A347533 Array A(n,k) where A(n,0) = n and A(n,k) = (k*n + 1)^2 - A(n,k-1), n > 0, read by ascending antidiagonals.

1, 2, 3, 3, 7, 6, 4, 13, 18, 10, 5, 21, 36, 31, 15, 6, 31, 60, 64, 50, 21, 7, 43, 90, 109, 105, 71, 28, 8, 57, 126, 166, 180, 151, 98, 36, 9, 73, 168, 235, 275, 261, 210, 127, 45, 10, 91, 216, 316, 390, 401, 364, 274, 162, 55, 11, 111, 270, 409, 525, 571, 560, 477, 351, 199, 66

Offset

1,2

Comments

A(n,k) is also the distance from A(n, k-1) to the earliest square greater than 3*A(n,k-1) - A(n,k-2).

In column k, every term is the arithmetic mean of its neighbors minus A000217(k).

Links

G. C. Greubel, Antidiagonals n = 1..50, flattened

Formula

A(n,k) = A000217(k)*n^2 + k*n + 1, for k odd.

A(n,k) = A000217(k)*n^2 + (k+1)*n = (k+1)*x*(k*n/2 + 1), for k even.

A(n,k) = (A(n,k-1) + A(n,k+1) + k*(k+1))/2, for any k.

A(n, 0) = A000027(n).

A(n, 1) = A002061(n+1).

A(n, 2) = A028896(n).

A(n, 3) = A085473(n).

From G. C. Greubel, Dec 25 2022: (Start)

A(n, k) = (1/2)*( (k*n+1)*(k*n+n+1) + (-1)^k*(n-1) ).

T(n, k) = (1/2)*( (k*(n-k)+1)*((k+1)*(n-k)+1) + (-1)^k*(n-k-1) ).

Sum_{k=0..n-1} T(n, k) = (1/120)*(2*n^5 + 5*n^4 + 20*n^3 + 25*n^2 + 98*n - 15*(1-(-1)^n)). (End)

Example

Array, A(n, k), begins:
  1  3   6  10  15   21   28   36   45 ... A000217;
  2  7  18  31  50   71   98  127  162 ... A195605;
  3 13  36  64 105  151  210  274  351 ...
  4 21  60 109 180  261  364  477  612 ...
  5 31  90 166 275  401  560  736  945 ...
  6 43 126 235 390  571  798 1051 1350 ...
  7 57 168 316 525  771 1078 1422 1827 ...
  8 73 216 409 680 1001 1400 1849 2376 ...
  9 91 270 514 855 1261 1764 2332 2997 ...
Antidiagonals, T(n, k), begin as:
   1;
   2,  3;
   3,  7,   6;
   4, 13,  18,  10;
   5, 21,  36,  31,  15;
   6, 31,  60,  64,  50,  21;
   7, 43,  90, 109, 105,  71,  28;
   8, 57, 126, 166, 180, 151,  98,  36;
   9, 73, 168, 235, 275, 261, 210, 127,  45;
  10, 91, 216, 316, 390, 401, 364, 274, 162,  55;

Mathematica

A[n_, 0]:= n; A[n_, k_]:= (k*n+1)^2 -A[n, k-1]; Table[Function[n, A[n, k]][m-k+1], {m, 0, 10}, {k, 0, m}]//Flatten (* Michael De Vlieger, Oct 27 2021 *)

Prog

(Magma)
A347533:= func< n, k | (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1)) >;
[A347533(n, k): k in [0..n-1], n in [1..13]]; // G. C. Greubel, Dec 25 2022
(SageMath)
def A347533(n, k): return (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1))
flatten([[A347533(n, k) for k in range(n)] for n in range(1, 14)]) # G. C. Greubel, Dec 25 2022

Crossrefs

Family of sequences (k*n + 1)^2: A016754 (k=2), A016778 (k=3), A016814 (k=4), A016862 (k=5), A016922 (k=6), A016994 (k=7), A017078 (k=8), A017174 (k=9), A017282 (k=10), A017402 (k=11), A017534 (k=12), A134934 (k=14).

Cf. A000027, A000217, A002061, A028896, A085473, A195605.

Sequence in context: A063670 A253357 A185909 * A193713 A171824 A143444

Adjacent sequences: A347530 A347531 A347532 * A347534 A347535 A347536

Keyword

nonn,tabl,easy

Author

Lamine Ngom, Sep 05 2021

Status

approved