A347533 - OEIS

%I #19 Dec 25 2022 08:31:06

%S 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,

%T 105,71,28,8,57,126,166,180,151,98,36,9,73,168,235,275,261,210,127,45,

%U 10,91,216,316,390,401,364,274,162,55,11,111,270,409,525,571,560,477,351,199,66

%N 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.

%C 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).

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

%H G. C. Greubel, <a href="/A347533/b347533.txt">Antidiagonals n = 1..50, flattened</a>

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

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

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

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

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

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

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

%F From _G. C. Greubel_, Dec 25 2022: (Start)

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

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

%F 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)

%e Array, A(n, k), begins:

%e   1  3   6  10  15   21   28   36   45 ... A000217;

%e   2  7  18  31  50   71   98  127  162 ... A195605;

%e   3 13  36  64 105  151  210  274  351 ...

%e   4 21  60 109 180  261  364  477  612 ...

%e   5 31  90 166 275  401  560  736  945 ...

%e   6 43 126 235 390  571  798 1051 1350 ...

%e   7 57 168 316 525  771 1078 1422 1827 ...

%e   8 73 216 409 680 1001 1400 1849 2376 ...

%e   9 91 270 514 855 1261 1764 2332 2997 ...

%e Antidiagonals, T(n, k), begin as:

%e    1;

%e    2,  3;

%e    3,  7,   6;

%e    4, 13,  18,  10;

%e    5, 21,  36,  31,  15;

%e    6, 31,  60,  64,  50,  21;

%e    7, 43,  90, 109, 105,  71,  28;

%e    8, 57, 126, 166, 180, 151,  98,  36;

%e    9, 73, 168, 235, 275, 261, 210, 127,  45;

%e   10, 91, 216, 316, 390, 401, 364, 274, 162,  55;

%t 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 *)

%o (Magma)

%o A347533:= func< n,k | (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1)) >;

%o [A347533(n,k): k in [0..n-1], n in [1..13]]; // _G. C. Greubel_, Dec 25 2022

%o (SageMath)

%o def A347533(n,k): return (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1))

%o flatten([[A347533(n,k) for k in range(n)] for n in range(1,14)]) # _G. C. Greubel_, Dec 25 2022

%Y 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).

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

%K nonn,tabl,easy

%O 1,2

%A _Lamine Ngom_, Sep 05 2021