The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A206603 Maximal apex value of an addition triangle whose base is a permutation of {k-n/2, k=0..n}. 2
 0, 0, 1, 4, 13, 36, 94, 232, 557, 1300, 2986, 6744, 15074, 33320, 73116, 159184, 344701, 742068, 1590898, 3395320, 7222550, 15308920, 32362276, 68213424, 143463378, 300999816, 630353764, 1317415792, 2748991012, 5726300880, 11911913912, 24742452128, 51331847709 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The base row of the addition triangle contains a permutation of the n+1 integers or half-integers {k-n/2, k=0..n}. Each number in a higher row is the sum of the two numbers directly below it. Rows above the base row contain only integers. The base row consists of integers iff n is even. Because of symmetry, a(n) is also the absolute value of the minimal apex value of an addition triangle whose base is a permutation of {k-n/2, k=0..n}. a(n) is odd iff n = 2^m and m > 0. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{k=0..n} C(n,floor(k/2)) * (k-n/2). G.f.: (1-sqrt(1-4*x^2)) / (2*(2*x-1)^2). a(n) = A189390(n)-A001787(n) = A001787(n)-A189391(n) = (A189390(n)-A189391(n))/2 = (A206604(n)-1)/2. EXAMPLE a(3) = 4: max: 4 min: -4 1 3 -1 -3 -1 2 1 1 -2 -1 -3/2 1/2 3/2 -1/2 3/2 -1/2 -3/2 1/2 a(4) = 13: 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 MAPLE a:= n-> add (binomial(n, floor(k/2))*(k-n/2), k=0..n): seq (a(n), n=0..40); # second Maple program: a:= proc(n) option remember; `if`(n<3, n*(n-1)/2, ((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}, {8(n+1)(n+2)y[n] - 4(n-1)(n+2)y[n+1] - (2n^2 + 12n + 12)y[n+2] + (n+1)(n+3)y[n+3] == 0, y[0] == 0, y[1] == 0, y[2] == 1, y[3] == 4}]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *) PROG (PARI) a(n) = sum(k=0, n, binomial(n, k\2)*(k-n/2)); \\ Michel Marcus, Dec 20 2020 CROSSREFS Cf. A001787, A189390, A189391, A206604. Sequence in context: A036643 A000299 A102301 * A271176 A031506 A251701 Adjacent sequences: A206600 A206601 A206602 * A206604 A206605 A206606 KEYWORD nonn AUTHOR Alois P. Heinz, Feb 10 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 30 05:38 EST 2022. Contains 358431 sequences. (Running on oeis4.)