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a(n) = Sum_{k=1..n} q(k+1-q(k)), where q(k) = A005185(k); partial sums of A283467.
3

%I #13 Mar 23 2017 11:43:26

%S 1,2,3,4,6,8,10,13,16,19,23,26,30,35,39,44,49,54,60,66,72,78,86,92,

%T 100,108,116,124,132,142,150,159,169,179,189,200,211,221,232,243,254,

%U 266,278,290,302,314,330,340,354,368,380,394,410,424,438,454,468,484,500,516,532,552,568,585,606,622,639,658,678,698,719,740

%N a(n) = Sum_{k=1..n} q(k+1-q(k)), where q(k) = A005185(k); partial sums of A283467.

%H Antti Karttunen, <a href="/A280706/b280706.txt">Table of n, a(n) for n = 1..10001</a>

%H <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>

%F a(1) = 1, for > 1, a(n) = A283467(n) + a(n-1).

%F A284173(n) = a(n) modulo n.

%t a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - a[n - 2]]; Accumulate@ Table[a[n + 1 - a[n]], {n, 72}] (* _Michael De Vlieger_, Mar 22 2017 *)

%o (Scheme)

%o ;; Code for A005185 given under that entry.

%o ;; With memoization-macro definec:

%o (definec (A280706 n) (if (= 1 n) 1 (+ (A280706 (- n 1)) (A283467 n))))

%o ;; As an explicit sum (slower):

%o (define (A280706 n) (add (lambda (k) (A005185 (- (+ k 1) (A005185 k)))) 1 n))

%o ;; Implements sum_{i=lowlim..uplim} intfun(i)

%o (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

%o (PARI) a(n) = if(n<3, 1, a(n - a(n - 1)) + a(n - a(n - 2)));

%o for(n=1, 72, print1(sum(k=1, n, a(k + 1 - a(k))),", ")) \\ _Indranil Ghosh_, Mar 22 2017

%Y Cf. A005185, A284173.

%Y Partial sums of A283467.

%K nonn

%O 1,2

%A _Antti Karttunen_ after _Altug Alkan_'s A284173, Mar 22 2017