A248800 a(n) = (2*n^2 + 3 + (-1)^n)/2.

2, 2, 6, 10, 18, 26, 38, 50, 66, 82, 102, 122, 146, 170, 198, 226, 258, 290, 326, 362, 402, 442, 486, 530, 578, 626, 678, 730, 786, 842, 902, 962, 1026, 1090, 1158, 1226, 1298, 1370, 1446, 1522, 1602, 1682, 1766, 1850, 1938, 2026, 2118

Offset: 0

Author

Paul Curtz, Oct 14 2014

Keywords

• nonn
• easy

Comments

Numbers belonging to A016825: a(n) = A016825( A002620(n) ). - Bruno Berselli, Oct 15 2014

For n>1, a(n) is the number of row vectors of length 2n with entries in [1,n], first entry 1, with maximum inner distance. That is, vectors where the modular distance between adjacent entries is at least (n-2)/2. Modular distance is the minimum of remainders of (x - y) and (y - x) when dividing by n. Geometrically, this metric counts how far the integers mod n are from each other if 1 and n are adjacent as on a circle. - Omar Aceval Garcia, Jan 30 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Omar Aceval Garcia, On the Number of Maximum Inner Distance Latin Squares, arXiv:2112.13912 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A000034, A000290, A002620, A016825, A042964, A080827, A168273, A005899, A069894.

Programs

Magma

[n^2+3/2+(-1)^n/2: n in [0..50]]; // Vincenzo Librandi, Oct 15 2014

Mathematica

Table[n^2 + 3/2 + (-1)^n/2, {n, 0, 50}] (* Bruno Berselli, Oct 15 2014 *)
CoefficientList[Series[2(x^3+x^2-x+1)/((1-x)^3 (x+1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2014 *)
LinearRecurrence[{2,0,-2,1},{2,2,6,10},60] (* Harvey P. Dale, Apr 08 2019 *)

PARI

Vec(-2*(x^3+x^2-x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, Oct 15 2014

Sage

[(2*n^2 +3 +(-1)^n)/2 for n in (0..50)] # G. C. Greubel, Dec 14 2021

Formula

a(n) = A000290(n) + A000034(n+1) = 4*A002620(n) + 2.

a(n+1) = 2*A080827(n+1) = (n+2)^2 - A042964(n+1) = a(n) + 2*n + 1 -(-1)^n.

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Colin Barker, Oct 15 2014

G.f.: 2*(1-x+x^2+x^3) / ((1-x)^3*(x+1)). - Colin Barker, Oct 15 2014

E.g.f.: cosh(x) + (1 + x + x^2)*exp(x). - G. C. Greubel, Dec 14 2021

a(2n) = A005899(n) for n > 0, a(2n+1) = A069894(n). - Omar Aceval Garcia, Dec 30 2021

Extensions

Typo in data fixed by Colin Barker, Oct 15 2014