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A122195 Numbers that are the sum of exactly 3 sets of Fibonacci numbers.

%I A122195 #24 Jan 05 2025 19:51:38
%S A122195 8,11,13,14,18,19,22,23,30,31,36,38,49,51,59,62,80,83,96,101,130,135,
%T A122195 156,164,211,219,253,266,342,355,410,431,554,575,664,698,897,931,1075,
%U A122195 1130,1452,1507,1740,1829,2350,2439,2816,2960,3803,3947,4557,4790,6154
%N A122195 Numbers that are the sum of exactly 3 sets of Fibonacci numbers.
%H A122195 G. C. Greubel, <a href="/A122195/b122195.txt">Table of n, a(n) for n = 1..1000</a>
%H A122195 M. Bicknell-Johnson & D. C. Fielder, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/37-1/bicknell.pdf">The number of Representations of N Using Distinct Fibonacci Numbers, Counted by Recursive Formulas</a>, Fibonacci Quart. 37.1 (1999) pp. 47 ff.
%H A122195 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrep.html#sumoffib">Sumthing about Fibonacci Numbers</a>
%H A122195 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1,0,0,1,-1).
%F A122195 G.f.: (8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(1-x-x^4+x^5-x^8+x^9).
%F A122195 a(n) = a(n-4) + a(n-8) + 1.
%F A122195 a(0)=8, a(1)=11, a(2)=13, a(3)=18, then: a(4n) = A022318(n+3) = 2*A000045(n+5) + A000045(n+3) - 1, a(4n+1) = A022406(n+2) = 4*A000045(n+4) - 1, a(4n+2) = A022308(n+4) = 2*A000045(n+4) + A000045(n+6) - 1, a(4n+3) = 3*A000045(n+4) - 1, for n>=1.
%F A122195 a(n) = a(n-1) +a(n-4) -a(n-5) +a(n-8) -a(n-9). - _G. C. Greubel_, Jul 13 2019
%e A122195 8 is the sum of only 3 sets of Fibonacci numbers: {8}, {3,5} and {1,2,5};
%e A122195 11 is the sum of only {3,8}, {1,2,8}, {1,2,3,5}.
%p A122195 # first N terms:
%p A122195 series((8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(x^9-x^8+x^5-x^4-x+1),x,N+1);
%t A122195 CoefficientList[Series[(8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(1 - x-x^4+x^5-x^8+x^9), {x, 0, 60}], x] (* _G. C. Greubel_, Jul 13 2019 *)
%o A122195 (PARI) my(x='x+O('x^60)); Vec((8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8 -3*x^9)/(1-x-x^4+x^5-x^8+x^9)) \\ _G. C. Greubel_, Jul 13 2019
%o A122195 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(1 - x-x^4+x^5-x^8+x^9) )); // _G. C. Greubel_, Jul 13 2019
%o A122195 (Sage) ((8+3*x+2*x^2+x^3-4*x^4-2*x^5+x^6-5*x^8-3*x^9)/(1 - x-x^4+x^5-x^8+x^9)).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 13 2019
%o A122195 (GAP) a:=[11,13,14,18,19,22,23,30];; for n in [9..60] do a[n]:=a[n-4]+a[n-8]+1; od; Concatenation([8], a); # _G. C. Greubel_, Jul 13 2019
%Y A122195 Cf. A000045, A000071, A013583, A000119, A122194.
%K A122195 nonn
%O A122195 1,1
%A A122195 _Ron Knott_, Aug 25 2006, corrected Aug 29 2006