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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. 0
 0, 1, 2, 3, 6, 10, 18, 30, 51, 84, 139, 227, 371, 603, 980, 1589, 2576, 4172, 6756, 10936, 17701, 28646, 46357, 75013, 121381, 196405, 317798, 514215, 832026, 1346254, 2178294, 3524562, 5702871, 9227448, 14930335, 24157799, 39088151, 63245967, 102334136, 165580121 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Same definition for k: k+b(n)+n is a square for each term b(n) of A097063 except the first; k+b(n)+n+1 is a square for each term b(n) of A007590 except the first; 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); k+b(n)+n is a triangular number for each term b(n) of A002378 (oblong numbers); k+b(n)+n is an oblong number for each term b(n) of A000217 (triangular numbers); 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, ... LINKS Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1). FORMULA a(n) = a(n-1) +a(n-2) +floor(n/2) -1 with n>1, a(0)=0, a(1)=1. From Bruno Berselli, Jul 31 2012: (Start) G.f.: x*(1-2*x^2+x^3+x^4)/((1+x)*(1-x)^2*(1-x-x^2)). a(n) = Fibonacci(n+2)-A004526(n+1) with n>0, a(0)=0. a(n) = A129696(n-1)+1 with n>1, a(0)=0, a(1)=1. (End) EXAMPLE 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 MATHEMATICA Join[{0}, LinearRecurrence[{2, 1, -3, 0, 1}, {1, 2, 3, 6, 10}, 39]] (* Jean-François Alcover, Oct 05 2017 *) PROG (Python) prpr = 0 prev = 1 fib = *100 for n in range(100):     fib[n] = prpr     curr = prpr+prev     prpr = prev     prev = curr a = 0 for n in range(1, 55):     print a,     b = c = 0     while c <= a:         c = fib[b] - a - n         b += 1     a=c print '\n\n0', prpr = 1 prev = 2 for n in range(3, 56):     print prpr,     curr = prpr+prev + n//2 - 1     prpr = prev     prev = curr (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 CROSSREFS Cf. A000045, A000217, A002378, A007590, A097063. Sequence in context: A075531 A066067 A121364 * A172516 A102702 A181532 Adjacent sequences:  A215003 A215004 A215005 * A215007 A215008 A215009 KEYWORD nonn,easy AUTHOR Alex Ratushnyak, Jul 31 2012 EXTENSIONS Definition corrected by David A. Corneth, Sep 03 2016 STATUS approved

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Last modified July 5 20:01 EDT 2020. Contains 335473 sequences. (Running on oeis4.)