A066629 -id:A066629 - cp's OEIS frontend

A066983 a(n+2) = a(n+1) + a(n) + (-1)^n, with a(1) = a(2) = 1.

1, 1, 1, 3, 3, 7, 9, 17, 25, 43, 67, 111, 177, 289, 465, 755, 1219, 1975, 3193, 5169, 8361, 13531, 21891, 35423, 57313, 92737, 150049, 242787, 392835, 635623, 1028457, 1664081, 2692537, 4356619, 7049155, 11405775, 18454929, 29860705, 48315633, 78176339

Offset: 1

Benoit Cloitre, Jan 27 2002

Length of strings given by a successive substitution of a "modified" Kolakoski-(3, 1) sequence. Starting with 1, using the rule "string begins with 1 if previous string ends with 3, string begins with 3 if previous string ends with 1" then applying the classical Kolakoski-(3,1) rule. This gives: 1 -> 3 -> 111 -> 313 -> 1113111 -> 313111313 -> 11131113131113111 and the length of string are 1, 1, 3, 3, 7, 9, 17, ... At step n, length = a(n+1). This substitution leads to two sequences: 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, ... and 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, ... - Benoit Cloitre, Jun 01 2004
Lengths of comparators in subsequent layers of correction network F_n. - Grzegorz Stachowiak (gst(AT)ii.uni.wroc.pl), Nov 28 2004
Convolution of F(n+1) and A105812(n). Action of inverse of sequence array for F(n-1)*(-1)^n on F(n+1). - Paul Barry, Oct 29 2006

Omur Deveci, The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite groups, Utilitas Mathematica, 98 (2015), 257-270.

Harry J. Smith, Table of n, a(n) for n = 1..250
K. Atanassov, D. Dimitrov and A. G. Shannon, A remark on psi-function and Pell-Padovan's sequence, Notes Number Theory Discrete Math., 15 (2009), no. 2, 1-44.
Michael Baake and Bernd Sing, Kolakoski-(3,1) is a (deformed) model set, arXiv:math/0206098 [math.MG], 2002-2003.
Taras Goy, S. V. Sharyn, A note on Pell-Padovan numbers and their connection with Fibonacci numbers, Carpathian Math. Publ. (2020) Vol. 12, No. 2, 280-288.
Zehra İşbilir and Nurten Gürses, Pell-Padovan generalized quaternions, Notes on Num. Theory and Disc. Math. (2021) Vol. 27, No. 1, 171—187.
G. Stachowiak, Fibonacci Correction Networks, SWAT 2000, LNCS 1851, 535-548.
G. Stachowiak, Lower Bounds on Correction Networks, ISAAC 2003, LNCS 2906, 221-229.
Dursun Tasci, Gaussian Padovan and Gaussian Pell-Padovan sequences, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 67 (2018), no. 2, 82-88. Sequence R_n.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000045, A001083, A042942, A064353, A066629.

a:=[1,1];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]+(-1)^n; od; a; # Muniru A Asiru, Aug 09 2018

[n le 2 select 1 else Self(n-1)+Self(n-2)+(-1)^n: n in [1..50]]; // Vincenzo Librandi, Aug 13 2018

seq(coeff(series(x*(1+x-x^2)/((1+x)*(1-x-x^2)), x,n+1),x,n),n=1..40); # Muniru A Asiru, Aug 09 2018

Table[ Floor[ GoldenRatio^(k-1) ] - Floor[ GoldenRatio^(k-1) / Sqrt[5] ], {k, 1, 100} ]  (* Federico Provvedi, Mar 26 2013 *)
LinearRecurrence[{0, 2, 1}, {1, 1, 1}, 40] (* Vincenzo Librandi, Aug 13 2018 *)

{ for (n=1, 250, if (n>2, a=a1 + a2 + (-1)^n; a2=a1; a1=a, a=a1=1; a=a2=1); write("b066983.txt", n, " ", a) ) } \\ Harry J. Smith, Apr 15 2010

vector(40, n, 2*fibonacci(n-2) + (-1)^n) \\ G. C. Greubel, Dec 26 2019

from sympy import fibonacci
def A066983(n): return (fibonacci(n-2)<<1)+(-1 if n&1 else 1) # Chai Wah Wu, May 05 2025

[2*fibonacci(n-2) + (-1)^n for n in (1..40)] # G. C. Greubel, Dec 26 2019

For n > 4, a(n-2) = floor(2 * phi^n/sqrt(5)) + (1 + (-1)^n)/2.
a(n) = 2 * Fibonacci(n-2) + (-1)^n. - Vladeta Jovovic, Mar 19 2003
G.f.: x*(1+x-x^2)/((1+x)*(1-x-x^2)). - Paul Barry, Oct 29 2006
a(n) = A066629(n-2) - A066629(n-3), n > 2. - R. J. Mathar, Jan 14 2009
a(n) = floor(phi^(n-1)) - floor(phi^(n-1)/sqrt(5)). - Federico Provvedi, Mar 26 2013
a(1) = a(2) = a(3) = 1; for n > 3, a(n) = 2*a(n-2) + a(n-3). - Taras Goy, Aug 03 2018
a(n) = (-1)^n + (-1 - 3/sqrt(5))*((1/2)*(1 - sqrt(5)))^n + (-1 + 3/sqrt(5))*((1/2)*(1 + sqrt(5)))^n. - Stefano Spezia, Jul 22 2019

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A154691 Expansion of (1+x+x^2) / ((1-x)*(1-x-x^2)).

Original entry on oeis.org

1, 3, 7, 13, 23, 39, 65, 107, 175, 285, 463, 751, 1217, 1971, 3191, 5165, 8359, 13527, 21889, 35419, 57311, 92733, 150047, 242783, 392833, 635619, 1028455, 1664077, 2692535, 4356615, 7049153, 11405771, 18454927, 29860701, 48315631, 78176335

Offset: 0

R. J. Mathar, Jan 14 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tomislav Došlić and Biserka Kolarec, On Log-Definite Tempered Combinatorial Sequences, Mathematics (2025) Vol. 13, Iss. 7, 1179.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A001595, A006355, A055389, A066629, A078642, A166863.

a154691 n = a154691_list !! n
a154691_list = 1 : zipWith (+)
                   a154691_list (drop 2 $ map (* 2) a000045_list)
-- Reinhard Zumkeller, Nov 17 2013

A154691:= func< n | 2*Fibonacci(n+3) - 3 >;
[A154691(n): n in [0..40]]; // G. C. Greubel, Jan 18 2025

A154691 := proc(n) coeftayl( (1+x+x^2)/(1-x-x^2)/(1-x),x=0,n) ; end proc:

Fibonacci[Range[3,60]]*2 -3 (* Vladimir Joseph Stephan Orlovsky, Mar 19 2010 *)
CoefficientList[Series[(1 + x + x^2)/((1 - x - x^2)(1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 18 2012 *)

Vec((1+x+x^2) / ((1-x-x^2)*(1-x)) + O(x^60)) \\ Colin Barker, Feb 01 2017

def A154691(n): return 2*fibonacci(n+3) - 3
print([A154691(n) for n in range(41)]) # G. C. Greubel, Jan 18 2025

a(n+1) - a(n) = A006355(n+3) = A055389(n+3).
a(n) = A066629(n-1) + A066629(n).
a(n) = A006355(n+4) - 3 = A078642(n+1) - 3.
a(n+1) = a(n) + 2*A000045(n+2). - Reinhard Zumkeller, Nov 17 2013
From Colin Barker, Feb 01 2017: (Start)
a(n) = -3 + (2^(1-n)*((1-r)^n*(-2+r) + (1+r)^n*(2+r))) / r where r=sqrt(5).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
a(n) = 2*Fibonacci(n+3) - 3. - Greg Dresden, Oct 10 2020
E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 3*exp(x). - Stefano Spezia, Apr 09 2025

A154327 Diagonal sums of number triangle A132046.

Original entry on oeis.org

1, 1, 2, 5, 8, 15, 24, 41, 66, 109, 176, 287, 464, 753, 1218, 1973, 3192, 5167, 8360, 13529, 21890, 35421, 57312, 92735, 150048, 242785, 392834, 635621, 1028456, 1664079, 2692536, 4356617, 7049154, 11405773, 18454928, 29860703, 48315632, 78176337, 126491970, 204668309, 331160280, 535828591, 866988872

Offset: 0

Paul Barry, Jan 07 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

A shifted version of A066629.

[0^n-(3+(-1)^n)/2+2*Fibonacci(n+1):n in [0..40]]; // Vincenzo Librandi, Sep 12 2016

Join[{1}, LinearRecurrence[{1, 2, -1, -1}, {1, 2, 5, 8}, 25]] (* G. C. Greubel, Sep 11 2016 *)
CoefficientList[Series[(1 - x^2 + 2 x^3 + x^4) / ((1 - x^2) (1 - x - x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 12 2016 *)

G.f.: (1 - x^2 + 2x^3 + x^4)/( (1-x^2)*(1-x-x^2) ).

a(n) = 0^n - (3 + (-1)^n)/2 + 2*Fibonacci(n+1).

A153864 Triangle read by rows, A000012 * A153860 * (A066983 * 0^(n-k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 6, 3, 1, 2, 2, 6, 6, 7, 2, 2, 2, 6, 6, 14, 9, 1, 2, 2, 6, 6, 14, 18, 17, 2, 2, 2, 6, 6, 14, 18, 34, 25, 1, 2, 2, 6, 6, 14, 18, 34, 50, 43, 2, 2, 2, 6, 6, 14, 18, 34, 50, 86, 67

Offset: 0

Gary W. Adamson, Jan 03 2009

Row sums = A066629: (1, 2, 5, 8, 15, 24, 41, 66, 109,...).

Right border = A066983: (1, 1, 1, 3, 3, 7, 9, 17,...).

			First few rows of the triangle =
1;
1, 1;
2, 2, 1;
1, 2, 2, 3;
2, 2, 2, 6, 3;
1, 2, 2, 6, 6, 7;
2, 2, 2, 6, 6, 14, 9;
1, 2, 2, 6, 6, 14, 18, 17;
2, 2, 2, 6, 6, 14, 18, 34, 25;
1, 2, 2, 6, 6, 14, 18, 34, 50, 43;
...

Cf. A153860, A066983, A066629

Triangle read by rows, A000012 * A153860 * (A066983 * 0^(n-k))
Given triangle A000012 * A153860 = partial sums of A153860 starting from the top.
(A066983 * 0^n-k) = an infinite lower triangular matrix with A066983 as the
main diagonal: (1, 1, 1, 3, 3, 7, 9, 17, 25,...) and the rest zeros.

A271729 Numbers n such that 2*Fibonacci(n+2)+((-1)^n-3)/2 is a prime.

Original entry on oeis.org

1, 2, 6, 8, 14, 16, 32, 40, 66, 88, 112, 120, 146, 158, 318, 514, 630, 638, 680, 710, 1054, 1630, 2198, 2466, 2696, 2994, 3138, 3654, 3958, 5558, 7008, 11416, 11632, 13510, 19752, 24480

Offset: 1

Vincenzo Librandi, Apr 13 2016

These numbers are the positions of prime numbers in A066629.

Cf. A066629.

[n: n in [0..2000] | IsPrime(2*Fibonacci(n+2)+((-1)^n-3) div 2)];

with(combinat): A271729:=n->`if`(isprime(2*fibonacci(n+2)+((-1)^n-3)/2), n, NULL): seq(A271729(n), n=1..3*10^3); # Wesley Ivan Hurt, Apr 13 2016

Select[Range[10000], PrimeQ[(2 Fibonacci[# + 2] + ((-1)^# - 3) / 2)] &]

lista(nn) = for(n=1, nn, if(ispseudoprime(2*fibonacci(n+2)+((-1)^n-3)/2), print1(n, ", "))); \\ Altug Alkan, Apr 13 2016

cp's OEIS Frontend

A078697 Duplicate of A066629.

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A066983 a(n+2) = a(n+1) + a(n) + (-1)^n, with a(1) = a(2) = 1.

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A154691 Expansion of (1+x+x^2) / ((1-x)*(1-x-x^2)).

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A154327 Diagonal sums of number triangle A132046.

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A153864 Triangle read by rows, A000012 * A153860 * (A066983 * 0^(n-k)).

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A271729 Numbers n such that 2*Fibonacci(n+2)+((-1)^n-3)/2 is a prime.

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