A206604 - OEIS

%I #32 Oct 28 2024 12:56:25

%S 1,1,3,9,27,73,189,465,1115,2601,5973,13489,30149,66641,146233,318369,

%T 689403,1484137,3181797,6790641,14445101,30617841,64724553,136426849,

%U 286926757,601999633,1260707529,2634831585,5497982025,11452601761,23823827825,49484904257

%N 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.

%C 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.

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

%H Alois P. Heinz, <a href="/A206604/b206604.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

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

%F a(n) ~ n*2^n * (1-2*sqrt(2)/sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 15 2014

%e a(3) =  9:   max:   20          min:   12

%e                   9   11             7   5

%e                 3   6   5          5   2   3

%e              1/2 5/2 7/2 3/2    7/2 3/2 1/2 5/2

%e [12, 13, ..., 20] contains 20-12+1 = 9 integers.

%e a(4) = 27:   max:   13          min:  -13

%e                    5  8              -5 -8

%e                  0  5  3            0 -5 -3

%e               -2  2  3  0         2 -2 -3  0

%e             -2  0  2  1 -1      2  0 -2 -1  1

%e [-13, -12, ..., 13] contains 13-(-13)+1 = 27 integers.

%p a:= n-> 1 +add(binomial(n, floor(k/2))*(2*k-n), k=0..n):

%p seq(a(n), n=0..40);

%p # second Maple program

%p a:= proc(n) option remember; `if`(n<3, 1+n*(n-1),

%p       (3*n^2-6*n+6+(2*n^2-6)*a(n-1)+4*(n-1)*(n-4)*a(n-2)

%p       -8*(n-1)*(n-2)*a(n-3)) / (n*(n-2)))

%p     end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Apr 25 2013

%t 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}]];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ *)

%o (PARI) a(n) = 1 + sum(k=0, n, binomial(n, k\2)*(2*k-n)); \\ _Michel Marcus_, Dec 20 2020

%o (Python)

%o from math import comb

%o def A206604(n): return sum(comb(n,k>>1)*((k<<1)-n) for k in range(n+1))+1 # _Chai Wah Wu_, Oct 28 2024

%Y Cf. A189390, A189391, A206603.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 10 2012