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

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

1,4

Comments

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

Links

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

Formula

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)

Example

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;
  ...

Maple

# 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

Mathematica

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

Prog

(Magma) [Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 14 2024
(SageMath) flatten([[(n-k+2)//2 for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 14 2024

Crossrefs

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

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

Sequence in context: A102523 A323023 A083415 * A326038 A122632 A176809

Adjacent sequences: A115511 A115512 A115513 * A115515 A115516 A115517

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 07 2006

Extensions

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

Status

approved