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A122195
Numbers that are the sum of exactly 3 sets of Fibonacci numbers.
5
8, 11, 13, 14, 18, 19, 22, 23, 30, 31, 36, 38, 49, 51, 59, 62, 80, 83, 96, 101, 130, 135, 156, 164, 211, 219, 253, 266, 342, 355, 410, 431, 554, 575, 664, 698, 897, 931, 1075, 1130, 1452, 1507, 1740, 1829, 2350, 2439, 2816, 2960, 3803, 3947, 4557, 4790, 6154
OFFSET
1,1
FORMULA
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).
a(n) = a(n-4) + a(n-8) + 1.
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.
a(n) = a(n-1) +a(n-4) -a(n-5) +a(n-8) -a(n-9). - G. C. Greubel, Jul 13 2019
EXAMPLE
8 is the sum of only 3 sets of Fibonacci numbers: {8}, {3,5} and {1,2,5};
11 is the sum of only {3,8}, {1,2,8}, {1,2,3,5}.
MAPLE
# first N terms:
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);
MATHEMATICA
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 *)
PROG
(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
(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
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ron Knott, Aug 25 2006, corrected Aug 29 2006
STATUS
approved