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) = A320486(a(n-2)+a(n-1)).
%C A321021 This must eventually enter a cycle, since there are only finitely many pairs of numbers that both have distinct digits. In fact, at step 171, enters a cycle of length 100 (see A321022).
%C A321021 Another entry into this cycle would be to start with 2, 1 and use the same rule, in which case the sequence would begin (2, 1, 3, 4, 7, 0, 7, 7, 14, 21, 35, 56, 91, 147, 238, 385, 623, ..., 40, 80, 120), a cycle of length 100 that repeats (cf. A321022).
%H A321021 N. J. A. Sloane, <a href="/A321021/b321021.txt">Table of n, a(n) for n = 0..1000</a>
%p A321021 f:= proc(n) local F, S;
%p A321021   F:= convert(n, base, 10);
%p A321021   S:= select(t -> numboccur(t, F)>1, [$0..9]);
%p A321021   if S = {} then return n fi;
%p A321021   F:= subs(seq(s=NULL, s=S), F);
%p A321021   add(F[i]*10^(i-1), i=1..nops(F))
%p A321021 end proc: # A320486
%p A321021 x:=0: y:=1: lprint(x); lprint(y);
%p A321021 for n from 2 to 500 do
%p A321021 z:=f(x+y); lprint(z); x:=y; y:=z; od:
%Y A321021 Cf. A000045 (Fibonacci), A320486 (Angelini's contraction), A321022.
%K A321021 nonn,base
%O A321021 0,4
%A A321021 _N. J. A. Sloane_, Nov 19 2018

%I A023439
%S A023439 0,1,1,2,3,5,8,13,21,34,54,87,139,223,357,572,916,1467,2349,3762,6024,
%T A023439 9647,15448,24738,39614,63436,101583,162670,260491,417137,667981,
%U A023439 1069670,1712913,2742969,4392446,7033832
%N A023439 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-9).
%H A023439 Floris P. van Doorn and Jasper Mulder, <a href="/A023439/b023439.txt">Table of n, a(n) for n = 0..2000</a>
%H A023439 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A023439 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A023439 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,0,-1).
%F A023439 G.f.: x / ( (x-1)*(1+x)*(x^7+x^5+x^3+x-1) ). - _R. J. Mathar_, Nov 29 2011
%Y A023439 See A000045 for the Fibonacci numbers.
%K A023439 nonn
%O A023439 0,4
%A A023439 _N. J. A. Sloane_

%I A023440
%S A023440 0,1,1,2,3,5,8,13,21,34,55,88,142,228,367,590,949,1526,2454,3946,6345,
%T A023440 10203,16406,26381,42420,68211,109682,176367,283595,456016,733266,
%U A023440 1179079,1895939,3048637,4902156,7882582
%N A023440 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-10).
%H A023440 Matthew House, <a href="/A023440/b023440.txt">Table of n, a(n) for n = 0..4825</a>
%H A023440 John H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A023440 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A023440 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,0,0,-1).
%F A023440 G.f.: x/((x - 1)*(x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 - 1)). - _R. J. Mathar_, Nov 29 2011
%Y A023440 See A000045 for the Fibonacci numbers.
%K A023440 nonn,easy
%O A023440 0,4
%A A023440 _N. J. A. Sloane_

%I A041247
%S A041247 1,1,2,3,5,8,13,21,475,496,971,1467,2438,3905,6343,10248,231799,
%T A041247 242047,473846,715893,1189739,1905632,3095371,5001003,113117437,
%U A041247 118118440,231235877,349354317,580590194,929944511,1510534705,2440479216,55201077457,57641556673
%N A041247 Denominators of continued fraction convergents to sqrt(135).
%H A041247 Vincenzo Librandi, <a href="/A041247/b041247.txt">Table of n, a(n) for n = 0..200</a>
%H A041247 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,488,0,0,0,0,0,0,0,-1).
%F A041247 G.f.: -(x^2-x-1)*(x^4+3*x^2+1)*(x^8+7*x^4+1) / (x^16-488*x^8+1). - _Colin Barker_, Nov 14 2013
%F A041247 a(n) = 488*a(n-8) - a(n-16). - _Vincenzo Librandi_, Dec 13 2013
%t A041247 Denominator[Convergents[Sqrt[135], 30]]  (* _Vincenzo Librandi_, Dec 13 2013 *)
%o A041247 (MAGMA) I:=[1,1,2,3,5,8,13,21,475,496,971,1467,2438,3905,6343,10248]; [n le 16 select I[n] else 488*Self(n-8)-Self(n-16): n in [1..40]]; // _Vincenzo Librandi_, Dec 13 2013
%Y A041247 Cf. A041246, A010194.
%K A041247 nonn,frac,easy
%O A041247 0,3
%A A041247 _N. J. A. Sloane_.
%E A041247 More terms from _Colin Barker_, Nov 14 2013

%I A042581
%S A042581 1,1,2,3,5,8,13,21,34,55,3114,3169,6283,9452,15735,25187,40922,66109,
%T A042581 107031,173140,9802871,9976011,19778882,29754893,49533775,79288668,
%U A042581 128822443,208111111,336933554,545044665,30859434794,31404479459,62263914253,93668393712
%N A042581 Denominators of continued fraction convergents to sqrt(819).
%H A042581 Vincenzo Librandi, <a href="/A042581/b042581.txt">Table of n, a(n) for n = 0..200</a>
%H A042581 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 3148, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).
%F A042581 G.f.: -(x^2-x-1)*(x^4-3*x^3+4*x^2-2*x+1)*(x^4-2*x^3+4*x^2-3*x+1)*(x^4+2*x^3+4*x^2+3*x+1)*(x^4+3*x^3+4*x^2+2*x+1) / (x^20-3148*x^10+1). - _Colin Barker_, Dec 18 2013
%t A042581 Denominator[Convergents[Sqrt[819], 30]] (* _Vincenzo Librandi_, Jan 25 2014 *)
%Y A042581 Cf. A042580, A040790.
%K A042581 nonn,frac,easy
%O A042581 0,3
%A A042581 _N. J. A. Sloane_.
%E A042581 More terms from _Colin Barker_, Dec 18 2013

%I A050762
%S A050762 1,1,2,3,5,8,13,21,34,89,610,987,1597,2584,4181,6765,10946,28657,
%T A050762 46368,75025,121393,196418,832040,1346269,2178309,3524578,14930352,
%U A050762 24157817,63245986,267914296,2971215073,4807526976,12586269025,86267571272
%N A050762 Fibonacci numbers (index numbers see A050761) containing no pair of consecutive equal digits (probably finite).
%C A050762 There are only 46 such numbers in the first 100,000 Fibonacci numbers. - _Harvey P. Dale_, Apr 01 2013
%H A050762 Harvey P. Dale, <a href="/A050762/b050762.txt">Table of n, a(n) for n = 1..46</a>
%t A050762 Select[Fibonacci[Range[100]],!MemberQ[Differences[IntegerDigits[#]],0]&] (* _Harvey P. Dale_, Apr 01 2013 *)
%Y A050762 Cf. A043096, A000045, A050761.
%Y A050762 See A000045 for the Fibonacci numbers.
%K A050762 nonn,base
%O A050762 1,3
%A A050762 _Patrick De Geest_, Sep 15 1999.

%I A065124
%S A065124 0,1,1,2,3,5,8,13,21,25,28,35,45,53,62,70,78,85,100,113,114,119,125,
%T A065124 136,144,154,163,173,183,194,206,220,228,232,244,251,261,269,278,295,
%U A065124 312,328,334,347,357,371,386,397,414,433,442,452,462,473,485,499,516
%N A065124 a(n) = (sum of digits of a(n-2)) + a(n-1); a(0) = 0 and a(1) = 1.
%H A065124 Harry J. Smith, <a href="/A065124/b065124.txt">Table of n, a(n) for n = 0..1000</a>
%p A065124 a[0]:=0:a[1]:=1:a[2]:=1:for n from 3 to 100 do d:=a[n-1]:s:=0:while d>0 do c:=d mod 10:s:=s+c:d:=(d-c)/10 od:e:=a[n-2]:t:=0:while e>0 do c:=e mod 10:t:=t+c:e:=(e-c)/10 od:a[n]:=a[n-1]+t od: seq(a[n], n=0..56); # _Zerinvary Lajos_, Mar 19 2009
%t A065124 a[0] = 0; a[1] = 1; a[n_] := a[n] = Apply[ Plus, IntegerDigits[ a[n - 2] ]] + a[n - 1]; Table[ a[n], {n, 0, 100} ]
%o A065124 (PARI) SumD(x)= { local(s=0); while (x>9, s+=x-10*(x\10); x\=10); return(s + x) } { for (n=0, 1000, if (n>1, a=SumD(a2) + a1; a2=a1; a1=a, if (n, a=a1=1, a=a2=0)); write("b065124.txt", n, " ", a) ) } \\ _Harry J. Smith_, Oct 10 2009
%Y A065124 Cf. A007612, A007953, A065076.
%K A065124 base,easy,nonn
%O A065124 0,4
%A A065124 _Robert G. Wilson v_, Nov 13 2001

%I A077373
%S A077373 0,1,1,2,3,5,8,13,21,34,55,89
%N A077373 Fibonacci numbers whose external digits as well as internal digits form a Fibonacci number.
%C A077373 Conjecture: sequence is finite. Is there any term > 89 in this sequence?
%H A077373 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/NonRecursions.html">Non Recursions</a>
%Y A077373 Cf. A077371, A077372.
%K A077373 base,more,nonn
%O A077373 1,4
%A A077373 _Amarnath Murthy_, Nov 06 2002

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Search: seq:1,1,2,3,5,8,13,21
Showing 41-50 of 92

%I A093091
%S A093091 1,1,2,3,5,8,13,21,34,55,89,14,4,10,3,18,14,13,21,32,27,34,53,59,61,
%T A093091 87,11,2,12,0,14,8,98,13,14,12,14,22,10,6,11,1,27,26,26,36,32,16,17,
%U A093091 12,28,53,52,62,68,48,33,29,40,81,10,5,11,4,13,0,11,6,81,62,69,12,1,91,15,16
%N A093091 "Fibonacci in pairs from left": start with a(1)=1, a(2)=1; repeatedly adjoin sum of previous two terms but chopped from the left into pairs of 2 digits.
%C A093091 Do all pairs of digits appear infinitely often? The sequence is not periodic.
%e A093091 ... a(11)=a(9)+a(10), a(12)=left pair of (a(10)+a(11)=55+89=14 4),
%e A093091 a(13)=right pair of (a(10)+a(11)=55+89=14 4),
%e A093091 a(14)=left pair of (a(11)+a(12)=89+14=10 3),
%e A093091 a(15)=right pair of (a(11)+a(12)=89+14=10 3), a(16)=a(12)+a(13) ...
%Y A093091 Cf. A093086, A093087, A093088, A093089, A093090.
%K A093091 nonn,base
%O A093091 1,3
%A A093091 _Bodo Zinser_, Mar 20 2004

%I A093332
%S A093332 0,1,1,2,3,5,8,13,21,34,56,92,152,251,414,684,1130,1868,3087,5102,
%T A093332 8433,13938,23038,38079,62940,104033,171955,284223,469789,776510,
%U A093332 1283487,2121464,3506550,5795947,9580072,15834821,26173243,43261534,71506628
%N A093332 a(0) = 0, a(1) = 1 and for n >=0, a(n+2) = int(sqrt(2 * (a(n)^2 + a(n+1)^2)))
%C A093332 First 10 terms are the same as the first 10 terms of the Fibonacci sequence (A000045).
%e A093332 a(5) = 5 because a(5) = int(sqrt(2* a(3)^2 + a(4)^2)) = int(sqrt(2*(2^2+3^2))) = int(sqrt(26) = 5.
%Y A093332 Cf. A000045, A093333, A093335.
%K A093332 easy,nonn
%O A093332 0,4
%A A093332 Robert A. Stump (rstump_2004(AT)yahoo.com), Apr 25 2004

%I A094102
%S A094102 1,1,1,1,1,1,2,1,1,1,1,2,3,2,1,1,1,1,2,3,5,3,2,1,1,1,1,2,3,5,8,5,3,2,
%T A094102 1,1,1,1,2,3,5,8,13,8,5,3,2,1,1,1,1,2,3,5,8,13,21,13,8,5,3,2,1,1,1,1,
%U A094102 2,3,5,8,13,21,34,21,13,8,5,3,2,1,1,1,1,2,3,5,8,13,21,34,55,34,21,13,8,5,3,2,1,1
%N A094102 Triangle read by rows in which row n contains Fib(1), ..., Fib(n-1), Fib(n), Fib(n-1), ..., Fib(1).
%e A094102 Triangle begins:
%e A094102 1
%e A094102 1 1 1
%e A094102 1 1 2 1 1
%e A094102 1 1 2 3 2 1 1
%e A094102 1 1 2 3 5 3 2 1 1
%e A094102 1 1 2 3 5 8 5 3 2 1 1
%Y A094102 Row sums are in A001911. Cf. A094103.
%K A094102 nonn,tabf
%O A094102 1,7
%A A094102 Alysia Veenhof (ladyluck1899(AT)hotmail.com), May 05 2004

%I A107358
%S A107358 0,1,1,2,3,5,8,13,21,34,55,89,144,233,376,608,982,1587,2564,4143,6694,
%T A107358 10816,17476,28237,45624,73717,119108,192449,310949,502416,811778,
%U A107358 1311630,2119265,3424201,5532650,8939375,14443788,23337539,37707610,60926041,98441202,159056294
%N A107358 Dying rabbits: a(n) = Fibonacci(n) for n <= 12; for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).
%C A107358 In the limit, the growth rate is 1.61575... per generation as opposed to 1.61803... for Fibonacci numbers. - _T. D. Noe_, Jan 22 2009
%C A107358 If the rabbits die after 12 months, then those who were there in month 1 should die in month 13, whence a(13) = 144 + 89 - 1 = 232 and not 233. In month 14, no rabbits die because the only pair which was there in month 2 already dies. Then in month 15, the one pair born in month 3 will die. In general, the number of rabbits which die in month n (because they are aged 12 months) is equal to the number of newborn rabbits in month n - 12, which is the number of rabbits present in month n - 14. (Recall that a(n - 12) = a(n - 13) + a(n - 14) - #(dying rabbits) = #(rabbits from previous month) + #(newborn rabbits) - #(dying rabbits).) So the recurrence should read a(n) = a(n - 1) + a(n - 2) - a(n - 14). - _M. F. Hasler_, Oct 06 2017
%H A107358 T. D. Noe, <a href="/A107358/b107358.txt">Table of n, a(n) for n = 0..500</a>
%H A107358 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A107358 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A107358 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,0,0,0,0,0,-1).
%F A107358 G.f.: x/((x-1)*(1+x)*(x^11+x^9+x^7+x^5+x^3+x-1)). - _R. J. Mathar_, Jul 27 2009
%p A107358 with(combinat); f:=proc(n) option remember; if n <= 12 then RETURN(fibonacci(n)); fi; f(n-1)+f(n-2)-f(n-13); end;
%t A107358 LinearRecurrence[{1,1,0,0,0,0,0,0,0,0,0,0,-1},Fibonacci[Range[0,12]],50] (* _Harvey P. Dale_, Feb 28 2013 *)
%o A107358 (PARI) Vec(x/(x^13-x^2-x+1)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 10 2011
%Y A107358 See A000045 for the Fibonacci numbers. This is a better version of A000044.
%K A107358 nonn,easy
%O A107358 0,4
%A A107358 _N. J. A. Sloane_, May 25 2005

%I A121104
%S A121104 1,1,2,3,5,8,13,21,5,24,15,21,18,1,27,43,11,38,63,49,44,32,88,2,83,59,
%T A121104 73,76,79,63,113,9,94,61,6,123,76,149,127,34,74,124,32,83,1,3,91,212,
%U A121104 204,232,85,76,171,141,148,234,145,38,2,271,208,179,194,73,14,127,177
%N A1