A038868 - OEIS

A038868

If a Fibonacci sequence is formed with first term = number of digits in n and second term = sum of decimal digits in n, then n itself occurs as a term in the sequence after the first two terms.

%I #15 Feb 18 2019 16:05:05

%S 16,21,25,50,66,102,115,154,193,291,471,573,675,777,879,2372,3668,

%T 4770,6867,22502,22790,32084,41666,46457,167151,331341,490740,1750051,

%U 2125176,2275226,2425276,2575326,2725376,2875426,22597419,73941113,167637057,188784525

%N If a Fibonacci sequence is formed with first term = number of digits in n and second term = sum of decimal digits in n, then n itself occurs as a term in the sequence after the first two terms.

%H Giovanni Resta, <a href="/A038868/b038868.txt">Table of n, a(n) for n = 1..1000</a>

%H Felice Russo, <a href="http://dx.doi.org/10.5281/zenodo.8795">A Set of New Smarandache Functions, Sequences and Conjectures in Number Theory</a>, Lupton, AZ: American Research Press, 2000.

%e 16 is a member because 2,7,9,16,25,... does contain 16.

%t aQ[n_] := Module[{d = IntegerDigits[n], a}, a[1] = Length[d]; a[2] = Total[d]; a[k_] := a[k] = a[k - 1] + a[k - 2]; k = 1; While[a[k] <= n, k++]; k--; k > 2 && a[k] == n]; Select[Range[1000], aQ] (* _Amiram Eldar_, Feb 17 2019 *)

%t gen[d_, s_, mx_] := Block[{a=d, b=s, v = {}, t}, While[True, t=a+b; If[t <= mx, If[d == IntegerLength@ t && s == Total@ IntegerDigits@ t, AppendTo[v, t]], Break[]]; a=b; b=t]; v]; upto[mxd_] := Sort@ Flatten@ Table[gen[d, s, 10^d], {d, mxd}, {s, 9*d}]; upto[20] (* terms < 10^20, _Giovanni Resta_, Feb 18 2019 *)

%Y Cf. A007629, A042983.

%K easy,nonn,base

%O 1,1

%A _Felice Russo_

%E Corrected and extended by Larry Reeves (larryr(AT)acm.org), Sep 28 2000

%E a(37)-a(38) from _Amiram Eldar_, Feb 17 2019