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 A122194 Numbers that are the sum of exactly two sets of Fibonacci numbers. 2
 3, 5, 6, 9, 10, 15, 17, 25, 28, 41, 46, 67, 75, 109, 122, 177, 198, 287, 321, 465, 520, 753, 842, 1219, 1363, 1973, 2206, 3193, 3570, 5167, 5777, 8361, 9348, 13529, 15126, 21891, 24475, 35421, 39602, 57313, 64078, 92735, 103681, 150049, 167760, 242785 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary. M. Bicknell-Johnson & D. C. Fielder, The number of Representations of N Using Distinct Fibonacci Numbers, Counted by Recursive Formulas, Fibonacci Quart. 37.1 (1999) pp. 47 ff. Ron Knott Sumthing about Fibonacci Numbers Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1). FORMULA a(2n-1) = A000032(n+2) - 1, a(2n) = 2*A000045(n+3) - 1. a(2n-1) = A001610(n+2), a(2n) = A001595(n+2). a(1)=3, a(2)=5, a(3)=6, a(4)=9, a(n) = a(n-2) + a(n-4) + 1, n > 4. G.f.: (3 + 2*x - 2*x^2 + x^3 - 3*x^4)/(1-x-x^2+x^3-x^4+x^5). EXAMPLE a(1)=3 as 3 is the sum of just 2 Fibonacci sets {3=Fibonacci(4)} and {1=Fibonacci(2), 2=Fibonacci(3)}; a(2)=5 as 5 is sum of Fibonacci sets {5} and {2,3} only. MAPLE fib:= combinat[fibonacci]: lucas:=n->fib(n-1)+fib(n+1): a:=n -> if n mod 2 = 0 then 2 *fib(n/2+3) -1 else lucas((n+1)/2+2)-1 fi: seq(a(n), n=1..50); MATHEMATICA LinearRecurrence[{1, 1, -1, 1, -1}, {3, 5, 6, 9, 10, 15}, 40] (* Vincenzo Librandi, Jul 25 2017 *) Table[If[Mod[n, 2]==0, 2*Fibonacci[(n+6)/2]-1, LucasL[(n+5)/2]-1], {n, 50}] (* G. C. Greubel, Jul 13 2019 *) PROG (PARI) vector(50, n, f=fibonacci; if(n%2==0, 2*f((n+6)/2)-1, f((n+7)/2) + f((n+3)/2)-1)) \\ G. C. Greubel, Jul 13 2019 (MAGMA) f:=Floor; [(n mod 2) eq 0 select 2*Fibonacci(f((n+6)/2))-1 else Lucas(f((n+5)/2))-1: n in [1..50]]; // G. C. Greubel, Jul 13 2019 (Sage) def a(n):     if (mod(n, 2)==0): return 2*fibonacci((n+6)/2) - 1     else: return lucas_number2((n+5)/2, 1, -1) -1 [a(n) for n in (1..50)] # G. C. Greubel, Jul 13 2019 (GAP) a:= function(n)     if n mod 2=0 then return 2*Fibonacci(Int((n+6)/2)) -1;     else return Lucas(1, -1, Int((n+5)/2))[2] -1;     fi;   end; List([1..50], n-> a(n) ); # G. C. Greubel, Jul 13 2019 CROSSREFS Cf. A000032, A000045, A000071, A000119, A013583, A122195. Sequence in context: A182050 A094598 A263654 * A225005 A053091 A324701 Adjacent sequences:  A122191 A122192 A122193 * A122195 A122196 A122197 KEYWORD nonn,easy AUTHOR Ron Knott, Aug 25 2006 STATUS approved

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Last modified October 16 05:23 EDT 2019. Contains 328043 sequences. (Running on oeis4.)
The text passage is an entry from the Online Encyclopedia of Integer Sequences (OEIS) describing a specific integer sequence (A122194) and providing various mathematical properties, formulas, and code implementations for generating the sequence.
no_question

the passage does not contain any explicit mathematical questions or problems to solve. it is primarily a reference entry providing information about a particular integer sequence, including its definition, formulas, examples, and code implementations in different programming languages. while there are mathematical concepts and equations present, they are not framed as questions or problems to be solved by the reader.
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 A122194 Numbers that are the sum of exactly two sets of Fibonacci numbers. 2
 3, 5, 6, 9, 10, 15, 17, 25, 28, 41, 46, 67, 75, 109, 122, 177, 198, 287, 321, 465, 520, 753, 842, 1219, 1363, 1973, 2206, 3193, 3570, 5167, 5777, 8361, 9348, 13529, 15126, 21891, 24475, 35421, 39602, 57313, 64078, 92735, 103681, 150049, 167760, 242785 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60th anniversary. M. Bicknell-Johnson & D. C. Fielder, The number of Representations of N Using Distinct Fibonacci Numbers, Counted by Recursive Formulas, Fibonacci Quart. 37.1 (1999) pp. 47 ff. Ron Knott Sumthing about Fibonacci Numbers Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1). FORMULA a(2n-1) = A000032(n+2) - 1, a(2n) = 2*A000045(n+3) - 1. a(2n-1) = A001610(n+2), a(2n) = A001595(n+2). a(1)=3, a(2)=5, a(3)=6, a(4)=9, a(n) = a(n-2) + a(n-4) + 1, n > 4. G.f.: (3 + 2*x - 2*x^2 + x^3 - 3*x^4)/(1-x-x^2+x^3-x^4+x^5). EXAMPLE a(1)=3 as 3 is the sum of just 2 Fibonacci sets {3=Fibonacci(4)} and {1=Fibonacci(2), 2=Fibonacci(3)}; a(2)=5 as 5 is sum of Fibonacci sets {5} and {2,3} only. MAPLE fib:= combinat[fibonacci]: lucas:=n->fib(n-1)+fib(n+1): a:=n -> if n mod 2 = 0 then 2 *fib(n/2+3) -1 else lucas((n+1)/2+2)-1 fi: seq(a(n), n=1..50); MATHEMATICA LinearRecurrence[{1, 1, -1, 1, -1}, {3, 5, 6, 9, 10, 15}, 40] (* Vincenzo Librandi, Jul 25 2017 *) Table[If[Mod[n, 2]==0, 2*Fibonacci[(n+6)/2]-1, LucasL[(n+5)/2]-1], {n, 50}] (* G. C. Greubel, Jul 13 2019 *) PROG (PARI) vector(50, n, f=fibonacci; if(n%2==0, 2*f((n+6)/2)-1, f((n+7)/2) + f((n+3)/2)-1)) \\ G. C. Greubel, Jul 13 2019 (MAGMA) f:=Floor; [(n mod 2) eq 0 select 2*Fibonacci(f((n+6)/2))-1 else Lucas(f((n+5)/2))-1: n in [1..50]]; // G. C. Greubel, Jul 13 2019 (Sage) def a(n):     if (mod(n, 2)==0): return 2*fibonacci((n+6)/2) - 1     else: return lucas_number2((n+5)/2, 1, -1) -1 [a(n) for n in (1..50)] # G. C. Greubel, Jul 13 2019 (GAP) a:= function(n)     if n mod 2=0 then return 2*Fibonacci(Int((n+6)/2)) -1;     else return Lucas(1, -1, Int((n+5)/2))[2] -1;     fi;   end; List([1..50], n-> a(n) ); # G. C. Greubel, Jul 13 2019 CROSSREFS Cf. A000032, A000045, A000071, A000119, A013583, A122195. Sequence in context: A182050 A094598 A263654 * A225005 A053091 A324701 Adjacent sequences:  A122191 A122192 A122193 * A122195 A122196 A122197 KEYWORD nonn,easy AUTHOR Ron Knott, Aug 25 2006 STATUS approved

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Last modified October 16 05:23 EDT 2019. Contains 328043 sequences. (Running on oeis4.)
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The text passage is an entry from the Online Encyclopedia of Integer Sequences (OEIS) describing a specific integer sequence (A122194) and providing various mathematical properties, formulas, and code implementations for generating the sequence.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems to solve. It is primarily a reference entry providing information about a particular integer sequence, including its definition, formulas, examples, and code implementations in different programming languages. While there are mathematical concepts and equations present, they are not framed as questions or problems to be solved by the reader.