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A283969 a(n) = n + 1 + Sum_{k=0..n} floor((n-k)/r), where r = (3+sqrt(5))/2.

%I A283969 #16 Dec 12 2023 14:19:27
%S A283969 1,4,10,18,29,43,59,78,99,123,150,179,211,246,283,323,365,410,458,508,
%T A283969 561,616,674,735,798,864,933,1004,1078,1154,1233,1315,1399,1486,1576,
%U A283969 1668,1763,1860,1960,2063,2168,2276,2386,2499,2615,2733,2854,2978,3104
%N A283969 a(n) = n + 1 + Sum_{k=0..n} floor((n-k)/r), where r = (3+sqrt(5))/2.
%C A283969 This is column 1 of the transposable interspersion A283938.
%H A283969 Clark Kimberling, <a href="/A283969/b283969.txt">Table of n, a(n) for n = 0..1000</a>
%F A283969 a(n) = n + 1 + Sum_{k=0..n} floor((n-k)/r), where r = (3+sqrt(5))/2.
%t A283969 r = GoldenRatio^2; z = 120;
%t A283969 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
%t A283969 Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A283968 *)
%t A283969 Table[s[n], {n, 0, z}] (* A283969 *)
%o A283969 (PARI) a(n) = if(n<1, 1, a(n - 1) + 1 + floor(n*(3 + sqrt(5))/2));
%o A283969 for(n = 0, 50, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 19 2017
%o A283969 (Python)
%o A283969 import math
%o A283969 from sympy import sqrt
%o A283969 def a(n):
%o A283969     return 1 if n<1 else a(n - 1) + 1 + int(math.floor(n*(3 + sqrt(5))/2))
%o A283969 print([a(n) for n in range(51)]) # _Indranil Ghosh_, Mar 19 2017
%Y A283969 Cf. A104457, A283968, A283938, A283961.
%Y A283969 Partial sums of A026352.
%K A283969 nonn,easy
%O A283969 0,2
%A A283969 _Clark Kimberling_, Mar 18 2017