A209927 -id:A209927 - cp's OEIS frontend

A006130 a(n) = a(n-1) + 3*a(n-2) for n > 1, a(0) = a(1) = 1.

1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, 14209, 32689, 75316, 173383, 399331, 919480, 2117473, 4875913, 11228332, 25856071, 59541067, 137109280, 315732481, 727060321, 1674257764, 3855438727, 8878212019, 20444528200, 47079164257, 108412748857

Offset: 0

N. J. A. Sloane

Counts walks of length n at the vertex of degree five in the graph with adjacency matrix A = [1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,0]. - Paul Barry, Oct 02 2004
Form the graph with matrix A = [0,1,1,1;1,1,0,0;1,0,1,0;1,0,0,1]. The sequence 0,1,1,4,... counts walks of length n between the vertex without loop and another vertex. - Paul Barry, Oct 02 2004
Length-n words with letters {0,1,2,3} where no two consecutive letters are nonzero, see fxtbook link below. - Joerg Arndt, Apr 08 2011
Hankel transform is the sequence [1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 10 2007
Let M = [1, sqrt(3); sqrt(3), 0] be a 2 X 2 matrix. Then A006130(n)={[M^n]A006130-A052533%20=%20A006130%20(shifted%20to%20the%20right%20one%20place,%20with%20first%20term%20=%200).%20-%20_L.%20Edson%20Jeffery">(1,1)}. Note that A006130-A052533 = A006130 (shifted to the right one place, with first term = 0). - _L. Edson Jeffery, Nov 25 2011 [Any matrix M = [1, y; 3/y, 0], with y not 0, will do it. - Wolfdieter Lang, Feb 18 2018]
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=2, 4*a(n-2) equals the number of 4-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
a(n) is the number of compositions (ordered partitions) of n into parts 1 and 2 where there are three sorts of part 2 (see the g.f.). - Joerg Arndt, Jan 16 2024
Number of pairs of rabbits when there are 3 pairs per litter and offspring reach parenthood after 2 gestation periods. - Robert FERREOL, Oct 28 2018
Numerators of stationary probabilities sequence for number of customers in steady state of M2/M/1 queue, whose g.f. is (1-z)/(3-3z-z(1-z^2)). - Igor Kleiner, Nov 03 2018
INVERT transform of (1, 0, 3, 0, 9, 0, 27, ...). - Gary W. Adamson, Jul 15 2019
Number of 3-compositions of n+2 with 1 not allowed as a part; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020
Number of ways to tile a strip of length n with 3 colors of dominoes and 1 color of squares. - Greg Dresden, Sep 01 2021
Number of 3-permutations of n elements avoiding the patterns 231, 312, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
a(n-1), with a(-1) = 0, appears in the formula for the powers of the fundamental (integer) algebraic number c = (1 + sqrt(13))/2 = A209927: c^n = A052533(n) + a(n-1)*c. With the formulas given below, and also in A052533, in terms of S-Chebyshev polynomials this is valid also for the powers of 1/c = (-1 + sqrt(13))/6 = A356033. - Wolfdieter Lang, Nov 26 2023

			G.f. = 1 + x + 4*x^2 + 7*x^3 + 19*x^4 + 40*x^5 + 97*x^6 + 217*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stephen Wolfram, 'The Mathematica Book,' Fourth Edition, Wolfram Media or Cambridge University Press, 1999, p. 96.

Vincenzo Librandi, G. C. Greubel and Robert Israel, Table of n, a(n) for n = 0..2485 (n = 0..149 from Vincenzo Librandi, n = 150..300 from G. C. Greubel)
Joerg Arndt, Matters Computational (The Fxtbook), pp.317-318.
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 436
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Thor Martinsen, Non-Fisherian generalized Fibonacci numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 2, 370-389. See pp. 387-389.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Arulalan Rajan, R. Vittal Rao, Ashok Rao and H. S. Jamadagni, Fibonacci Sequence, Recurrence Relations, Discrete Probability Distributions and Linear Convolution, arXiv preprint arXiv:1205.5398 [math.PR], 2012.
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, 2015, No. 2, 35-42.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
Fatih Yılmaz and Mustafa Özkan, On the Generalized Gaussian Fibonacci Numbers and Horadam Hybrid Numbers: A Unified Approach, Axioms (2022) Vol. 11, No. 6, 255.
Paul Thomas Young, p-adic congruences for generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 32, No. 1, 1994.
Index entries for linear recurrences with constant coefficients, signature (1,3).
Index entries for sequences related to Chebyshev polynomials.

Cf. A006131, A015440, A049310, A052533, A140167, A175291 (Pisano periods), A099232 (partial sums), A274977.

Cf. A209927, A356033.

a := [1, 1];; for n in [3..30] do a[n] := a[n-1] + 3*a[n-2]; od; a; # Muniru A Asiru, Feb 18 2018

[n le 2 select 1 else Self(n-1) + 3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 17 2012

a := n -> add(binomial(n-k, k)*3^k, k=0..n): seq(a(n), n=0..29); # Zerinvary Lajos, Sep 30 2006
f:= gfun:-rectoproc({a(n) = a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 1},a(n),remember):
map(f, [$0..100]); # Robert Israel, Aug 31 2015

a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + 3a[n - 2]; Table[ a[n], {n, 0, 30}]
LinearRecurrence[{1, 3}, {1, 1}, 30] (* Vincenzo Librandi, Oct 17 2012 *)
RecurrenceTable[{a[n]== a[n-1] + 3*a[n-2], a[0]== 1, a[1]== 1}, a, {n,0,30}] (* G. C. Greubel, Aug 30 2015 *)
a[0] := 1; a[n_] := Hypergeometric2F1[1/2-n/2, -n/2, -n, -12]; Table[a[n], {n, 0, 29}] (* Peter Luschny, Feb 18 2018 *)
a[ n_] := With[{s = Sqrt[-1/3]}, ChebyshevU[n, s/2] / s^n] // Simplify; (* Michael Somos, Nov 04 2018 *)

Vec(1/(1-x-3*x^2+O(x^66))) \\ Franklin T. Adams-Watters, May 26 2011

an = an1 = 1
while an<10**5:
   print(an)
   an1 += an*3
   an = an1 - an*3   # Alex Ratushnyak, Apr 20 2012

from sage.combinat.sloane_functions import recur_gen2
it = recur_gen2(1,1,1,3)
[next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008

[lucas_number1(n,1,-3) for n in range(1, 31)] # Zerinvary Lajos, Apr 22 2009

O.g.f.: 1/(1 - x - 3*x^2). - Simon Plouffe in his 1992 dissertation.
a(n) = (( (1 + sqrt(13))/2 )^(n+1) - ( (1 - sqrt(13))/2 )^(n+1))/sqrt(13).
a(n) = Sum_{k = 0..ceiling(n/2)} 3^k*C(n-k, k). - Benoit Cloitre, Philippe Deléham, Mar 07 2004
a(0) = 1; a(1) = 1; for n >= 1, a(n+1) = (a(n)^2 - (-3)^n) / a(n-1). - Philippe Deléham, Mar 07 2004
The i-th term of the sequence is the (1, 2) entry in the i-th power of the 2 X 2 matrix M = ((-1, 1), (1, 2)). - Simone Severini, Oct 15 2005
a(n) = lower right term in the 2 X 2 matrix [0,3; 1,1]^n. - Gary W. Adamson, Mar 02 2008
a(n) = Sum_{k = 0..n} A109466(n,k)*(-3)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = Product_{k = 1..floor((n - 1)/2)} (1 + 12*cos(k*Pi/n)^2). - Roger L. Bagula and Gary W. Adamson, Nov 21 2008
Limiting ratio = (1 + sqrt(13))/2 = 2.30277563.. = A098316 - 1. - Roger L. Bagula and Gary W. Adamson, Nov 21 2008
G.f.: G(0)/(2-x), where G(k)= 1 + 1/(1 - x*(13*k - 1)/(x*(13*k + 12) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 18 2013
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + 3*x)/( x*(4*k+3 + 3*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
a(n) = ( Sum_{1 <= k <= n+1, k odd} C(n+1,k)*13^((k-1)/2) )/2^n. - Vladimir Shevelev, Feb 05 2014
E.g.f.: (1/(a - b))*(a*exp(a*x) - b*exp(b*x)), where 2*a = 1 + sqrt(13) and 2*b = 1 - sqrt(13). - G. C. Greubel, Aug 30 2015
a(n) = ((i*sqrt(3))^n)*S(n, (-i/sqrt(3))), with the imaginary unit i and the Chebyshev S polynomials (coefficients in A049310). - Wolfdieter Lang, Feb 18 2018
a(n) = hypergeom([(1-n)/2, -n/2], [-n], -12) for n >= 1. - Peter Luschny, Feb 18 2018
a(n) = 3 * (-3)^n * a(-2-n) for all n in Z. - Michael Somos, Nov 04 2018
a(n) = ( sqrt(3) )^n * Fibonacci(n+1, 1/sqrt(3)). - G. C. Greubel, Dec 26 2019
a(n) = numerator of the continued fraction 1 + 3/(1 + 3/(1 + 3/ ... + 3/1)) with exactly n 1's, for n>0. - Greg Dresden and Alexander Mark, Aug 16 2020
With an initial 0 prepended, the sequence [0, 1, 1, 4, 7, 19, 40, ...] satisfies the congruences a(n*p^k) == e*a(n*p^(k-1)) (mod p^k) for positive integers k and n and all primes p, where e = +1 for the primes p listed in A296937, e = 0 if p = 13, otherwise e = -1. See Young, Theorem 1, Corollary 1 (i). - Peter Bala, Dec 28 2022
From Wolfdieter Lang, Nov 27 2023: (Start)
a(n) = sqrt(-3)^n*S(n, 1/sqrt(-3)) with the S-Chebyshev polynomials (see A049310), also valid for negative n, using S(-n, x) = -S(n-2, x), for n >= 2, and S(-1, x) = 0. See above the formula in terms of Fibonacci polynomials).
a(n) = A052533(n+2)/3, for n >= 0. (End)
From Peter Bala, Jun 27 2025: (Start)
G.f.: Sum_{n >= 0} x^n * Product_{k = 1..n} (k + 3*x)/(1 + k*x) = Sum_{n >= 0} x^n * Product_{k = 1..n} (1 + 3*k*x)/(1 + 3*k*x^2).
The following products telescope:
Product_{k >= 0} (1 + 3^k/a(2*k+1)) = 1 + sqrt(13).
Product_{k >= 1} (1 - 3^k/a(2*k+1)) = 1/14 * (1 + sqrt(13)).
Product_{k >= 0} (1 + (-3)^k/a(2*k+1)) = (1/13) * (13 + sqrt(13)).
Product_{k >= 1} (1 - (-3)^k/a(2*k+1)) = (1/14) * (13 + sqrt(13)). (End)

A010470 Decimal expansion of square root of 13.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 3 followed by {1, 1, 1, 1, 6} repeated. - Harry J. Smith, Jun 02 2009
The convergents to sqrt(13) are given in A041018/A041019. - Wolfdieter Lang, Nov 23 2017
The fundamental algebraic (integer) number in the field Q(sqrt(13)) is (1 + sqrt(13))/2  = A209927. - Wolfdieter Lang, Nov 21 2023

			3.605551275463989293119221267470495946251296573845246212710453056227166...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.4, p. 201.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2.

Cf. A010122 (continued fraction), A041018/A041019 (convergents), A248242 (Egyptian fraction), A171983 (Beatty sequence).

Cf. A020770 (reciprocal), A209927, A295330, A344069.

RealDigits[N[Sqrt[13],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(13); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010470.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

A295330 Decimal expansion of sqrt(13)/2.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 20 2017

In a regular hexagon inscribed in a circle of radius R the largest distance between any vertex and a midpoint of a side, after division of R, is sqrt(13)/2. The two smaller distance ratios are sqrt(7)/2 = A242703 and 1/2.
The regular period 6 continued fraction of sqrt(13)/2 is [1; 1, 4, 14, 4, 1, 2], and the convergents are given in A295331/A295332.
Essentially the same as A223139, A209927, A098316 and A085550. - R. J. Mathar, Nov 23 2017

			1.8027756377319946465596106337352479731256482869226231063552265281135834741465...

Index entries for algebraic numbers, degree 2.

Cf. A010470, A242703, A295331/A295332.

RealDigits[Sqrt@13/2, 10, 111][[1]] (* Robert G. Wilson v, Nov 20 2017 *)

sqrt(13)/2 \\ Michel Marcus, Nov 21 2017

A052533 Expansion of (1-x)/(1-x-3*x^2).

Original entry on oeis.org

1, 0, 3, 3, 12, 21, 57, 120, 291, 651, 1524, 3477, 8049, 18480, 42627, 98067, 225948, 520149, 1197993, 2758440, 6352419, 14627739, 33684996, 77568213, 178623201, 411327840, 947197443, 2181180963, 5022773292, 11566316181, 26634636057

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Form the graph with matrix A=[0,1,1,1;1,1,0,0;1,0,1,0;1,0,0,1]. A052533 counts closed walks of length n at the vertex without loop. - Paul Barry, Oct 02 2004
Let M = [0, sqrt(3); sqrt(3), 1] be a 2 X 2 matrix. Then A052533 = {[M^n](1,1)}. Note also that {[M^n](2,2)} = A006130. - L. Edson Jeffery, Nov 25 2011
Pisano period lengths:  1, 3, 1, 6, 24, 3, 24, 6, 1, 24,120, 6,156, 24, 24, 12, 16, 3, 90, 24, ... - R. J. Mathar, Aug 10 2012
a(n) appears in the formula for powers of the fundamental algebraic number c = (1 + sqrt(13))/2 = A209927 of the quadratic number field Q(sqrt(13)): c^n = a(n) + A006130(n-1), for n >=0, with  A006130(-1) = 0. The formulas given below and in A006130 in terms of S-Chebyshev polynomials are valid also for c^(-n), for n >= 0, with 1/c = (-1 + sqrt(13))/2 = A356033. - Wolfdieter Lang, Nov 26 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 463
Index entries for linear recurrences with constant coefficients, signature (1,3).

Cf. A006130, A049310, A209927, A274977, A356033.

a:=[1,0];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, May 09 2019

I:=[1,0]; [n le 2 select I[n] else Self(n-1)+3*Self(n-2): n in [1..40]]; // G. C. Greubel, May 09 2019

R:=PowerSeriesRing(Integers(), 33); Coefficients(R!( (1-x)/(1-x-3*x^2))); // Marius A. Burtea, Jan 15 2020

spec := [S,{S=Sequence(Prod(Z,Union(Z,Z,Z),Sequence(Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(coeff(series((1-x)/(1-x-3*x^2), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 15 2020

CoefficientList[Series[(1-x)/(1-x-3x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 07 2013 *)
LinearRecurrence[{1,3}, {1,0}, 40] (* G. C. Greubel, May 09 2019 *)

my(x='x+O('x^30)); Vec((1-x)/(1-x-3*x^2)) \\ G. C. Greubel, May 09 2019

[lucas_number1(n+1,1,-3) -lucas_number1(n,1,-3) for n in (0..40)] # G. C. Greubel, May 09 2019

G.f.: (1 - x)/(1 - x - 3*x^2).
a(n) = A006130(n) - A006130(n-1).
a(n) = a(n-1) + 3*a(n-2), with a(0)=1, a(1)=0.
a(n) = Sum_{alpha = RootOf(-1+x+3*x^2)} (1/13)*(-1 + 7*alpha)* alpha^(-n-1).
a(n) = Sum_{k=0..floor(n/2)} C(n-k-1,n-2*k)*3^k. - Paul Barry, Mar 16 2010
If p[1]=0, and p[i]=3, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010
G.f.: (Q(0) -1)*(1-x)/x, where Q(k) = 1 + 3*x^2 + (k+2)*x - x*(k+1 + 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
a(n) = 3^(n/2) * Fibonacci(n-1, 1/sqrt(3)). - G. C. Greubel, Jan 15 2020
From Wolfdieter Lang, Nov 27 2023: (Start)
a(n) = 3*A006130(n-2), with A006130(-2) = 1/3 and A006130(-1) = 0.
a(n) = 3*sqrt(-3)^(n-2)*S(n-2, 1/sqrt(-3)), with the S Chebyshev polynomials (see A049310), valid also for negative indices n, using S(-n, x) = - S(n-2, x), for n>= 2, and S(-1, x) = 0. (End)

More terms from James Sellers, Jun 06 2000

A274977 a(n) = a(n-1) + 3*a(n-2) with n>1, a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 9, 27, 54, 135, 297, 702, 1593, 3699, 8478, 19575, 45009, 103734, 238761, 549963, 1266246, 2916135, 6714873, 15463278, 35607897, 81997731, 188821422, 434814615, 1001278881, 2305722726, 5309559369, 12226727547, 28155405654, 64835588295, 149301805257, 343808570142

Offset: 0

Bruno Berselli, Sep 13 2016

a(n)/a(n+1) converges to 1/A209927 as n approaches infinity.

			Table of similar sequences (not extendable on the left side) where this recurrence can be applied to the first two terms:
----------------------------------------------------------------------
(*)      -  -  1, -1,  2, -1,  5,   2,  17,  23,   74,  143,  365, ...
A052533: -  -  1,  0,  3,  3, 12,  21,  57, 120,  291,  651, 1524, ...
(^)      -  0, 1,  1,  4,  7, 19,  40,  97, 217,  508, 1159, 2683, ...
A006138: -  -  1,  2,  5, 11, 26,  59, 137, 314,  725, 1667, 3842, ...
A105476: -  -  1,  3,  6, 15, 33,  78, 177, 411,  942, 2175, 5001, ...
(^)      0, 1, 1,  4,  7, 19, 40,  97, 217, 508, 1159, 2683, 6160, ...
A105963: -  -  1,  5,  8, 23, 47, 116, 257, 605, 1376, 3191, 7319, ...
A274977: -  -  1,  6,  9, 27, 54, 135, 297, 702, 1593, 3699, 8478, ...
A075118: -  2, 1,  7, 10, 31, 61, 154, 337, 799, 1810, 4207, 9637, ...
----------------------------------------------------------------------
(*) see version A140165.
(^) see A006130 and the signed versions A140167, A182228.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3).

Cf. A006130, A006138, A052533, A075118, A105476, A105963.

a:=[1,6];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Jan 15 2020

[n le 2 select 5*n-4 else Self(n-1)+3*Self(n-2): n in [1..40]];

R:=PowerSeriesRing(Integers(), 32); Coefficients(R!((1 + 5*x)/(1- x-3*x^2))); // Marius A. Burtea, Jan 15 2020

seq(coeff(series((1+5*x)/(1-x-3*x^2), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 15 2020

RecurrenceTable[{a[n]==a[n-1] +3a[n-2], a[0]==1, a[1]==6}, a, {n,0,40}]
Table[Round[Sqrt[3]^(n-1)*(Sqrt[3]*Fibonacci[n+1, 1/Sqrt[3]] + 5*Fibonacci[n, 1/Sqrt[3]])], {n,0,40}] (* G. C. Greubel, Jan 15 2020 *)
LinearRecurrence[{1,3},{1,6},40] (* Harvey P. Dale, Jul 11 2023 *)

v=vector(40); v[1]=1; v[2]=6; for(n=3, #v, v[n]=v[n-1]+3*v[n-2]); v

from sage.combinat.sloane_functions import recur_gen2
a = recur_gen2(1, 6, 1, 3)
[next(a) for n in range(40)]

G.f.: (1 + 5*x)/(1 - x - 3*x^2).
a(n) = ((13 + 11*sqrt(13))*(1 + sqrt(13))^n + (13 - 11*sqrt(13))*(1 - sqrt(13))^n)/(26*2^n).
3*a(n) + a(n+1) =  9*A105476(n+1).
3*a(n) - a(n+1) = 27*A006130(n-3) with n>1, A006130(-1) = 0.
a(n+1) - a(n)   = 27*A105476(n-3) with n>2.
a(n) = 3^((n-1)/2)*( sqrt(3)*Fibonacci(n+1, 1/sqrt(3)) + 5*Fibonacci(n, 1/sqrt(3)) ). - G. C. Greubel, Jan 15 2020
E.g.f.: (1/13)*exp(x/2)*(13*cosh((sqrt(13)*x)/2) + 11*sqrt(13)*sinh((sqrt(13)*x)/2)). - Stefano Spezia, Jan 15 2020

A223139 Decimal expansion of (sqrt(13) - 1)/2.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Apr 02 2013

Apart from a(1) the same as A209927 and A085550. [Joerg Arndt, Sep 17 2013]
Decimal expansion of sqrt(3 - sqrt(3 - sqrt(3 - sqrt(3 - ... )))).
Sequence with a(1) = 2 is decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))) - A209927.
This is the positive root of x^2 + x - 3, and the negative one is -A209927. - Wolfdieter Lang, Aug 29 2022

			1.3027756377319946465...

B. Sury, Nothing Lucky about 13, Mathematics Magazine, Vol. 83, No. 4 (October 2010), pp. 289-293.

Cf. A098316, A209927, A085550, A209927.

CF. A122553 (continued fraction).

Mathematica
```
RealDigits[(Sqrt[13] - 1)/2, 10, 130]
```

(sqrt(13)-1)/2 \\ Altug Alkan, Oct 02 2018

Closed form: (sqrt(13) - 1)/2 = A209927-1 = A098316-2.
sqrt(3 - sqrt(3 - sqrt(3 - sqrt(3 - ... )))) + 1 = sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))). See A209927.
Equals 2*(cos(2*Pi/13) + cos(6*Pi/13) + cos(8*Pi/13)) (see Sury link). - Michel Marcus, Aug 21 2015

A188943 Decimal expansion of (7 + sqrt(13))/6.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 14 2011

Decimal expansion of the shape (= length/width = (7+sqrt(13))/6) of the greater (7/3)-contraction rectangle.
See A188738 for an introduction to lesser and greater r-contraction rectangles, their shapes, and partitioning these rectangles into a sets of squares in a manner that matches the continued fractions of their shapes.
From Wolfdieter Lang, Aug 29 2022: (Start)
This constant t is an element of the quadratic number field Q(sqrt(13)) with (monic) polynomial x^2 - (7/3)*x + 1, and the negative root is -A188942.
The constant t - 1 = (1 + sqrt(13))/6 = A209927/3 has minimal polynomial x^2 - x/3 - 1/3, with negative root -(-1 + sqrt(13))/6 = -A223139/3 = -A356033.
(End)

			1.7675918792439982155198702112450826577085494289742...

Cf. A188738, A188942, A209927, A223139, A356033.

r = 7/3; t = (r + (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

A235162 Decimal expansion of (sqrt(33) + 1) / 2.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 06 2014

Solution of y^2 - y - 8 = 0.
Decimal expansion of sqrt(8 + sqrt(8 + sqrt(8 + sqrt(8 + ... )))).
The sequence with a(1) = 2 is decimal expansion of sqrt(8 - sqrt(8 - sqrt(8 - sqrt(8 - ... )))).
A basis for the integers of the real quadratic number field K(sqrt(33)) is
  <1, omega(33)>, where omega(33) = (1 + sqrt(33))/2. - Wolfdieter Lang, Feb 11 2020

			3.37228132326901432992530573410946465911013222899139618384993873528...

Christopher M. Conrey, Table of n, a(n) for n = 1..10000

Cf. A001622, A010488, A209927, A222132, A222134, A223140.

val = vpa((sqrt(sym(33))+1)/2,10001); list = char(val)-'0'; list = list([1,3:end-1]); % Christopher M. Conrey, Jan 26 2022

Mathematica
```
RealDigits[(1 + Sqrt[33])/2, 10, 130]
```

A356033 Decimal expansion of (-1 + sqrt(13))/6 = A223139/3.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Aug 29 2022

This constant r, an algebraic integer of the quadratic number field Q(13), is the positive root of its monic minimal polynomial x^2 + x/3 - 1/3. The negative root is -(1 + sqrt(13))/6 = -A209927/3 = -(A188943 - 1).

r^n = A052533(-n) + A006130(-(n+1))*r, for n >= 0, with A052533(-n) = 3*sqrt(-3)^(-n-2)*Snx(-n-2,1/sqrt(-3)), and A006130(-(n+1)) = sqrt(-3)^(-(n+1))*Snx(-(n+1), 1/sqrt(-3)), with the S-Chebyshev polynomials (see A049310), with S(-n, x) = -S(n-2, x), for n>=2, and S(-1, x) = 0. - Wolfdieter Lang, Nov 27 2023

			0.4342585459106648821865368779117493243752160956408743687850755...

Cf. A006130, A049310, A052533, A188943, A209927, A223139.

First[RealDigits[x/.N[Last[Solve[3x^2+x-1==0,x]],78]]] (* Stefano Spezia, Aug 29 2022 *)

r = (-1 + sqrt(13))/6 = A223139/3 = 1/A209927.

A209924 Concatenation of the months' names spelled in English, with 1=A, 2=B,..., 26=Z.

Original entry on oeis.org

10, 1, 14, 21, 1, 18, 25, 6, 5, 2, 18, 21, 1, 18, 25, 13, 1, 18, 3, 8, 1, 16, 18, 9, 12, 13, 1, 25, 10, 21, 14, 5, 10, 21, 12, 25, 1, 21, 7, 21, 19, 20, 19, 5, 16, 20, 5, 13, 2, 5, 18, 15, 3, 20, 15, 2, 5, 18, 14, 15, 22, 5, 13, 2, 5, 18, 4, 5, 3, 5, 13, 2, 5, 18

Offset: 1

M. F. Hasler, Mar 15 2012

Cf. A209926 (same for French), A209927 (same for German).

Cf. A031139, A049004, A033986, A089746, A049005, A069195, A103742, A134456, A161376.

LetterNumber/@Table[DateString[{2024,m,1},"MonthName"],{m,12}]//Flatten (* Harvey P. Dale, Jul 28 2024 *)

Vec( Vecsmall( "JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember" ))%32

cp's OEIS Frontend

A006130 a(n) = a(n-1) + 3*a(n-2) for n > 1, a(0) = a(1) = 1.

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A010470 Decimal expansion of square root of 13.

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A295330 Decimal expansion of sqrt(13)/2.

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A052533 Expansion of (1-x)/(1-x-3*x^2).

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A274977 a(n) = a(n-1) + 3*a(n-2) with n>1, a(0)=1, a(1)=6.

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A223139 Decimal expansion of (sqrt(13) - 1)/2.

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