A296844 - OEIS

%I #9 Jan 12 2018 23:43:39

%S 4,4,6,5,2,0,5,2,9,3,4,2,0,7,5,5,5,3,6,4,1,2,6,7,8,3,7,3,0,1,3,0,3,6,

%T 0,4,6,4,1,5,6,3,1,6,5,1,8,4,3,6,4,9,4,0,1,2,2,3,2,7,5,6,0,2,4,7,3,1,

%U 7,4,6,3,8,9,3,0,4,6,8,4,0,7,6,1,0,0

%N Decimal expansion of ratio-sum for A296843; see Comments.

%C Suppose that A = (a(n)), for n >= 0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A296843, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425..A296434 for related ratio-sums and A296452..A296461 for related limiting power-ratios.

%e ratio-sum = 4.465205293420755536412678373013036046415...

%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];

%t j = 1; While[j < 16, k = a[j] - j - 1;

%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

%t u = Table[a[n], {n, 0, k}];  (* A296843 *)

%t GoldenRatio; s = N[Sum[-g + a[n]/a[n - 1], {n, 1, 1000}], 200];

%t StringJoin[StringTake[ToString[s], 41], "..."]

%t Take[RealDigits[s, 10][[1]], 100] (* A296844 *)

%Y Cf. A001622, A296843, A296845.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Jan 12 2018