A128623 -id:A128623 - cp's OEIS frontend

A128624 Row sums of A128623.

1, 4, 12, 24, 45, 72, 112, 160, 225, 300, 396, 504, 637, 784, 960, 1152, 1377, 1620, 1900, 2200, 2541, 2904, 3312, 3744, 4225, 4732, 5292, 5880, 6525, 7200, 7936, 8704, 9537, 10404, 11340, 12312, 13357, 14440, 15600, 16800, 18081, 19404, 20812, 22264, 23805

Offset: 1

Gary W. Adamson, Mar 14 2007

Also the number of (w,x,y) with all terms in {0,...,n-1} and w <= R <= x, where R = max(w,x,y)-min(w,x,y), see A212959. - Clark Kimberling, Jun 10 2012

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A128621, A128623, A212959.

Cf. A094728 (diagonal row sums).

[n*((n+1)^2-1+(n mod 2))/4: n in [1..50]]; // G. C. Greubel, Mar 12 2024

Table[n*(n^2 +2*n +Mod[n,2])/4, {n,50}] (* G. C. Greubel, Mar 12 2024 *)

Vec(x*(1+2*x+3*x^2)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 31 2016

[n*((n+1)^2-1+(n%2))//4 for n in range(1,51)] # G. C. Greubel, Mar 12 2024

G.f.: x*(1+2*x+3*x^2) / ((1+x)^2*(1-x)^4). - R. J. Mathar, Jun 27 2012
From Colin Barker, Jan 31 2016: (Start)
a(n) = n*(2*n^2 + 4*n + 1 - (-1)^n)/8.
a(n) = n^2*(n + 2)/4 for n even.
a(n) = n*(n^2 + 2*n + 1)/4 for n odd. (End)
From G. C. Greubel, Mar 12 2024: (Start)
a(n) = Sum_{k=0..floor((n-1)/2)} A094728(n, k).
E.g.f.: (1/8)*x*(exp(-x) + (7 + 10*x + 2*x^2)*exp(x)). (End)

Incorrect formula removed by R. J. Mathar, Jun 27 2012

A115514 Triangle read by rows: row n >= 1 lists first n positive terms of A004526 (integers repeated) in decreasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 1, 4, 3, 3, 2, 2, 1, 1, 4, 4, 3, 3, 2, 2, 1, 1, 5, 4, 4, 3, 3, 2, 2, 1, 1, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Roger L. Bagula, Mar 07 2006

T(n,k) = number of 2-element subsets of {1,2,...,n+2} such that the absolute difference of the elements is k+1, where 1 <= k < = n. E.g., T(7,3) = 3, the subsets are {1,5}, {2,6}, and {3,7}. - Christian Barrientos, Jun 27 2022

			Triangle begins as, for n >= 1, 1 <= k <= n,
  1;
  1, 1;
  2, 1, 1;
  2, 2, 1, 1;
  3, 2, 2, 1, 1;
  3, 3, 2, 2, 1, 1;
  4, 3, 3, 2, 2, 1, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000012, A000073, A004526, A008805, A008967.

Cf. A002620 (row sums), A008805 (diagonal sums), A142150 (alternating row sums)

[Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 14 2024

# Assuming offset 0:
Even := n -> (1 + (-1)^n)/2: # Iverson's even.
p := n -> add(add(Even(k)*x^j, j = 0..n-k), k = 0..n):
for n from 0 to 9 do seq(coeff(p(n), x, k), k=0..n) od; # Peter Luschny, Jun 03 2021

Table[Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

flatten([[(n-k+2)//2 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 14 2024

Sum_{k=1..n} T(n, k) = A002620(n+1) (row sums). - Gary W. Adamson, Oct 25 2007
T(n, k) = [x^k] p(n), where p(n) are partial Gaussian polynomials (A008967) defined by p(n) = Sum_{k=0..n} Sum_{j=0..n-k} even(k)*x^j, and even(k) = 1 if k is even and otherwise 0. We assume offset 0. - Peter Luschny, Jun 03 2021
T(n, k) = floor((n+2-k)/2). - Christian Barrientos, Jun 27 2022
From G. C. Greubel, Mar 14 2024: (Start)
T(n, k) = A128623(n, k)/n.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A142150(n+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A008805(n-1).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A002265(n+3). (End)

Edited by N. J. A. Sloane, Mar 23 2008 and Dec 15 2017

cp's OEIS Frontend

A128624 Row sums of A128623.

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