|
%I #29 Dec 20 2020 03:36:38
%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
%Y Cf. A189390, A189391, A206603.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Feb 10 2012
| 