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A189162 The maximum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 1 to n, and each number in a higher row is the sum of the two numbers directly below it. 5
1, 3, 9, 24, 61, 148, 350, 808, 1837, 4116, 9130, 20056, 43746, 94760, 204188, 437712, 934525, 1987252, 4212338, 8900344, 18756886, 39426168, 82693924, 173071024, 361567186, 753984648, 1569877860, 3263572848, 6775522852, 14047800016 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The maximum is attained by the triangle with base 1, 3, 5, ..., 2*ceiling(n/2)-1, 2*floor(n/2), ..., 6, 4, 2 (i.e., odd numbers increasing, followed by even numbers decreasing).

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = 2^(n-1) + A189390(n-1).

D-finite with recurrence (-n+1)*a(n) +4*(n-1)*a(n-1) -12*a(n-2) +16*(-n+4)*a(n-3) +16*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 17 2021

EXAMPLE

For n = 5 consider the triangle:

         61

       29  32

     12  17  15

    4   8   9   6

  1   3   5   4   2

This triangle has 61 at its apex and no other such triangle with the numbers 1 - 5 on its base has a larger apex value, so a(5) = 61.

MAPLE

a:=proc(n)return 2^(n-1) + add((4*k+1)*binomial(n-1, k), k=0..floor(n/2)-1) + `if`(n mod 2=1, (n-1)*binomial(n-1, (n-1)/2), 0):end:

seq(a(n), n=1..50);

CROSSREFS

Cf. A066411, A099325, A189390, A189391.

Sequence in context: A228820 A335470 A003262 * A079282 A117585 A317474

Adjacent sequences:  A189159 A189160 A189161 * A189163 A189164 A189165

KEYWORD

easy,nonn

AUTHOR

Nathaniel Johnston, Apr 20 2011

STATUS

approved

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Last modified September 27 04:10 EDT 2021. Contains 347673 sequences. (Running on oeis4.)