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A283525 Remainder when sum of first n terms of A004001 is divided by 3*n.

%I A283525 #12 Mar 14 2017 00:26:47
%S A283525 1,2,4,6,9,13,17,21,26,2,6,10,15,20,25,30,36,43,51,0,6,13,21,29,38,47,
%T A283525 56,66,76,86,3,10,18,27,37,48,60,72,85,99,114,3,16,30,44,59,74,89,105,
%U A283525 122,139,1,16,31,47,63,79,95,112,129,146,163,180,5,20,36,53,71,90,110,130,151,173,196,220,16,38,61,85,109
%N A283525 Remainder when sum of first n terms of A004001 is divided by 3*n.
%C A283525 Sequence represents b(n, 3) where b(n, i) = (Sum_{k=1..n} A004001(k)) mod (n*i). See also A282891, A283501 and corresponding illustrations in Links section.
%H A283525 Altug Alkan, <a href="/A283525/b283525.txt">Table of n, a(n) for n = 1..10000</a>
%H A283525 Altug Alkan, <a href="/A283525/a283525.png">Illustration of Residue Classes Modulo 12</a>
%H A283525 Altug Alkan, <a href="/A283525/a283525_1.png">Alternative Scatterplot of A283525</a>
%F A283525 a(n) = (Sum_{k=1..n} A004001(k)) mod (3*n).
%p A283525 A004001:= proc(n) option remember; procname(procname(n-1)) +procname(n-procname(n-1)) end proc:
%p A283525 A004001(1):= 1: A004001(2):= 1:
%p A283525 L:= ListTools[PartialSums](map(A004001, [$1..1000])):
%p A283525 seq(L[i] mod (3*i), i=1..1000); # after _Robert Israel_ at A282891
%t A283525 b[1] = 1; b[2] = 1; b[n_] := b[n] = b[b[n - 1]] + b[n - b[n - 1]]; a[n_] := Mod[Sum[b[k], {k, n}], 3 n]; Array[a, 80] (* _Robert G. Wilson v_, Mar 13 2017 *)
%o A283525 (PARI) a=vector(1000); a[1]=a[2]=1; for(n=3, #a, a[n]=a[a[n-1]]+a[n-a[n-1]]); vector(#a, n, sum(k=1, n, a[k]) % (3*n))
%Y A283525 Cf. A004001, A282891, A283501.
%K A283525 nonn
%O A283525 1,2
%A A283525 _Altug Alkan_, Mar 10 2017