A064843 -id:A064843 - cp's OEIS frontend

A084265 a(n) = (n^2 + 3*n + 1 + (-1)^n) / 2.

1, 2, 6, 9, 15, 20, 28, 35, 45, 54, 66, 77, 91, 104, 120, 135, 153, 170, 190, 209, 231, 252, 276, 299, 325, 350, 378, 405, 435, 464, 496, 527, 561, 594, 630, 665, 703, 740, 780, 819, 861, 902, 946, 989, 1035, 1080, 1128, 1175, 1225, 1274, 1326, 1377, 1431, 1484

Offset: 0

Paul Barry, May 31 2003

Previous name was: Modified triangular numbers.
Binomial transform is A084266.
Partial sums give A064843. - N. J. A. Sloane, Jul 20 2008
Starting with "1" = triangle A171608 * the odd integers, (1, 3, 5, ...). - Gary W. Adamson, Dec 12 2009

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

Cf. A084263.
Cf. A171608.
Cf. A000384, A014105, A014107.

[(n^2+3*n+1)/2+(-1)^n/2: n in [0..60]]; // Vincenzo Librandi, Aug 15 2013

A084265:=n->(n^2+3*n+1)/2+(-1)^n/2: seq(A084265(n),n=0..100); # Wesley Ivan Hurt, Mar 21 2015

CoefficientList[Series[(-1 - 2 x^2 + x^3) / ((1 + x) (x - 1)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 15 2013 *)

vector(100,n,(n^2+n-1-(-1)^n)/2) \\ Derek Orr, Mar 22 2015

a(n) = A000217(n)+A059841(n)+n.
E.g.f.: cosh(x) + exp(x)*(2x+x^2/2).
a(n) = (n^2+3*n+1)/2+(-1)^n/2.
G.f.: ( -1-2*x^2+x^3 ) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Nov 26 2012
From Wesley Ivan Hurt, Mar 21 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = Sum_{i=0..n+1} i-(-1)^i. (End)
a(2*n) = A000384(n+1); a(2*n-1) = A014105(n)-1; a(2*n-1) = A014107(n+1), for all integers n. - Hartmut F. W. Hoft, Feb 02 2022

New name from Joerg Arndt, Aug 15 2013

A064842 Maximal value of Sum_{i=1..n} (p(i) - p(i+1))^2, where p(n+1) = p(1), as p ranges over all permutations of {1, 2, ..., n}.

Original entry on oeis.org

0, 2, 6, 18, 36, 66, 106, 162, 232, 322, 430, 562, 716, 898, 1106, 1346, 1616, 1922, 2262, 2642, 3060, 3522, 4026, 4578, 5176, 5826, 6526, 7282, 8092, 8962, 9890, 10882, 11936, 13058, 14246, 15506, 16836, 18242, 19722, 21282, 22920, 24642, 26446, 28338, 30316

Offset: 1

N. J. A. Sloane, Oct 25 2001

			a(4) = 18 because the values of the sum for the permutations of {1, 2, 3, 4} are 10 (8 times), 12 (8 times) and 18 (8 times).

G. L. Cohen and E. Tonkes, Dartboard arrangements, Elect. J. Combin., 8(2) (2001), #R4.
Vasile Mihai and Michael Woltermann, Problem 10725: The Smoothest and Roughest Permutations, Amer. Math. Monthly, 108 (2001), 272-273.
Keith Selkirk, Re-designing the dartboard, Math. Gaz., 60 (1976), 171-178.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A064843.

a:=proc(n) if n mod 2 = 0 then (n^3-4*n)/3+2 else (n^3-4*n)/3+1 fi end: seq(a(n),n=1..41); # Emeric Deutsch

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 2, 6, 18, 36}, 45] (* Jean-François Alcover, Apr 01 2020 *)

If n mod 2 = 0, then n^3/3 - 4*n/3 + 2 else n^3/3 - 4*n/3 + 1.
a(n) = 2 * A064843(n).
G.f.: -2*x^2*(-1 + x^3 - 2*x^2) / ((1 + x)*(x - 1)^4). - R. J. Mathar, Nov 26 2012
a(n) = (2*n^3 - 8*n + 3*(-1)^n + 9)/6. - Luce ETIENNE, Jul 08 2014
E.g.f.: (2 - x + x^2 + x^3/3)*cosh(x) + (1 - x + x^2 + x^3/3)*sinh(x) - 2. - Stefano Spezia, Apr 13 2024

Edited by Emeric Deutsch, Jul 30 2005

cp's OEIS Frontend

A084265 a(n) = (n^2 + 3*n + 1 + (-1)^n) / 2.

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