|
%I #36 Sep 15 2023 16:30:23
%S 1,2,2,4,4,4,7,7,7,7,11,11,11,11,11,16,16,16,16,16,16,22,22,22,22,22,
%T 22,22,29,29,29,29,29,29,29,29,37,37,37,37,37,37,37,37,37,46,46,46,46,
%U 46,46,46,46,46,46,56,56,56,56,56,56,56,56,56,56,56
%N Integers are repeated in runs of 1, 2, 3, ... Each new integer (following a run) is given the value of its sequence index value.
%C Omitting repeats yields the triangular numbers plus 1 sequence A000124.
%F G.f.: x*y*(1 + 2*x^4*y^2 - x*(1 + y) - 2*x^3*y*(1 + y) + x^2*(1 + y + y^2))/((1 - x)^3*(1 - x*y)^3). - _Stefano Spezia_, Sep 02 2023
%F Sum_{k=1..n} k = T(n,k) = A006528(n). - _Alois P. Heinz_, Sep 15 2023
%e Illustrated as a triangle begins:
%e 1;
%e 2, 2;
%e 4, 4, 4;
%e 7, 7, 7, 7;
%e 11, 11, 11, 11, 11;
%e 16, 16, 16, 16, 16, 16;
%e 22, 22, 22, 22, 22, 22, 22;
%e ...
%p T:= (n, k)-> n*(n-1)/2+1:
%p seq(seq(T(n,k), k=1..n), n=1..11); # _Alois P. Heinz_, Aug 31 2023
%o (PARI) a(n) = my(t=(sqrtint(8*n-1)-1)\2); t*(t+1)/2+1 \\ _Thomas Scheuerle_, Aug 10 2023
%o (Python)
%o from math import isqrt
%o def A364843(n): return ((t:=isqrt((n<<3)-1)-1>>1)*(t+1)>>1)+1 # _Chai Wah Wu_, Sep 15 2023
%Y Cf. A000124, A002024, A006528.
%Y Row sums give A006000(n-1).
%K easy,nonn,tabl
%O 1,2
%A _Peter Woodward_, Aug 10 2023
|
