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A215006
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a(0)=0, a(n+1) is the least k>a(n) such that k+a(n)+n+1 is a Fibonacci number.
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0
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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
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OFFSET
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0,3
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COMMENTS
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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, ...
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LINKS
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Table of n, a(n) for n=0..39.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).
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FORMULA
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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)
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EXAMPLE
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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
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MATHEMATICA
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Join[{0}, LinearRecurrence[{2, 1, -3, 0, 1}, {1, 2, 3, 6, 10}, 39]] (* Jean-François Alcover, Oct 05 2017 *)
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PROG
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(Python)
prpr = 0
prev = 1
fib = [0]*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
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CROSSREFS
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Cf. A000045, A000217, A002378, A007590, A097063.
Sequence in context: A075531 A066067 A121364 * A172516 A102702 A181532
Adjacent sequences: A215003 A215004 A215005 * A215007 A215008 A215009
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KEYWORD
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nonn,easy
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AUTHOR
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Alex Ratushnyak, Jul 31 2012
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EXTENSIONS
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Definition corrected by David A. Corneth, Sep 03 2016
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STATUS
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approved
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