*(n+1)/2 else (n+1)^2/2-1; fi;

f[n_] := If[EvenQ@ n, n (n + 1)/2, (n + 1)^2/2 - 1]; Array[f, 54] (* Michael De Vlieger, Mar 17 2015 *)
Table[(- 1 + (-1)^n - (- 3 + (-1)^n) n + 2 n^2) / 4, {n, 0, 60}] (* Vincenzo Librandi, Mar 18 2015 *)
CoefficientList[ Series[(-x - 2x^2 - 2x^3 + x^4)/((-1 + x)^3 (1 + x)^2), {x, 0, 54}], x] (* Robert G. Wilson v, Nov 16 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{0,1,3,7,10},60] (* Harvey P. Dale, Mar 17 2020 *)

main(size)={my(n,m,v=vector(size),i);for(i=0,size-1,v[i+1]=if(i%2==0,i*(i+1)/2,(i+1)^2/2-1));return(v);} /* Anders Hellström, Jul 14 2015 */

a(n) = (-1+(-1)^n-(-3+(-1)^n)*n+2*n^2)/4. a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). G.f.: x*(x^3-2*x^2-2*x-1) / ((x-1)^3*(x+1)^2). - Colin Barker, Oct 16 2013

a(n) = A053439(n) - 1. - Peter Kagey, Nov 16 2016

Edited by N. J. A. Sloane, Dec 06 2007

A362652 Expansion of g.f. x(-2 - 2x + x^2 - x^3)/((1 + x)^2 *(-1 + x)^3).

Original entry on oeis.org

2, 4, 7, 12, 16, 24, 29, 40, 46, 60, 67, 84, 92, 112, 121, 144, 154, 180, 191, 220, 232, 264, 277, 312, 326, 364, 379, 420, 436, 480, 497, 544, 562, 612, 631, 684, 704, 760, 781, 840, 862, 924, 947, 1012, 1036, 1104, 1129, 1200, 1226, 1300, 1327

Offset: 1

Jonathon Priestley, Apr 28 2023

a(n) gives the number of vertices encountered along the shortest walk that encounters every edge at least once on the graph with n vertices where the graph is both complete and every node also has an edge to itself.

a(n) can be thought of as the length of a list made up using n distinct elements where every element is next to every other element (including a copy of itself) at least once. Such a list could be used forwards and backward when kerning a font as a way to minimize the number of characters typed in total.

			G.f.: 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 16*x^5 + 24*x^6 + 29*x^7 + 40*x^8 + 46*x^9 + ...

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A053439.

CoefficientList[Series[x(-2-2x+x^2-x^3)/((1+x)^2(-1+x)^3), {x, 0, 50}], x]
(* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {2, 4, 7, 12, 16}, 50]

def a(n: int): return n + (n & 1) + n * ( n >> 1 )

a(n) = n + (n mod 2) + (n * (n - (n mod 2)))/2.
a(2*n) = 2*n + 2*n^2;
a(2*n - 1) = 1 - n + 2*n^2.
E.g.f.: (2 + x)*(exp(x)*x + sinh(x))/2. - Stefano Spezia, May 07 2023

A278299 a(n) is the tile count of the smallest polyomino with an n-coloring such that every color is adjacent to every other distinct color at least once.

Original entry on oeis.org

2, 4, 6, 9, 12, 15, 19, 24, 30, 34

Offset: 2

Alec Jones and Peter Kagey, Nov 17 2016

Only edge-to-edge adjacencies are considered.
The sequence is bounded above by A053439(n-1).
a(n) is bounded below by n * ceiling((n - 1)/4). This bound is achieved for n=2, n=6, and n=10.

			Example: for n = 4, the following diagram gives a minimal polyomino of a(4) = 6 tiles:
      +---+---+
      | 1 | 4 |
  +---+---+---+
  | 4 | 3 | 2 |
  +---+---+---+
          | 1 |
          +---+
Example: for n = 10, the following diagram gives a minimal polyomino of a(10) = 30 tiles. Note that redundant adjacencies, e.g., between 2 and 7, can exist in minimal diagrams.
              +---+---+
              | 8 | 6 |
          +---+---+---+---+---+
          | 3 | 2 | 5 | 9 | 4 |
  +---+---+---+---+---+---+---+---+
  | 2 | 7 | 5 | 1 | 4 | 2 | 10| 9 |
  +---+---+---+---+---+---+---+---+
  | 6 | 9 | 8 | 3 | 6 | 7 | 8 | 1 |
  +---+---+---+---+---+---+---+---+
  | 10| 3 | 4 | 7 | 1 | 10| 5 |
  +---+---+---+---+---+---+---+
From _Ryan Lee_, May 14 2019: (Start)
Example for n = 11:
  +---+---+---+---+---+
  | 9 | 11| 2 | 5 | 8 |
  +---+---+---+---+---+---+
  | 1 | 5 | 10| 9 | 2 | 1 |
  +---+---+---+---+---+---+
  | 4 | 6 | 11| 8 | 7 | 3 |
  +---+---+---+---+---+---+
  | 3 | 9 | 7 | 10| 6 | 2 |
  +---+---+---+---+---+---+
  | 11| 4 | 5 | 3 | 8 | 4 |
  +---+---+---+---+---+---+
  | 1 | 10|   | 6 | 1 | 7 |
  +---+---+   +---+---+---+
(End)

Cf. A053439.

a(11) from Ryan Lee, May 14 2019

cp's OEIS Frontend

A128223 a(n) = if n mod 2 = 0 then n*(n+1)/2 otherwise (n+1)^2/2-1.

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A362652 Expansion of g.f. x(-2 - 2x + x^2 - x^3)/((1 + x)^2 *(-1 + x)^3).

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A278299 a(n) is the tile count of the smallest polyomino with an n-coloring such that every color is adjacent to every other distinct color at least once.

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cp's OEIS Frontend

A128223 a(n) = if n mod 2 = 0 then n*(n+1)/2 otherwise (n+1)^2/2-1.

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Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A362652 Expansion of g.f. x*(-2 - 2*x + x^2 - x^3)/((1 + x)^2 *(-1 + x)^3).

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A278299 a(n) is the tile count of the smallest polyomino with an n-coloring such that every color is adjacent to every other distinct color at least once.

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Author

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Extensions

A362652 Expansion of g.f. x(-2 - 2x + x^2 - x^3)/((1 + x)^2 *(-1 + x)^3).