A014445 -id:A014445 - cp's OEIS frontend

A134489 a(n) = Fibonacci(5*n + 2).

1, 13, 144, 1597, 17711, 196418, 2178309, 24157817, 267914296, 2971215073, 32951280099, 365435296162, 4052739537881, 44945570212853, 498454011879264, 5527939700884757, 61305790721611591, 679891637638612258

Offset: 0

Artur Jasinski, Oct 28 2007

The o.g.f. of {F(m*n + 2)}_{n>=0}, for m = 1, 2, ..., is

G(m,x) = (1 + F(m - 2)*x) / (1 - L(m)*x + (-1)^m*x^2), with F = A000045 and F(-1) = 1, and L = A000032. - Wolfdieter Lang, Feb 06 2023

Index entries for linear recurrences with constant coefficients, signature (11,1).

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888-A033891, A102312, A099100, A134490-A134495, A103134, A134497-A134504.

[Fibonacci(5*n+2): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011

Table[Fibonacci[5n + 2], {n, 0, 30}]
LinearRecurrence[{11,1},{1,13},20] (* Harvey P. Dale, May 05 2022 *)

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-1-2*x) / (-1 + 11*x + x^2).
a(n) = 2*A049666(n) + A049666(n+1). (End)
a(n) = A000045(A016873(n)). - Michel Marcus, Nov 05 2013

A360958 Decimal expansion of Sum_{i>=1} 1/Fibonacci(3*i).

Original entry on oeis.org

6, 6, 3, 5, 0, 2, 1, 3, 8, 9, 3, 3, 0, 2, 8, 1, 9, 7, 1, 3, 5, 8, 8, 1, 0, 9, 5, 9, 4, 9, 9, 9, 3, 2, 9, 5, 7, 7, 5, 2, 6, 6, 2, 5, 1, 6, 2, 4, 5, 2, 9, 5, 2, 8, 3, 0, 3, 1, 0, 8, 4, 2, 5, 6, 8, 0, 3, 2, 9, 1, 6, 0, 4, 1, 4, 2, 6, 3, 3, 5, 0, 5, 1, 9, 3, 5, 4, 5, 3, 9, 3, 4, 3, 5, 4, 0, 8, 5, 0, 9, 5, 3, 2, 2, 8

Offset: 0

Kevin Ryde, Feb 28 2023

Sum of reciprocals of the even Fibonacci numbers, so Sum_{i>=1} 1/A014445(i)

			.66350213893302819713588109594999329...

Kevin Ryde, Table of n, a(n) for n = 0..10000

Cf. A014445, A079586, A153386, A360957.

Equals A079586 - A360957.

A020712 Pisot sequences E(5,8), P(5,8).

Original entry on oeis.org

5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141

Offset: 0

David W. Wilson

Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60,.. - R. J. Mathar, Aug 10 2012

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).

Subsequence of A020701 and hence A020695, A000045. See A008776 for definitions of Pisot sequences.

Trisections: A015448, A014445, A033887.

CoefficientList[Series[(-3 z - 5)/(z^2 + z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{1,1},{5,8},40] (* Harvey P. Dale, Dec 28 2013 *)

a(n)=fibonacci(n+5) \\ Charles R Greathouse IV, Jan 17 2012

a(n) = Fib(n+5). a(n) = a(n-1) + a(n-2).
O.g.f.: (5+3x)/(1-x-x^2). a(n)=A020701(n+1). - R. J. Mathar, May 28 2008
a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-11+5*sqrt(5))+(1+sqrt(5))^n*(11+5*sqrt(5))))/sqrt(5). - Colin Barker, Jun 05 2016

A087579 a(n) = (1/6)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*6^k.

Original entry on oeis.org

1, 8, 93, 976, 10505, 112344, 1203397, 12885152, 137979729, 1477507240, 15821470061, 169419470448, 1814178395353, 19426591805816, 208023907911765, 2227562425662784, 23853192734743457, 255424852222168392, 2735141407084907389, 29288451971122142480

Offset: 0

Benoit Cloitre, Oct 25 2003

Index entries for linear recurrences with constant coefficients, signature (8,29).

Cf. A014445, A057088, A015553.

Join[{b=1},a=0;Table[c=8*b+29*a;a=b;b=c,{n,30}]] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
LinearRecurrence[{8,29},{1,8},20] (* Harvey P. Dale, Mar 22 2019 *)

a(n) = 8*a(n-1) + 29*a(n-2).

G.f.: 1 / (-29*x^2-8*x+1). - Colin Barker, Aug 08 2013

More terms from Colin Barker, Aug 08 2013

A087584 a(n) = (1/7)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*7^k.

Original entry on oeis.org

1, 9, 122, 1467, 18205, 223992, 2762333, 34044669, 419657674, 5172750495, 63760719089, 785929242096, 9687552661513, 119411072879553, 1471889315038010, 18142857823403763, 223633182327192277, 2756555811704284776, 33977962780753446341, 418820453306656692885

Offset: 0

Benoit Cloitre, Oct 25 2003

Index entries for linear recurrences with constant coefficients, signature (9,41).

Cf. A014445, A057088, A015553.

a(n) = 9*a(n-1) + 41*a(n-2).

G.f.: 1 / (-41*x^2-9*x+1). - Colin Barker, Aug 08 2013

More terms from Colin Barker, Aug 08 2013

A138859 Smallest prime factor of Fibonacci(3n)-1, i.e., A020639(A000071(3n)).

Original entry on oeis.org

7, 3, 11, 3, 3, 5, 199, 13, 13, 3, 37, 3, 3, 89, 5, 11, 7, 3, 59, 3, 3, 11, 11, 37, 37, 3, 370248451, 3, 3, 139, 13, 5, 5, 3, 5, 3, 3, 59, 709, 557, 127, 3, 11, 3, 3, 5, 9375829, 73, 7, 3, 29, 3, 3, 2789, 5, 11, 47, 3, 13, 3, 3, 11, 11, 13, 7, 3, 809, 3, 3, 953, 47927441, 5, 5

Offset: 2

M. F. Hasler, Apr 05 2008

nonn

F(n)-1 is even if n is not a multiple of 3.

Amiram Eldar, Table of n, a(n) for n = 2..934 (terms 2..379 from Chai Wah Wu)

Cf. A000071, A014445, A020639.

Table[FactorInteger[Fibonacci[3n]-1][[1,1]],{n,2,80}] (* Harvey P. Dale, Aug 08 2017 *)

A138859(n)=factor(fibonacci(3*n)-1)[1,1] /* A138859 is defined only for n>1 ! */

a(n) = A020639(A000071(3n)).

A269500 a(n) = Fibonacci(10*n).

Original entry on oeis.org

0, 55, 6765, 832040, 102334155, 12586269025, 1548008755920, 190392490709135, 23416728348467685, 2880067194370816120, 354224848179261915075, 43566776258854844738105, 5358359254990966640871840, 659034621587630041982498215, 81055900096023504197206408605

Offset: 0

Ilya Gutkovskiy, Mar 03 2016

More generally, the ordinary generating function for the Fibonacci(k*n) is F(k)*x/(1 - L(k)*x + (-1)^k*x^2), where F(k) is the k-th Fibonacci number (A000045), L(k) is the k-th Lucas number (A000032), or (phi^k - (-1/phi)^k)*x/(sqrt(5)*(1 - (phi^k + (-1/phi)^k)*x + (-1)^k*x^2)), where phi is the golden ratio (A001622).

Eric Weisstein's World of Mathematics, Fibonacci Number
Index entries for linear recurrences with constant coefficients, signature (123,-1)

Cf. similar sequences of the form Fibonacci(k*n): A000045 (k = 1), A001906 (k = 2), A014445 (k = 3), A033888 (k = 4), A102312 (k = 5), A134492 (k = 6), A134498 (k = 7), A138473 (k = 8), A138590 (k = 9), this sequence (k = 10), A167398 (k = 11), A214855 (k = 15).

Cf. A000032 (Lucas numbers), A001622 (golden ratio).

Fibonacci[10Range[0, 14]]
FullSimplify[Table[(((1 + Sqrt[5])/2)^(10 n) - (2/(1 + Sqrt[5]))^(10 n))/Sqrt[5], {n, 0, 12}]]
LinearRecurrence[{123, -1}, {0, 55}, 15]

a(n) = fibonacci(10*n); \\ Michel Marcus, Mar 03 2016

concat(0, Vec(55*x/(1-123*x+x^2) + O(x^100))) \\ Altug Alkan, Mar 04 2016

G.f.: 55*x/(1 - 123*x + x^2).
a(n) = 123*a(n-1) - a(n-2).
a(n) = A000045(10*n).
Lim_{n -> infinity} a(n + 1)/a(n) = phi^10 = 122.9918693812442…

A133012 Even imperfect numbers.

Original entry on oeis.org

2, 4, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118

Offset: 1

Omar E. Pol, Oct 20 2007

Cf. A005843, A014445, A014494. Imperfect numbers: A132999.

A133016 Even imperfect numbers, divided by 2.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64

Offset: 1

Omar E. Pol, Oct 20 2007

Cf. A005843, A014335, A014445, A014494, A026503, A028334. Imperfect numbers: A132999.

A171731 Triangle T : T(n,k)= binomial(n,k)Fibonacci(n-k)= A007318(n,k)A000045(n-k).

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 2, 3, 3, 0, 3, 8, 6, 4, 0, 5, 15, 20, 10, 5, 0, 8, 30, 45, 40, 15, 6, 0, 13, 56, 105, 105, 70, 21, 7, 0, 21, 104, 224, 280, 210, 112, 28, 8, 0, 34, 189, 468, 672, 630, 378, 168, 36, 9, 0, 55, 340, 945, 1560, 1680, 1260, 630, 240, 45, 10, 0

Offset: 0

Philippe Deléham, Dec 16 2009

Diagonal sums : A112576.

Essentially the same as A094440. - Peter Bala, Jan 06 2015

			Triangle begins :
0 ;
1,0 ;
1,2,0 ;
2,3,3,0 ;
3,8,6,4,0 ;
5,15,20,10,5,0 ;
...

Cf. A000045, A007318, A094440.

Flatten[Table[Binomial[n,k]Fibonacci[n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jan 16 2013 *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000045(n), A001906(n), A093131(n), A099453(n-1), A081574(n), A081575(n) for x = 0,1,2,3,4,5 respectively. Sum_{k, 0<=k<=n} T(n,k)*2^(n-k) = A014445(n).

cp's OEIS Frontend

A134489 a(n) = Fibonacci(5*n + 2).

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A360958 Decimal expansion of Sum_{i>=1} 1/Fibonacci(3*i).

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A020712 Pisot sequences E(5,8), P(5,8).

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A087579 a(n) = (1/6)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*6^k.

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A087584 a(n) = (1/7)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*7^k.

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A138859 Smallest prime factor of Fibonacci(3n)-1, i.e., A020639(A000071(3n)).

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Mathematica

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A269500 a(n) = Fibonacci(10*n).

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PARI

PARI

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A133012 Even imperfect numbers.

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A133016 Even imperfect numbers, divided by 2.

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A087579 a(n) = (1/6)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*6^k.

A087584 a(n) = (1/7)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*7^k.

A171731 Triangle T : T(n,k)= binomial(n,k)Fibonacci(n-k)= A007318(n,k)A000045(n-k).