A253902 - OEIS

%I #23 Nov 05 2024 09:09:59

%S 1,4,3,2,1,9,8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,

%T 25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,36,

%U 35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1

%N Write numbers 1, then 2^2 down to 1, then 3^2 down to 1, then 4^2 down to 1 and so on.

%C Triangle read by rows in which row n lists the first n^2 positive integers in decreasing order, n >= 1. - _Omar E. Pol_, Jan 20 2015

%F For 1 <= n <= 650, a(n) = -n + (t + 2)*(2*t^2 - t + 3)/6, where t = floor((3*n)^(1/3)+1/2).

%F a(n) = k*(2k^2-9k+13)/6-n where k = floor((3n)^(1/3))+3 if n>A000330(floor((3n)^(1/3)) and k = floor((3n)^(1/3))+2 otherwise. - _Chai Wah Wu_, Nov 05 2024

%e From _Omar E. Pol_, Jan 20 2015: (Start)

%e Written as an irregular triangle in which row lengths are successive squares, the sequence begins:

%e    1;

%e    4,  3,  2,  1;

%e    9,  8,  7,  6,  5,  4,  3, 2, 1;

%e   16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;

%e   ...

%e (End)

%p T:= n-> (t-> seq(t-i, i=0..t-1))(n^2):

%p seq(T(n), n=1..6);  # _Alois P. Heinz_, Nov 05 2024

%t a253902[n_] := Flatten@ Table[Reverse[Range[i^2]], {i, n}]; a253902[6] (* _Michael De Vlieger_, Jan 19 2015 *)

%o (PARI) lista(nn=10) = {for (n=1, nn, forstep(k=n^2, 1, -1, print1(k, ", ");););} \\ _Michel Marcus_, Jan 20 2015

%o (Python)

%o from sympy import integer_nthroot

%o def A253902(n): return (k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1))+2)*(k*((k<<1)-9)+13)//6-n # _Chai Wah Wu_, Nov 05 2024

%Y Cf. A000290, A064866, A074279.

%K nonn,easy,tabf,changed

%O 1,2

%A _Mikael Aaltonen_, Jan 18 2015