A189391 -id:A189391 - cp's OEIS frontend

A206603 Maximal apex value of an addition triangle whose base is a permutation of {k-n/2, k=0..n}.

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

Offset: 0

Alois P. Heinz, Feb 10 2012

nonn

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.

			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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001787, A189390, A189391, A206604.

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

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 *)

a(n) = sum(k=0, n, binomial(n, k\2)*(k-n/2)); \\ Michel Marcus, Dec 20 2020

from math import comb
def A206603(n): return sum(comb(n,k>>1)*((k<<1)-n) for k in range(n+1))>>1 # Chai Wah Wu, Oct 28 2024

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.

A206604 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.

Original entry on oeis.org

1, 1, 3, 9, 27, 73, 189, 465, 1115, 2601, 5973, 13489, 30149, 66641, 146233, 318369, 689403, 1484137, 3181797, 6790641, 14445101, 30617841, 64724553, 136426849, 286926757, 601999633, 1260707529, 2634831585, 5497982025, 11452601761, 23823827825, 49484904257

Offset: 0

Alois P. Heinz, Feb 10 2012

nonn

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.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A189390, A189391, A206603.

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

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 *)

a(n) = 1 + sum(k=0, n, binomial(n, k\2)*(2*k-n)); \\ Michel Marcus, Dec 20 2020

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

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) = 1 + A189390(n)-A189391(n).
a(n) ~ n*2^n * (1-2*sqrt(2)/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 15 2014

A281862 Riordan transform of the triangular number sequence A000217 with the Chebyshev S matrix A049310.

Original entry on oeis.org

0, 1, 3, 4, 1, -6, -11, -6, 9, 21, 14, -12, -34, -25, 15, 50, 39, -18, -69, -56, 21, 91, 76, -24, -116, -99, 27, 144, 125, -30, -175, -154, 33, 209, 186, -36, -246, -221, 39, 286, 259, -42, -329, -300, 45, 375

Offset: 0

Wolfdieter Lang, Feb 18 2017

For the analogous sequence with the inverse S Riordan matrix A053121 see A189391.

Cf. A000217, A049310, A128504, A189391.

CoefficientList[Series[x (1 + x^2)/(1 - x + x^2)^3, {x, 0, 45}], x] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = Sum_{m=0..n} A049310(n,m)*A000217(m), n >= 0.
a(n) = b(n-1) + b(n-3), n >= 0 with b(-3) = b(-2) = b(-1) = 0 and  b(n) = A128504(n) for n >= 0.
G.f.: (1/(1+x^2))*Tri(x/(1+x^2)), with Tri(x) = x/(1-x)^3 (g.f. of A000217).
G.f. x*(1 + x^2)/(1 - x + x^2)^3.

cp's OEIS Frontend

A206603 Maximal apex value of an addition triangle whose base is a permutation of {k-n/2, k=0..n}.

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A206604 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A281862 Riordan transform of the triangular number sequence A000217 with the Chebyshev S matrix A049310.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula