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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

CROSSREFS

Cf. A189390, A189391, A206603.

Sequence in context: A103828 A110740 A042938 * A084707 A193703 A289658

Adjacent sequences:  A206601 A206602 A206603 * A206605 A206606 A206607

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 10 2012

STATUS

approved

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Last modified December 5 13:26 EST 2019. Contains 329751 sequences. (Running on oeis4.)