A128222 -id:A128222 - 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

A128221 A128174 * A127701.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 1, 1, 3, 1, 5, 1, 2, 1, 4, 1, 6, 1, 1, 3, 1, 5, 1, 7, 1, 2, 1, 4, 1, 6, 1, 8, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = A024206: (1, 3, 5, 8, 11, 15, 19, ...). A128222 = A127701 * A128174.

Table T(n,k) = n, if k is odd, 1 if k is even; n, k > 0, read by antidiagonals. -Boris Putievskiy, Jan 30 2013

			From _Boris Putievskiy_, Jan 30 2013: (Start)
The start of the sequence as a table:
  1, 1, 1, 1, 1, 1, 1, ...
  2, 1, 2, 1, 2, 1, 2, ...
  3, 1, 3, 1, 3, 1, 3, ...
  4, 1, 4, 1, 4, 1, 4, ...
  5, 1, 5, 1, 5, 1, 5, ...
  6, 1, 6, 1, 6, 1, 6, ...
  7, 1, 7, 1, 7, 1, 7, ...
  ...
(End)
First few rows of the triangle are:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 1, 4;
  1, 1, 3, 1, 5;
  1, 2, 1, 4, 1, 6;
  1, 1, 3, 1, 5, 1, 7;
  ...

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A128174, A127701, A024206, A128222.

a128221[n_, k_] := If[EvenQ[n-k], k, 1]/;1<=k<=n
a128221[r_] := Table[a128221[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128221[7]] (* triangle *)
Flatten[a128221[10]] (* data *) (* Hartmut F. W. Hoft, Mar 08 2017 *)
t[r_, c_] := If[ OddQ@ c, r, 1]; Table[t[k, n - k + 1], {n, 13}, {k, n}] // Flatten (* Robert G. Wilson v, Mar 09 2017 *)

A128174 * A127701 as infinite lower triangular matrices. By columns, k-th column = k, 1, k, ...; k=1,2,3,...
From Boris Putievskiy, Jan 30 2013: (Start)
As table T(n,k) = (1+(-1)^k)/2 - (-1+(-1)^k)*n/2.
As linear sequence a(n) = (1+(-1)^A004736(n))/2 - (-1+(-1)^A004736(n))*A002260(n)/2. a(n) = (1+(-1)^j)/2 - (-1+(-1)^j)*i/2,
where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

More terms from Robert G. Wilson v, Mar 09 2017

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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