A006130 -id:A006130 - cp's OEIS frontend

A053404 Expansion of 1/((1+3x)(1-4*x)).

1, 1, 13, 25, 181, 481, 2653, 8425, 40261, 141361, 624493, 2320825, 9814741, 37664641, 155441533, 607417225, 2472715621, 9761722321, 39434309773, 156574977625, 629786694901, 2508686426401, 10066126765213, 40170363882025

Offset: 0

Barry E. Williams, Jan 07 2000

Hankel transform is := 1,12,0,0,0,... - Philippe Deléham, Nov 02 2008

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, 13*a(n-2) equals the number of 13-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (1,12).

Cf. A000045, A001045, A006130, A006131, A015440 - A015443, A015445 - A015447, A053404, A053428, A168579, A350468, A350469.

[((4^(n+1)) - (-3)^(n+1))/7: n in [0..30]]; // G. C. Greubel, Jan 16 2018

seq(simplify(hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48)), n = 1..40); # Peter Bala, Jul 05 2025

CoefficientList[Series[1/((1 + 3 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 06 2014 *)

a(n)=([0,1; 12,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,1,-12) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

a(n) = ((4^(n+1))-(-3)^(n+1))/7.
a(n) = a(n-1) + 12*a(n-2), n > 1; a(0)=1, a(1)=1.
From Paul Barry, Jul 30 2004: (Start)
Convolution of 4^n and (-3)^n.
G.f.: 1/((1+3x)(1-4x)); a(n) = Sum_{k=0..n, 4^k*(-3)^(n-k)} = Sum_{k=0..n, (-3)^k*4^(n-k)}. (End)
a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*(-12)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = (sum_{1<=k<=n+1, k odd} C(n+1,k)*7^(k-1))/2^n. - Vladimir Shevelev, Feb 05 2014
From Peter Bala, Jun 27 2025: (Start)
a(n) = hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48) for n >= 1.
The following products telescope:
Product_{k >= 0} (1 + 12^k/a(2*k+1)) = 8.
Product_{k >= 1} (1 - 12^k/a(2*k+1)) = 4/25.
Product_{k >= 0} (1 + (-12)^k/a(2*k+1)) = 8/7.
Product_{k >= 1} (1 - (-12)^k/a(2*k+1)) = 28/25. (End)

More terms from James Sellers, Feb 02 2000

A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5x^2)/(1-3x-x^2+6*x^3).

Original entry on oeis.org

1, 4, 8, 22, 50, 124, 290, 694, 1628, 3838, 8978, 21004, 48962, 114022, 265004, 615262, 1426658, 3305212, 7650722, 17697430, 40911740, 94528318, 218312114, 503994220, 1163124866, 2683496134, 6189647948, 14273690782

Offset: 0

Johannes W. Meijer, Aug 06 2010, Aug 10 2010

a(n) represents the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
For the central square the 512 elephants lead to 46 different elephant sequences, see the cross-references for examples.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the corner squares to A175654.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences central square [decimal value A[5]]: A000007 [0], A000012 [16], A000045 [1], A011782 [2], A000079 [3], A003945 [42], A099036 [11], A175656 [7], A105476 [69], A168604 [26], A045891 [19], A078057 [21], A151821 [170], A175657 [43], 4*A172481 [15; n>=-1], A175655 [71, this sequence], 4*A026597 [325; n>=-1], A033484 [58], A087447 [27], A175658 [23], A026150 [85], A175661 [171], A036563 [186], A098156 [59], A046717 [341], 2*A001792 [187; n>=1 with a(0)=1], A175659 [343].

Cf. A000244, A006130, A075118, A175654.

I:=[1, 4, 8]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013

with(LinearAlgebra): nmax:=27; m:=5; A[5]:= [0,0,1,0,0,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

CoefficientList[Series[(1 + x - 5 x^2) / (1 - 3 x - x^2 + 6 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{3,1,-6},{1,4,8},40] (* Harvey P. Dale, Dec 25 2024 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;4;8])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=4 and a(2)=8.
a(n) = ((10+8*A)*A^(-n-1) + (10+8*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n)-A006130(n-1)*sqrt(13)).
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 5*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Jan 31 2023

A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 5, 7, 3, 1, 11, 16, 12, 4, 1, 21, 41, 34, 18, 5, 1, 43, 94, 99, 60, 25, 6, 1, 85, 219, 261, 195, 95, 33, 7, 1, 171, 492, 678, 576, 340, 140, 42, 8, 1, 341, 1101, 1692, 1644, 1106, 546, 196, 52, 9, 1

Offset: 0

Wolfdieter Lang, Aug 02 2002

The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(1+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.
Riordan array (1/(1-x-2*x^2), x/(1-x-2*x^2)). - Paul Barry, Mar 15 2005
Subtriangle (obtained by dropping the first column) of the triangle given by (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 19 2013
The number of ternary words of length n having k letters equal 2 and 0,1 avoid runs of odd lengths. - Milan Janjic, Jan 14 2017

			Triangle begins as:
    1;
    1,   1;
    3,   2,   1;
    5,   7,   3,   1;
   11,  16,  12,   4,   1;
   21,  41,  34,  18,   5,   1;
   43,  94,  99,  60,  25,   6,   1;
   85, 219, 261, 195,  95,  33,   7,   1;
  171, 492, 678, 576, 340, 140,  42,   8,   1;
The triangle (0, 1, 2, -2, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  3,  2,  1;
  0,  5,  7,  3,  1;
  0, 11, 16, 12,  4,  1;
  0, 21, 41, 34, 18,  5,  1; - _Philippe Deléham_, Feb 19 2013

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
W. Lang, First 10 rows.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Columns: A001045 (k=0), A073371 (k=1), A073372 (k=2), A073373 (k=3), A073374 (k=4), A073375 (k=5), A073376 (k=6), A073377 (k=7), A073378 (k=8), A073379 (k=9).
Cf. A002605 (row sums), A006130 (diagonal sums), A073399, A073400.
Cf. A000045, A077957.

A073370:= func< n,k | (&+[Binomial(n-j,k)*Binomial(n-k-j,j)*2^j: j in [0..Floor((n-k)/2)]]) >;
[A073370(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 01 2022

# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> (2^n - (-1)^n) / 3); # Peter Luschny, Oct 07 2022

T[n_, k_]:= T[n, k]= Sum[Binomial[n-j,k]*Binomial[n-k-j,j]*2^j, {j,0,Floor[(n- k)/2]}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 01 2022 *)

def A073370(n,k): return binomial(n,k)*sum( 2^j * binomial(2*j,j) * binomial(n-k,2*j)/binomial(n,j) for j in range(1+(n-k)//2))
flatten([[A073370(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 01 2022

T(n, m) = Sum_{k=0..floor((n-m)/2)} binomial(n-k, m)*binomial(n-m-k, k)*2^k, if n > m, else 0.
Sum_{k=0..n} T(n, k) = A002605(n+1).
T(n, m) = (1*(n-m+1)*T(n, m-1) + 2*2*(n+m)*T(n-1, m-1))/((1^2 + 4*2)*m), n >= m >= 1, T(n, 0) = A001045(n+1), n >= 0, else 0.
T(n, m) = (p(m-1, n-m)*1*(n-m+1)*T(n-m+1) + q(m-1, n-m)*2*(n-m+2)*T(n-m))/(m!*9^m), n >= m >= 1, with T(n) = T(n, m=0) = A001045(n+1), else 0; p(k, n) = Sum_{j=0..k} (A(k, j)*n^(k-j) and q(k, n) = Sum_{j=0..k} B(k, j)*n^(k-j), with the number triangles A(k, m) = A073399(k, m) and B(k, m) = A073400(k, m).
G.f.: 1/(1-(1+2*x)*x)^(m+1) = 1/((1+x)*(1-2*x))^(m+1), m >= 0, for column m (without leading zeros).
T(n, 0) = A001045(n), T(1, 1) = 1, T(n, k) = 0 if k>n, T(n, k) = T(n-1, k-1) + 2*T(n-2, k) + T(n-1, k) otherwise. - Paul Barry, Mar 15 2005
G.f.: (1+x)*(1-2*x)/(1-x-2*x^2-x*y) for the triangle including the 1, 0, 0, 0, 0, ... column. - R. J. Mathar, Aug 11 2015
From Peter Bala, Oct 07 2019: (Start)
Recurrence for row polynomials: R(n,x) = (1 + x)*R(n-1,x) + 2*R(n-2,x) with R(0,x) = 1 and R(1,x) = 1 + x.
The row reverse polynomial x^n*R(n,1/x) is equal to the numerator polynomial of the finite continued fraction 1 + x/(1 - 2*x/(1 + ... + x/(1 - 2*x/(1)))) (with 2*n partial numerators). Cf. A110441. (End)
From G. C. Greubel, Oct 01 2022: (Start)
T(n, k) = binomial(n,k)*Sum_{j=0..floor((n-k)/2)} 2^j*binomial(2*j, j)*binomial(n-k, 2*j)/binomial(n, j).
T(n, k) = binomial(n, k)*Hypergeometric2F1([(k-n)/2, (k-n+1)/2], [-2*n], -8).
Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006130(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000045(n+1). (End)

A091914 a(n) = 2a(n-1) + 12a(n-2).

Original entry on oeis.org

1, 2, 16, 56, 304, 1280, 6208, 27776, 130048, 593408, 2747392, 12615680, 58200064, 267788288, 1233977344, 5681414144, 26170556416, 120518082560, 555082842112, 2556382674944, 11773759455232, 54224111009792, 249733335482368

Offset: 0

Paul Barry, Feb 12 2004

Binomial transform of 1, 1, 13, 13, 169, 169, ....

The inverse binomial transform of 2^n*c(n), where c(n) is the solution to c(n) = c(n-1) + k*c(n-2), a(0)=1, a(1)=1 is 1, 1, 4k+1, 4k+1, (4k+1)^2, ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,12).

Cf. A003683, A063727.

a := [1,2];; for n in [3..30] do a[n] := 2*a[n-1] + 12*a[n-2]; od; a; # Muniru A Asiru, Jan 31 2018

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/(1-2*x-12*x^2))) // G. C. Greubel, Jan 30 2018

a := proc(n) option remember: if n=0 then 1 elif n=1 then 2 elif n>=2 then 2*procname(n-1) + 12*procname(n-2) fi; end: # Muniru A Asiru, Jan 31 2018

LinearRecurrence[{2,12},{1,2},30] (* or *) With[{s=Sqrt[13]},Table[ Simplify[ -(((13+s)((1-s)^n-(1+s)^n))/(26(1+s)))],{n,30}]] (* Harvey P. Dale, May 25 2013 *)

my(x='x+O('x^30)); Vec(1/(1-2*x-12*x^2)) \\ G. C. Greubel, Jan 30 2018

[lucas_number1(n,2,-12) for n in range(1, 30)] # Zerinvary Lajos, Apr 22 2009

a(n) = A000079(n)*A006130(n).
G.f.: 1/(1-2*x-12*x^2).
a(n) = ((1+sqrt(13))*(1+sqrt(13))^n - (1-sqrt(13))*(1-sqrt(13))^n) /(2*sqrt(13)).
a(n) = Sum_{k=0..floor(n/2)} C(n+1,2*k+1) * 13^k. - Paul Barry, Jan 15 2007

A052924 Expansion of g.f.: (1-x)/(1 - 3*x - x^2).

Original entry on oeis.org

1, 2, 7, 23, 76, 251, 829, 2738, 9043, 29867, 98644, 325799, 1076041, 3553922, 11737807, 38767343, 128039836, 422886851, 1396700389, 4612988018, 15235664443, 50319981347, 166195608484, 548906806799, 1812916028881

Offset: 0

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

Euler encountered this sequence when finding the largest root of z^2 - 3z - 1 = 0. - V. Frederick Rickey (fred-rickey(AT)usma.edu), Aug 20 2003
Let M = a triangle with the Pell series A000129 (1, 2, 5, 12, ...) in each column, with the leftmost column shifted upwards one row. A052924 starting (1, 2, 7, 23, ...) = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 31 2010
a(n) is the number of compositions of n when there are 2 types of 1 and 3 types of other natural numbers. - Milan Janjic, Aug 13 2010
Equals partial sums of A108300 prefaced with a 1: (1, 1, 5, 16, 53, 175, 578, ...). - Gary W. Adamson, Feb 15 2012

L. Euler, Introductio in analysin infinitorum, 1748, section 338. English translation by John D. Blanton, Introduction to Analysis of the Infinite, 1988, Springer, p. 286.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sergio Falcón, The k-Fibonacci difference sequences, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153-157.
Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 909
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,1).

A108300 (first differences), A006190 (partial sums), A355981 (primes).

Cf. A000032, A001077, A078343, A164581, A179237, A180148, A329723.

a:=[1,2];; for n in [3..30] do a[n]:=3*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Jun 09 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x-x^2) )); // G. C. Greubel, Jun 09 2019

spec:= [S,{S=Sequence(Prod(Sequence(Z),Union(Z,Z,Prod(Z,Z))))}, unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);
seq(coeff(series((1-x)/(1-3*x-x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019

CoefficientList[Series[(1-x)/(1-3*x-x^2), {x,0,30}], x] (* G. C. Greubel, Jun 09 2019 *)

Vec((1-x)/(1-3*x-x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011

((1-x)/(1-3*x-x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 09 2019

a(n) = 3*a(n-1) + a(n-2).
a(n) = Sum_{alpha=RootOf(-1+3*x+x^2)} (1/13)*(1+5*alpha)*alpha^(-1-n).
With offset 1: a(1)=1; for n > 1, a(n) = Sum_{i=1..3*n-4} a(ceiling(i/3)). - Benoit Cloitre, Jan 04 2004
Binomial transform of A006130. a(n) = (1/2 - sqrt(13)/26)*(3/2 - sqrt(13)/2)^n + (1/2 + sqrt(13)/26)*(3/2 + sqrt(13)/2)^n. - Paul Barry, Jul 20 2004
From Creighton Dement, Nov 04 2004: (Start)
a(n) = A006190(n+1) - A006190(n);
4*a(n) = 9*A006190(n+1) - A006497(n+1) - 2*A003688(n+1). (End)
Numerators in continued fraction [1, 2, 3, 3, 3, ...], where the latter = 0.69722436226...; the length of an inradius of a right triangle with legs 2 and 3. - Gary W. Adamson, Dec 19 2007
If p[1]=2, 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-1) = det A. - Milan Janjic, Apr 29 2010
a(n) = A006190(n) + A003688(n). - R. J. Mathar, Jul 06 2012
a(n) = Sum_{k=0..n-2} A168561(n-2,k)*3^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - R. J. Mathar, Feb 14 2024
From Peter Bala, Jul 08 2025: (Start)
The following series telescope:
Sum_{n >= 1} 1/(a(n) + 3*(-1)^(n+1)/a(n)) = 1/2, since 1/(a(n) + 3*(-1)^(n+1)/a(n)) = b(n) - b(n+1), where b(n) = (1/3) * (a(n) + a(n-1)) / (a(n)*a(n-1)).
Sum_{n >= 1} (-1)^(n+1)/(a(n) + 3*(-1)^(n+1)/a(n)) = 1/6, since 1/(a(n) + 3*(-1)^(n+1)/a(n)) = c(n) + c(n+1), where c(n) = (1/3) * (a(n) - a(n-1)) / (a(n)*a(n-1)). (End)

More terms from James Sellers, Jun 06 2000

A006138 a(n) = a(n-1) + 3*a(n-2).

Original entry on oeis.org

1, 2, 5, 11, 26, 59, 137, 314, 725, 1667, 3842, 8843, 20369, 46898, 108005, 248699, 572714, 1318811, 3036953, 6993386, 16104245, 37084403, 85397138, 196650347, 452841761, 1042792802, 2401318085, 5529696491, 12733650746, 29322740219, 67523692457, 155491913114

Offset: 0

N. J. A. Sloane

The binomial transform of a(n) is b(n) = A006190(n+1), which satisfies b(n) = 3*b(n-1) + b(n-2). - Paul Barry, May 21 2006
Partial sums of A105476. - Paul Barry, Feb 02 2007
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 69, 261, 321 and 324, lead to this sequence. For the central square these vectors lead to the companion sequence A105476 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Equals the INVERTi transform of A063782: (1, 3, 10, 32, 104, ...). - Gary W. Adamson, Aug 14 2010
Pisano period lengths: 1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, ... - R. J. Mathar, Aug 10 2012
The sequence is the INVERT transform of A016116: (1, 1, 2, 2, 4, 4, 8, 8, ...). - Gary W. Adamson, Jul 17 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
N. T. Gridgeman, A new look at Fibonacci generalization, Fib,. Quart., 11 (1973), 40-55.
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
Index entries for linear recurrences with constant coefficients, signature (1,3).

Cf. A063782. - Gary W. Adamson, Aug 14 2010

Cf. A006130, A006190, A016116, A105476, A122950, A175654, A274977.

a:=[1,2];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Nov 19 2019

[n le 2 select n else Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Sep 15 2016

A006138:=-(1+z)/(-1+z+3*z**2); # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(1+z)/(1-z-3*z^2), {z,0,40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
Table[(I*Sqrt[3])^(n-1)*(I*Sqrt[3]*ChebyshevU[n, 1/(2*I*Sqrt[3])] + ChebyshevU[n-1, 1/(2*I*Sqrt[3])]), {n, 0, 40}]//Simplify (* G. C. Greubel, Nov 19 2019 *)
LinearRecurrence[{1,3},{1,2},40] (* Harvey P. Dale, May 29 2025 *)

main(size)={my(v=vector(size),i);v[1]=1;v[2]=2;for(i=3,size,v[i]=v[i-1]+3*v[i-2]);return(v);} /* Anders Hellström, Jul 17 2015 */

def A006138_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x)/(1-x-3*x^2)).list()
A006138_list(40) # G. C. Greubel, Nov 19 2019

a(n) = Sum_{k=0..n+1} A122950(n+1,k)*2^(n+1-k). - Philippe Deléham, Jan 04 2008
G.f.: (1+x)/(1-x-3*x^2). - Paul Barry, May 21 2006
a(n) = Sum_{k=0..n} C(floor((2n-k)/2),n-k)*3^floor(k/2). - Paul Barry, Feb 02 2007
a(n) = A006130(n) + A006130(n-1). - R. J. Mathar, Jun 22 2011
a(n) = (i*sqrt(3))^(n-1)*(i*sqrt(3)*ChebyshevU(n, 1/(2*i*sqrt(3))) + ChebyshevU(n-1, 1/(2*i*sqrt(3)))), where i=sqrt(-1). - G. C. Greubel, Nov 19 2019

Typo in formula corrected by Johannes W. Meijer, Aug 15 2010

A111006 Another version of Fibonacci-Pascal triangle A037027.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144

Offset: 0

Philippe Deléham, Oct 02 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.
Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, ...).
Triangle read by rows: T(n,n-k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)). - Philippe Deléham, Feb 17 2014
Diagonal sums are A013979(n). - Philippe Deléham, Feb 17 2014
T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and 1 X 2 tiles. - Emeric Deutsch, Aug 14 2014

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 1, 5,  5;
  0, 0, 0, 3, 10,  8;
  0, 0, 0, 1,  9, 20, 13;
  0, 0, 0, 0,  4, 22, 38,  21;
  0, 0, 0, 0,  1, 14, 51,  71,  34;
  0, 0, 0, 0,  0,  5, 40, 111, 130,  55;
  0, 0, 0, 0,  0,  1, 20, 105, 233, 235,  89;
  0, 0, 0, 0,  0,  0,  6,  65, 256, 474, 420, 144;

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000045, A000129, A001045, A037027, A038112, A038149, A084938, A128100 (reversed version).

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A114197, A162741, A228074.

a111006 n k = a111006_tabl !! n !! k
a111006_row n = a111006_tabl !! n
a111006_tabl =  map fst $ iterate (\(us, vs) ->
   (vs, zipWith (+) (zipWith (+) ([0] ++ us ++ [0]) ([0,0] ++ us))
                    ([0] ++ vs))) ([1], [0,1])
-- Reinhard Zumkeller, Aug 15 2013

T(0, 0) = 1, T(n, k) = 0 for k < 0 or for n < k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).
T(n, k) = A037027(k, n-k). T(n, n) = A000045(n+1). T(3n, 2n) = (n+1)*A001002(n+1) = A038112(n).
G.f.: 1/(1-yx(1-x)-x^2*y^2). - Paul Barry, Oct 04 2005
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 02 2006
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 respectively. - Philippe Deléham, Feb 17 2014

A209927 Decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))).

Original entry on oeis.org

2, 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

Alonso del Arte, Mar 17 2012

The number x given by the infinitely nested radical for n = 3 is such that x^2 = x + 3, bearing some similarity to the golden ratio phi with its property that phi^2 = phi + 1. Also, 3/x = x - 1.
The mentioned polynomial x^2 - x - 3 has the present number as positive root, and the negative one is -A223139. - Wolfdieter Lang, Aug 29 2022
It is the spectral radius of the bull-graph (see Seeger and Sossa, 2023). - Stefano Spezia, Sep 19 2023
c^n = A006130(n) + A006130(n-1) * d, where c = (1 + sqrt(13))/2 and d = (-1 + sqrt(13))/2. - Gary W. Adamson, Nov 25 2023
c^n = A052533(n) + A006130(n-1)*c, with A006130(-1) = 0. This is also valid for powers of 1/c = A356033, with A052533 and A006130 given there in terms of S-Chebyshev polynomials (A049310), used for negative indices. - Wolfdieter Lang, Nov 26 2023

			2.30277563773...

Alberto Seeger and David Sossa, Infinite families of connected graphs with equal spectral radius, Australas. J. Combin. 87 (2) (2023), 258-276. See p. 274.
Index entries for algebraic numbers, degree 2.

Cf. A157532 (continued fraction), A001622, A010470, A085550, A098316, A188943, A223139.

Cf. A006130, A049310, A052533, A209927.

Digits:=140:
evalf((sqrt(13)+1)/2);  # Alois P. Heinz, Sep 19 2023

RealDigits[(1 + Sqrt[13])/2, 10, 130][[1]]
RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[3, {n, 168}]], 10, 111][[1]] (* Robert G. Wilson v, Oct 02 2018 *)

(sqrt(13)+1)/2 \\ Altug Alkan, Oct 03 2018

Closed form: (sqrt(13) + 1)/2 = A098316-1 = A085550+2 = 3*(A188943-1).

A228683 T(n,k)=Number of nXk binary arrays with no two ones adjacent horizontally, diagonally or antidiagonally.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 8, 19, 17, 16, 13, 40, 77, 41, 32, 21, 97, 216, 313, 99, 64, 34, 217, 809, 1152, 1277, 239, 128, 55, 508, 2529, 6737, 6160, 5215, 577, 256, 89, 1159, 8832, 28977, 56549, 32928, 21305, 1393, 512, 144, 2683, 28793, 152048, 333517, 475809, 176032

Offset: 1

R. H. Hardin Aug 30 2013

Table starts
...2....3......5.......8........13.........21...........34............55
...4....7.....19......40........97........217..........508..........1159
...8...17.....77.....216.......809.......2529.........8832.........28793
..16...41....313....1152......6737......28977.......152048........699833
..32...99...1277....6160.....56549.....333517......2644336......17124415
..64..239...5215...32928....475809....3837761.....46125216.....419022831
.128..577..21305..176032...4008817...44171841....806190208...10258304689
.256.1393..87049..941056..33795201..508425617..14105294112..251170142257
.512.3363.355685.5030848.284980061.5852202757.246929287360.6150224353031

			Some solutions for n=4 k=4
..0..0..0..1....1..0..0..0....0..0..1..0....0..0..1..0....1..0..0..0
..0..1..0..1....1..0..0..0....0..0..1..0....0..0..1..0....1..0..0..0
..0..0..0..0....0..0..0..1....1..0..1..0....0..0..1..0....1..0..0..0
..0..0..0..1....0..1..0..1....1..0..0..0....0..0..1..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1057

Column 1 is A000079
Column 2 is A001333(n+1)
Diagonal is A067963
Row 1 is A000045(n+2)
Row 2 is A006130(n+1)

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 2*a(n-1) +a(n-2)
k=3: a(n) = 5*a(n-1) -3*a(n-2) -3*a(n-3)
k=4: a(n) = 6*a(n-1) -2*a(n-2) -8*a(n-3)
k=5: a(n) = 12*a(n-1) -27*a(n-2) -32*a(n-3) +49*a(n-4) +20*a(n-5) -5*a(n-6)
k=6: [order 7]
k=7: [order 12]
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-2)
n=2: a(n) = a(n-1) +3*a(n-2)
n=3: a(n) = 2*a(n-1) +6*a(n-2) -5*a(n-3)
n=4: a(n) = 2*a(n-1) +16*a(n-2) -7*a(n-3) -18*a(n-4)
n=5: [order 7]
n=6: [order 10]
n=7: [order 16]

A084386 Number of pairs of rabbits when there are 3 pairs per litter and offspring reach parenthood after 3 gestation periods; a(n) = a(n-1) + 3*a(n-3), with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 4, 7, 10, 22, 43, 73, 139, 268, 487, 904, 1708, 3169, 5881, 11005, 20512, 38155, 71170, 132706, 247171, 460681, 858799, 1600312, 2982355, 5558752, 10359688, 19306753, 35983009, 67062073, 124982332, 232931359, 434117578, 809064574

Offset: 0

Merrill Jensen (mpjensen(AT)mninter.net), Jun 23 2003

This comment covers an infinite family of growth sequences, where a(n) = a(n-1) + k*a(n-m). k is number of pairs per litter and m is periods until adulthood. G.f. = 1/(1-x-k*x^m). For example, A000930 has k=1 and m=3 while A006130 has k=3 and m=2.
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 >= 3, 4*a(n-3) equals the number of 4-colored compositions of n with all parts >= 3, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
a(n+2) equals the number of words of length n on alphabet {0,1,2,3}, having at least two zeros between every two successive nonzero letters. - Milan Janjic, Feb 07 2015
Number of compositions of n into one sort of part 1 and three sorts of part 3 (see the g.f.). - Joerg Arndt, Feb 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet , J. Int. Seq. 19 (2016) # 16.1.3, Example 9
Merrill Jensen, Generating Functions
Index entries for linear recurrences with constant coefficients, signature (1,0,3).

Partial sums of A052900. Also A052900/3.

Cf. A000930, A001045, A006130.

I:=[1,1,1]; [n le 3 select I[n] else Self(n-1)+3*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 28 2017

seq(add(binomial(n-2*k,k)*3^k,k=0..floor(n/3)),n=0..34); # Zerinvary Lajos, Apr 03 2007

a[0]=a[1]=a[2]=1; a[n_] := a[n]=a[n-1]+3a[n-3]; Table[a[n], {n, 0, 34}]
LinearRecurrence[{1, 0, 3}, {1, 1, 1}, 37] (* Robert G. Wilson v, Jul 12 2014 *)

a(n)=([0,1,0; 0,0,1; 3,0,1]^n*[1;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = a(n-1) + 3*a(n-3). a(n) = A052900(n+3)/3.
G.f.: 1/(1-x-3*x^3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-2*k, k)*3^k. - Paul Barry, Nov 18 2003
G.f.: W(0)/2, where W(k) = 1 + 1/(1 - x*(1 + 3*x^2)/(x*(1 + 3*x^2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
Starting (1 + x + 4*x^2 + ...), is the INVERT transform of (1 + 3*x^2). - Gary W. Adamson, Mar 27 2017
a(m+n) = a(m)*a(n) + 3*a(m-1)*a(n-2) + 3*a(m-2)*a(n-1). - Michael Tulskikh, Jun 23 2020

Edited by Dean Hickerson, Jun 24 2003

Recurrence appended to the name by Antti Karttunen, Mar 28 2017

cp's OEIS Frontend

A053404 Expansion of 1/((1+3*x)*(1-4*x)).

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A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3).

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A073370 Convolution triangle of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0.

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A091914 a(n) = 2*a(n-1) + 12*a(n-2).

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A052924 Expansion of g.f.: (1-x)/(1 - 3*x - x^2).

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A006138 a(n) = a(n-1) + 3*a(n-2).

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A053404 Expansion of 1/((1+3x)(1-4*x)).

A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5x^2)/(1-3x-x^2+6*x^3).

A091914 a(n) = 2a(n-1) + 12a(n-2).