A206604 Number of integers in the smallest interval containing both minimal and maximal possible apex values of an addition triangle whose base is a permutation of n+1 consecutive integers.

1, 1, 3, 9, 27, 73, 189, 465, 1115, 2601, 5973, 13489, 30149, 66641, 146233, 318369, 689403, 1484137, 3181797, 6790641, 14445101, 30617841, 64724553, 136426849, 286926757, 601999633, 1260707529, 2634831585, 5497982025, 11452601761, 23823827825, 49484904257

Offset

0,3

Comments

For n>0 the base row of the addition triangle may contain a permutation of any set {b+k, k=0..n} where b is an integer or a half-integer. Each number in a higher row is the sum of the two numbers directly below it. Rows above the base row contain only integers.

a(n) = 3 (mod 4) if n = 2^m with m > 0 and a(n) = 1 (mod 4) else.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = 1 + Sum_{k=0..n} C(n,floor(k/2)) * (2*k-n).

G.f.: 1/(1-x) + (1-sqrt(1-4*x^2)) / (2*x-1)^2.

a(n) = 1 + 2*A206603(n).

a(n) = 1 + A189390(n)-A189391(n).

a(n) ~ n*2^n * (1-2*sqrt(2)/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 15 2014

Example

a(3) =  9:   max:   20          min:   12
                  9   11             7   5
                3   6   5          5   2   3
             1/2 5/2 7/2 3/2    7/2 3/2 1/2 5/2
[12, 13, ..., 20] contains 20-12+1 = 9 integers.
a(4) = 27:   max:   13          min:  -13
                   5  8              -5 -8
                 0  5  3            0 -5 -3
              -2  2  3  0         2 -2 -3  0
            -2  0  2  1 -1      2  0 -2 -1  1
[-13, -12, ..., 13] contains 13-(-13)+1 = 27 integers.

Maple

a:= n-> 1 +add(binomial(n, floor(k/2))*(2*k-n), k=0..n):
seq(a(n), n=0..40);
# second Maple program
a:= proc(n) option remember; `if`(n<3, 1+n*(n-1),
      (3*n^2-6*n+6+(2*n^2-6)*a(n-1)+4*(n-1)*(n-4)*a(n-2)
      -8*(n-1)*(n-2)*a(n-3)) / (n*(n-2)))
    end:
seq(a(n), n=0..40); # Alois P. Heinz, Apr 25 2013

Mathematica

a = DifferenceRoot[Function[{y, n}, {(-2n^2 - 12n - 12) y[n+2] - 3n^2 + 8(n+1)(n+2) y[n] - 4(n-1)(n+2) y[n+1] + (n+1)(n+3) y[n+3] - 12n - 15 == 0, y[0] == 1, y[1] == 1, y[2] == 3, y[3] == 9}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Prog

(PARI) a(n) = 1 + sum(k=0, n, binomial(n, k\2)*(2*k-n)); \\ Michel Marcus, Dec 20 2020

Crossrefs

Cf. A189390, A189391, A206603.

Sequence in context: A110740 A348555 A042938 * A084707 A193703 A289658

Adjacent sequences: A206601 A206602 A206603 * A206605 A206606 A206607

Keyword

nonn

Author

Alois P. Heinz, Feb 10 2012

Status

approved