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) ~ 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
(Python)
from math import comb
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 10 2012
STATUS
approved