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A364843 Integers are repeated in runs of 1, 2, 3, ... Each new integer (following a run) is given the value of its sequence index value.

%I A364843 #36 Sep 15 2023 16:30:23
%S A364843 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 A364843 22,22,29,29,29,29,29,29,29,29,37,37,37,37,37,37,37,37,37,46,46,46,46,
%U A364843 46,46,46,46,46,46,56,56,56,56,56,56,56,56,56,56,56
%N A364843 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 A364843 Omitting repeats yields the triangular numbers plus 1 sequence A000124.
%F A364843 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 A364843 Sum_{k=1..n} k = T(n,k) = A006528(n). - _Alois P. Heinz_, Sep 15 2023
%e A364843 Illustrated as a triangle begins:
%e A364843    1;
%e A364843    2,  2;
%e A364843    4,  4,  4;
%e A364843    7,  7,  7,  7;
%e A364843   11, 11, 11, 11, 11;
%e A364843   16, 16, 16, 16, 16, 16;
%e A364843   22, 22, 22, 22, 22, 22, 22;
%e A364843   ...
%p A364843 T:= (n, k)-> n*(n-1)/2+1:
%p A364843 seq(seq(T(n,k), k=1..n), n=1..11);  # _Alois P. Heinz_, Aug 31 2023
%o A364843 (PARI) a(n) = my(t=(sqrtint(8*n-1)-1)\2); t*(t+1)/2+1 \\ _Thomas Scheuerle_, Aug 10 2023
%o A364843 (Python)
%o A364843 from math import isqrt
%o A364843 def A364843(n): return ((t:=isqrt((n<<3)-1)-1>>1)*(t+1)>>1)+1 # _Chai Wah Wu_, Sep 15 2023
%Y A364843 Cf. A000124, A002024, A006528.
%Y A364843 Row sums give A006000(n-1).
%K A364843 easy,nonn,tabl
%O A364843 1,2
%A A364843 _Peter Woodward_, Aug 10 2023