A042947 Gilda's numbers: numbers k such that if a Fibonacci sequence is formed with first term = a certain absolute value between decimal digits in k (A007953) and second term = sum of decimal digits in k (A040997), then k itself occurs as a term in the sequence.
0, 29, 49, 78, 110, 152, 220, 314, 330, 364, 440, 550, 628, 660, 683, 770, 880, 990, 997, 2207, 5346, 13064, 30254, 35422, 37862, 38006, 65676, 73805, 143662, 202196, 933138, 977909, 3120796, 3242189, 3363582, 3606368, 3727761, 3849154, 3970547, 4484776, 4848955
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..632 (terms below 10^100)
- Felice Russo, A Set of New Smarandache Functions, Sequences and Conjectures in Numer Theory., Lupton, AZ: American Research Press, 2000.
Programs
-
Mathematica
check[abs_, sum_, max_] := Module[{s = {}, a = abs, b = sum, c}, c = b; While[c <= max, id = IntegerDigits[c]; If[c > 10 && abs == Abs[id[[1]] - Total[Rest@id]] && sum == Total[id ], AppendTo[s, c]]; c = a + b; a = b; b = c]; s]; seq[digmax_] := Module[{s = {}}, Do[s = Join[s, check[a, b, 10^digmax]], {a, 0, 10*digmax}, {b, 1, 10*digmax}]; Join[{0}, Sort[s]]]; seq[7] (* Amiram Eldar, Jul 07 2021 *) -
PARI
for(n=0,10000000,s=eval(Vec(Str(n)));f1=sum(i=1,#s,s[i]);f0=abs(2*s[1]-f1);f=f0+f1;while(f<=n,if(f==n,print1(n",");break);f0=f1;f1=f;f=f0+f1)) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008
Formula
Let [x1.x2.x3. ... .xi] be the decimal expansion of n. Then define F(0) = |x1-x2-...-xi|, F(1) = x1 + x2 + x3 + ... + xi, and for k>1, F(k) = F(k-1) + F(k-2). If F(k)=n for some k, then n belongs to the sequence.
Extensions
Corrected and extended by Larry Reeves (larryr(AT)acm.org), Sep 28 2000
2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008
Offset corrected by Amiram Eldar, Jul 07 2021
Comments