A222132 -id:A222132 - cp's OEIS frontend

A006131 a(n) = a(n-1) + 4*a(n-2), a(0) = a(1) = 1.

1, 1, 5, 9, 29, 65, 181, 441, 1165, 2929, 7589, 19305, 49661, 126881, 325525, 833049, 2135149, 5467345, 14007941, 35877321, 91909085, 235418369, 603054709, 1544728185, 3956947021, 10135859761, 25963647845, 66507086889, 170361678269

Offset: 0

N. J. A. Sloane

Length-n words with letters {0,1,2,3,4} where no two consecutive letters are nonzero, see fxtbook link below. - Joerg Arndt, Apr 08 2011
Equals INVERTi transform of A063727: (1, 2, 8, 24, 80, 256, 832, ...). - Gary W. Adamson, Aug 12 2010
a(n) is equal to the permanent of the n X n Hessenberg matrix with 1's along the main diagonal, 2's along the superdiagonal and the subdiagonal, and 0's everywhere else. - John M. Campbell, Jun 09 2011
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, 5*a(n-2) equals the number of 5-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
Pisano period lengths: 1, 1, 8, 1, 6, 8, 48, 2, 24, 6,120, 8, 12, 48, 24, 4,136, 24, 18, 6, ... - R. J. Mathar, Aug 10 2012
This is one of only two Lucas-type sequences whose 8th term is a square. The other one is A097705. - Michel Marcus, Dec 07 2012
Numerators of stationary probabilities for the M2/M/1 queue. In this queue, customers arrives in groups of 2. Intensity of arrival = 1. Service rate = 4. There is only one server and an infinite queue. - Igor Kleiner, Nov 02 2018
Number of 4-compositions of n+2 with 1 not allowed as a part; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020
From M. Eren Kesim, May 13 2021: (Start)
a(n) is equal to the number of n-step walks from a universal vertex to another (itself or the other) on the diamond graph. It is also equal to the number of (n+1)-step walks from vertex A to vertex B on the graph below.
    B--C
    | /|
    |/ |
    A--D
(End)
From Wolfdieter Lang, Jan 03 2024: (Start)
This sequence {a(n-1)}, with a(-1) = 0, appears in the formula for powers of phi17 := (1 + sqrt(17))/2 = A222132, the fundamental (integer) algebraic number of Q(sqrt(17)): phi17^n = A052923(n) + a(n-1)*phi17, for n >= 0.
Limit_{n->oo} a(n+1)/a(n) = phi17. (End)

			G.f. = 1 + x + 5*x^2 + 9*x^3 + 29*x^4 + 65*x^5 + 181*x^6 + 441*x^7 + 1165*x^8 + ...

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..1000
Ilya Amburg, Krishna Dasaratha, Laure Flapan, Thomas Garrity, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse, and Matthew Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015.
Joerg Arndt, Matters Computational (The Fxtbook), pp.317-318.
Jolanta Borowska and Lena Łacińska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix", J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for tridiagonal Toeplitz matrices a=1, b=2.
Andrew Bremner and Nikos Tzanakis, Lucas sequences whose 8th term is a square, arXiv:math/0408371 [math.NT], 2004.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 437
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
A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 2, (2015), 35-42.
A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
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 (1,4).
Index entries for sequences related to Chebyshev polynomials.

Cf. A006130, A015440, A026581, A026583, A026597, A026599, A052923, A097705, A102446, A063727, A344236, A344261.

a:=[1,1];; for n in [3..30] do a[n]:=a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+4*Self(n-2): n in [1..40] ]; // Vincenzo Librandi, Aug 19 2011

A006131:=-1/(-1+z+4*z**2); # conjectured by Simon Plouffe in his 1992 dissertation
seq( simplify((2/I)^n*ChebyshevU(n, I/4)), n=0..30); # G. C. Greubel, Dec 26 2019

m = 16; f[n_] = Product[(1 + m*Cos[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 15}]; N[%] (* Roger L. Bagula, Nov 21 2008 *)
a[n_]:=(MatrixPower[{{1,4},{1,0}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{1, 4}, {1, 1}, 29] (* Jean-François Alcover, Sep 25 2017 *)
Table[2^n*Fibonacci[n+1, 1/2], {n,0,30}] (* G. C. Greubel, Dec 26 2019 *)

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

vector(31, n, (2/I)^(n-1)*polchebyshev(n-1, 2, I/4) ) \\ G. C. Greubel, Dec 26 2019

def A006131_list(n):
    list = [1, 1] + [0] * (n - 2)
    for i in range(2, n):
        list[i] = list[i - 1] + 4 * list[i - 2]
    return list
print(A006131_list(29)) # M. Eren Kesim, Jul 19 2021

[lucas_number1(n,1,-4) for n in range(1, 30)] # Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1 - x - 4*x^2).
a(n) = (((1+sqrt(17))/2)^(n+1) - ((1-sqrt(17))/2)^(n+1))/sqrt(17).
a(n+1) = Sum_{k=0..ceiling(n/2)} 4^k*binomial(n-k, k). - Benoit Cloitre, Mar 06 2004
a(n) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*2^(n-k)/2. - Paul Barry, Aug 28 2005
a(n) = A102446(n)/2. - Zerinvary Lajos, Jul 09 2008
a(n) = Sum_{k=0..n} A109466(n,k)*(-4)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = Product_{k=1..floor((n - 1)/2)} (1 + 16*cos(k*Pi/n)^2). - Roger L. Bagula, Nov 21 2008
Limiting ratio a(n+1)/a(n) is (1 + sqrt(17))/2 = 2.561552812... - Roger L. Bagula, Nov 21 2008
The fraction b(n) = a(n)/2^n satisfies b(n) = 1/2 b(n-1) + b(n-2); g.f. 1/(1-x/2-x^2); b(n) = (( (1+sqrt(17))/4 )^(n+1) - ( (1-sqrt(17))/4 )^(n+1))*2/sqrt(17). - Franklin T. Adams-Watters, Nov 30 2009
G.f.: G(0)/(2-x), where G(k) = 1 + 1/(1 - x*(17*k-1)/(x*(17*k+16) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + 4*x)/( x*(4*k+3 + 4*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(k+1 + 4*x)/( x*(k+3/2 + 4*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013
G.f.: 1 / (1 - x / (1 - 4*x / (1 + 4*x))). - Michael Somos, Sep 15 2013
a(n) = (Sum_{1<=k<=n+1, k odd} C(n+1,k)*17^((k-1)/2))/2^n. - Vladimir Shevelev, Feb 05 2014
a(n) = 2^n*Fibonacci(n+1, 1/2) = (2/i)^n*ChebyshevU(n, i/4). - G. C. Greubel, Dec 26 2019
E.g.f.: exp(x/2)*(sqrt(17)*cosh(sqrt(17)*x/2) + sinh(sqrt(17)*x/2))/sqrt(17). - Stefano Spezia, Dec 27 2019
a(n) = A344236(n) + A344261(n). - M. Eren Kesim, May 13 2021
With an initial 0 prepended, the sequence [0, 1, 1, 5, 9, 29, 65, ...] 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 A296938, e = 0 when p = 17, otherwise e = -1. - Peter Bala, Dec 28 2022
a(n) = A052923(n+2)/4. - Wolfdieter Lang, Jan 03 2024
From Peter Bala, Jun 27 2025: (Start)
The following products telescope:
Product_{k >= 0} (1 + 4^k/a(2*k+1)) = 1 + sqrt(17).
Product_{k >= 1} (1 - 4^k/a(2*k+1)) = 1/18 * (1 + sqrt(17)).
Product_{k >= 0} (1 + (-4)^k/a(2*k+1)) = (1/17) * (17 + sqrt(17)).
Product_{k >= 1} (1 - (-4)^k/a(2*k+1)) = (1/18) * (17 + sqrt(17)). (End)

More terms from Roger L. Bagula, Sep 26 2006

A010473 Decimal expansion of square root of 17.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 4 followed by {8} repeated. - Harry J. Smith, Jun 05 2009
The spiral of Theodorus is an agglomeration of right triangles each having a hypotenuse with a length that is the square root of an integer. The original spiral stops at sqrt(17). - Alonso del Arte, Apr 30 2015
The fundamental algebraic (integer) number in the field Q(sqrt(17)) is (1 + sqrt(17))/2 = A222132. - Wolfdieter Lang, Nov 21 2023

			4.123105625617660549821409855974077025147199225373620434398633573...

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 275.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 58.

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

Cf. A040012 (continued fraction), A222132.

RealDigits[N[Sqrt[17], 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(17); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010473.txt", n, " ", d));  \\ Harry J. Smith, Jun 03 2009

A052923 Expansion of (1-x)/(1 - x - 4*x^2).

Original entry on oeis.org

1, 0, 4, 4, 20, 36, 116, 260, 724, 1764, 4660, 11716, 30356, 77220, 198644, 507524, 1302100, 3332196, 8540596, 21869380, 56031764, 143509284, 367636340, 941673476, 2412218836, 6178912740, 15827788084, 40543439044, 103854591380

Offset: 0

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

First differences of A006131.

This sequence {a(n)} appears in the formula for powers of c = (1 + sqrt(17))/2 = A222132, the fundamental (integer) algebraic number of Q(sqrt(17)): c^n = a(n) + A006131(n-1)*c. This is also valid for positive powers of 1/c = (-1 + sqrt(17)) /8. See the formula below and in A006131 in terms of Chebyshev or Fibonacci polynomials. - Wolfdieter Lang, Nov 27 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 908
Index entries for linear recurrences with constant coefficients, signature (1,4).
Index entries for sequences related to Chebyshev polynomials.

Cf. A006131, A026581, A049310, A222132.

a:=[1,0];; for n in [3..30] do a[n]:=a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Oct 16 2019
a := n -> -(2*I)^n*ChebyshevU(n-2, -I/4):
seq(simplify(a(n)), n = 0..28);  # Peter Luschny, Dec 03 2023

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1 -x -4*x^2) )); // G. C. Greubel, Oct 16 2019

spec := [S,{S=Sequence(Prod(Sequence(Z),Z,Union(Z,Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(coeff(series((1-x)/(1 -x -4*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019

LinearRecurrence[{1,4}, {1,0}, 30] (* G. C. Greubel, Oct 16 2019 *)

my(x='x+O('x^30)); Vec((1-x)/(1 -x -4*x^2)) \\ G. C. Greubel, Oct 16 2019

def A052923_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1 -x -4*x^2)).list()
A052923_list(30) # G. C. Greubel, Oct 16 2019

G.f.: (1-x)/(1 - x - 4*x^2).
a(n) = a(n-1) + 4*a(n-2), with a(0)=1, a(1)=0.
a(n) = Sum_{alpha=RootOf(-1+z+4*z^2)} (1/17)*(-1+9*alpha)*alpha^(-1-n).
If p[1]=0, and p[i]=4, ( 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
From Wolfdieter Lang, Nov 27 2023: (Start)
a(n) = 4*A006131(n-2), with A006131(-2) = 1/4 and A006131(-1) = 0.
a(n) = -(-2*i)^n*S(n-2, i/2), with i = sqrt(-1), and the S-Chebyshev polynomials (see A049310). S(-n, x) = -S(n-2, x). The Fibonacci polynomials are F(n, x) = (-i)^(n-1)*S(n-1, i*x). (End)

More terms from James Sellers, Jun 06 2000

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]
```

A222133 Decimal expansion of sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 08 2013

Sequence with a(1) = 2 is the decimal expansion of sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))) = - A222132.

This is the positive root of the minimal polynomial x^2 + x - 4, with negative root -A222132. - Wolfdieter Lang, Dec 10 2022

			1.561552812808830274910704...

Cf. A178255, A082486, A222132.

Mathematica
```
RealDigits[(Sqrt[17] - 1)/2, 10, 130]
```

Closed form: (sqrt(17) - 1)/2 = A178255-2 = A082486-3 = A222132-1.

sqrt(4 - sqrt(4 - sqrt(4 - sqrt(4 - ... )))) + 1 = sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... )))). See A222132.

A277236 Number of strings of length n composed of symbols from the circular list [1,2,3,4] such that adjacent symbols in the string must be adjacent in the list. No runs of length 2 or more are allowed for symbols 1 and 3.

Original entry on oeis.org

1, 4, 10, 26, 66, 170, 434, 1114, 2850, 7306, 18706, 47930, 122754, 314474, 805490, 2063386, 5285346, 13538890, 34680274, 88835834, 227556930, 582900266, 1493127986, 3824729050, 9797240994, 25096157194, 64285121170, 164669749946, 421810234626, 1080489234410, 2767730172914

Offset: 0

Stefan Hollos, Oct 06 2016

To generalize to strings composed of symbols from the circular list [1,2,3,...2m], m>=2, with no runs of 2 or more allowed for symbols 1,3,5,...2m-1, use the same recurrence given below with initial values a(1)=2m, a(2)=5m, see A277237 for the m=3 case.

			For n=3 the 26 strings are 121, 122, 123, 141, 143, 144, 212, 214, 221, 222, 223, 232, 234, 321, 322, 323, 341, 343, 344, 412, 414, 432, 434, 441, 443, 444.
For n=4 the 66 strings are 1212, 1214, 1221, 1222, 1223, 1232, 1234, 1412, 1414, 1432, 1434, 1441, 1443, 1444, 2121, 2122, 2123, 2141, 2143, 2144, 2212, 2214, 2221, 2222, 2223, 2232, 2234, 2321, 2322, 2323, 2341, 2343, 2344, 3212, 3214, 3221, 3222, 3223, 3232, 3234, 3412, 3414, 3432, 3434, 3441, 3443, 3444, 4121, 4122, 4123, 4141, 4143, 4144, 4321, 4322, 4323, 4341, 4343, 4344, 4412, 4414, 4432, 4434, 4441, 4443, 4444.

Index entries for linear recurrences with constant coefficients, signature (1,4).

Cf. A222132 (z1), A277237.

CoefficientList[Series[(1 + 3 x + 2 x^2)/(1 - x - 4 x^2), {x, 0, 30}], x] (* Michael De Vlieger, Oct 07 2016 *)
LinearRecurrence[{1,4},{1,4,10},40] (* Harvey P. Dale, Jul 12 2025 *)

Vec((1+3*z+2*z^2)/(1-z-4*z^2) + O(z^40)) \\ Michel Marcus, Oct 06 2016

G.f.: (1+3*x+2*x^2)/(1-x-4*x^2).
For n>=3, the recurrence is a(n) = a(n-1) + 4*a(n-2), a(1)=4, a(2)=10.
a(n) = ((13+3*sqrt(17))*z1^n-(13-3*sqrt(17))*z2^n)/(4*sqrt(17)) where z1=(1+sqrt(17))/2 and z2=(1-sqrt(17))/2.

A305834 Triangle read by rows: T(0,0)= 1; T(n,k)= T(n-1,k) + 4*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, 4, 1, 8, 1, 12, 16, 1, 16, 48, 1, 20, 96, 64, 1, 24, 160, 256, 1, 28, 240, 640, 256, 1, 32, 336, 1280, 1280, 1, 36, 448, 2240, 3840, 1024, 1, 40, 576, 3584, 8960, 6144, 1, 44, 720, 5376, 17920, 21504, 4096

Offset: 0

Shara Lalo, Jun 11 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013611 ((1+4*x)^n).
The coefficients in the expansion of 1/(1-x-4*x^2) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.5615528128...: A222132 (sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... ))))), when n approaches infinity.

			Triangle begins:
1;
1;
1,  4;
1,  8;
1, 12,   16;
1, 16,   48;
1, 20,   96,    64;
1, 24,  160,   256;
1, 28,  240,   640,    256;
1, 32,  336,  1280,   1280;
1, 36,  448,  2240,   3840,   1024;
1, 40,  576,  3584,   8960,   6144;
1, 44,  720,  5376,  17920,  21504,    4096;
1, 48,  880,  7680,  32256,  57344,   28672;
1, 52, 1056, 10560,  53760, 129024,  114688,   16384;
1, 56, 1248, 14080,  84480, 258048,  344064,  131072;
1, 60, 1456, 18304, 126720, 473088,  860160,  589824,  65536;
1, 64, 1680, 23296, 183040, 811008, 1892352, 1966080, 589824;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 371, 372.

Shara Lalo, Right justified triangle
Shara Lalo, Skew diagonals in triangle A013611

Row sums give A006131.
Cf. A000012 (column 0), A008586 (column 1), A035008 (column 2), A141478 (column 3), A120054 (column 4).
Cf. A013611.
Cf. A222132.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 4 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten

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

Column k is binomial (n + k - 1, k) * 4^k.

cp's OEIS Frontend

A329592 Decimal expansion of sqrt(34 - 2*sqrt(17))/4 = sqrt(9 - A222132)/2.

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A329591 Decimal expansion of sqrt(34 + 2*sqrt(17))/4 = sqrt(8 + A222132)/2.

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A006131 a(n) = a(n-1) + 4*a(n-2), a(0) = a(1) = 1.

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A010473 Decimal expansion of square root of 17.

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Mathematica

PARI

A052923 Expansion of (1-x)/(1 - x - 4*x^2).

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GAP

Magma

Maple

Mathematica

PARI

Sage

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

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MATLAB

Mathematica