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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

Table of n, a(n) for n=0..39.

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-Fran├žois Alcover, Oct 05 2017 *)

PROG

(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

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.)