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A215006 a(0)=0, a(n+1) is the least k>a(n) such that k+a(n)+n+1 is a Fibonacci number.

%I A215006 #44 Sep 08 2022 08:46:03
%S A215006 0,1,2,3,6,10,18,30,51,84,139,227,371,603,980,1589,2576,4172,6756,
%T A215006 10936,17701,28646,46357,75013,121381,196405,317798,514215,832026,
%U A215006 1346254,2178294,3524562,5702871,9227448,14930335,24157799,39088151,63245967,102334136,165580121
%N A215006 a(0)=0, a(n+1) is the least k>a(n) such that k+a(n)+n+1 is a Fibonacci number.
%C A215006 Same definition for k:
%C A215006 k+b(n)+n is a square for each term b(n) of A097063 except the first;
%C A215006 k+b(n)+n+1 is a square for each term b(n) of A007590 except the first;
%C A215006 k+b(n)+n is a cube for each term b(n) of the sequence 0, 7, 18, 43, 78, 133, 204, 301, 420, 571, 750, 967, 1218, 1513, 1848, 2233, 2664, 3151, 3690, 4291, 4950, 5677, 6468, 7333, ... (last digit repeats with period 10);
%C A215006 k+b(n)+n is a triangular number for each term b(n) of A002378 (oblong numbers);
%C A215006 k+b(n)+n is an oblong number for each term b(n) of A000217 (triangular numbers);
%C A215006 k+b(n)+n is a prime for each term b(n) of the sequence 0, 1, 2, 6, 7, 11, 12, 18, 21, 23, 26, 30, 31, 35, 40, 42, 43, 47, 48, 60, 69, 73, 78, 80, 87, 99, 102, 104, 107, 115, 118, 120, 125, 135, ...
%H A215006 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,0,1).
%F A215006 a(n) = a(n-1) +a(n-2) +floor(n/2) -1 with n>1, a(0)=0, a(1)=1.
%F A215006 From _Bruno Berselli_, Jul 31 2012: (Start)
%F A215006 G.f.: x*(1-2*x^2+x^3+x^4)/((1+x)*(1-x)^2*(1-x-x^2)).
%F A215006 a(n) = Fibonacci(n+2)-A004526(n+1) with n>0, a(0)=0.
%F A215006 a(n) = A129696(n-1)+1 with n>1, a(0)=0, a(1)=1. (End)
%e A215006 For n + 1 = 7, a(n + 1) = 30 is the least k > a(n) = a(6) = 18 such that k + a(n) + n + 1 = 30 + 18 + 6 + 1 = 55 is a Fibonacci number. - _David A. Corneth_, Sep 03 2016
%t A215006 Join[{0}, LinearRecurrence[{2, 1, -3, 0, 1}, {1, 2, 3, 6, 10}, 39]] (* _Jean-François Alcover_, Oct 05 2017 *)
%o A215006 (Python)
%o A215006 prpr = 0
%o A215006 prev = 1
%o A215006 fib = [0]*100
%o A215006 for n in range(100):
%o A215006     fib[n] = prpr
%o A215006     curr = prpr+prev
%o A215006     prpr = prev
%o A215006     prev = curr
%o A215006 a = 0
%o A215006 for n in range(1,55):
%o A215006     print(a, end=',')
%o A215006     b = c = 0
%o A215006     while c <= a:
%o A215006         c = fib[b] - a - n
%o A215006         b += 1
%o A215006     a=c
%o A215006 (Python)
%o A215006 print(0, end=',')
%o A215006 prpr = 1
%o A215006 prev = 2
%o A215006 for n in range(3,56):
%o A215006     print(prpr, end=',')
%o A215006     curr = prpr+prev + n//2 - 1
%o A215006     prpr = prev
%o A215006     prev = curr
%o A215006 (Magma) [n le 3 select n else Self(n)+Self(n-1)+Floor(n/2)-1: n in [0..40]]; // _Bruno Berselli_, Jul 31 2012
%Y A215006 Cf. A000045, A000217, A002378, A007590, A097063.
%K A215006 nonn,easy
%O A215006 0,3
%A A215006 _Alex Ratushnyak_, Jul 31 2012
%E A215006 Definition corrected by _David A. Corneth_, Sep 03 2016