A109466 -id:A109466 - cp's OEIS frontend

A057084 Scaled Chebyshev U-polynomials evaluated at sqrt(2).

1, 8, 56, 384, 2624, 17920, 122368, 835584, 5705728, 38961152, 266043392, 1816657920, 12404916224, 84706066432, 578409201664, 3949625081856, 26969727041536, 184160815677440, 1257528709087232, 8586943147278336

Offset: 0

Wolfdieter Lang, Aug 11 2000

From Kival Ngaokrajang, Dec 14 2014 (Start):
-2*a(n-1) is the irrational part of the integer in Q(sqrt 2) giving the length of a Levy C-curve variant L(n)=(2*(2- sqrt 2))^n at iteration step n. The length of this C-curve is an integer in the real quadratic number field Q(sqrt 2), namely L(n) = A(n)+B(n)*sqrt(2) with A(n) = A084130(n) and B(n) = -2*a(n-1). See the construction rule and the illustration in the links.
The fractal dimension of the Levy C-curve is 2, but for this modified case it is log(4)/log(2 + sqrt 2) = 1.1289527...
(End)
For lim_{n->oo} a(n+1)/a(n) = 2*(2 + sqrt(2)) = 6.82842... see A365823. - Wolfdieter Lang, Nov 15 2023

			The first pairs [A(n),B(n)] determining the length L(n) are : [1, 0], [4, -2], [24, -16], [160, -112], [1088, -768], [7424, -5248], [50688, -35840], [346112, -244736], [2363392, -1671168], [16138240, -11411456], ... _Kival Ngaokrajang_, Dec 14 2014

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=8, q=-8.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=8.
Kival Ngaokrajang, Illustration of construction rule and initial terms
Wikipedia, Lévy C curve
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-8).

Cf. A001109, A002315, A049310, A084130, A109466, A251732, A251733, A365823.

Join[{a=1,b=8},Table[c=8*b-8*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8,-8},{1,8},30] (* Harvey P. Dale, Feb 07 2015 *)

x='x+O('x^50); Vec(1/(1-8*x+8*x^2)) \\ G. C. Greubel, Jul 03 2017

[lucas_number1(n,8,8) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

a(n) = 8*(a(n-1)-a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, 2*sqrt(2))*(2*sqrt(2))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(2*k) = A002315(k)*8^k, a(2*k+1) = A001109(k+1)*8^(k+1).
G.f.: 1/(1-8*x+8*x^2).
a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*8^k. [Philippe Deléham, Oct 28 2008]
Binomial transform of A002315. [Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009]

A015445 Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).

Original entry on oeis.org

1, 1, 10, 19, 109, 280, 1261, 3781, 15130, 49159, 185329, 627760, 2295721, 7945561, 28607050, 100117099, 357580549, 1258634440, 4476859381, 15804569341, 56096303770, 198337427839, 703204161769, 2488241012320, 8817078468241, 31211247579121, 110564953793290

Offset: 0

Olivier Gérard

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, 10*a(n-2) equals the number of 10-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Borowska, L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=1, b=3.
Index entries for linear recurrences with constant coefficients, signature (1,9).

Cf. A015443, A015442, A026595, A128099.

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+9*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

m:=25; S:=series(1/(1-x-9*x^2), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 18 2020

CoefficientList[Series[1/(1-x-9*x^2), {x,0,25}], x] (* or *) LinearRecurrence[{1,9}, {1,1}, 25] (* G. C. Greubel, Apr 30 2017 *)

a(n)=([0,1; 9,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,1,-9) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

a(n) = (((1+sqrt(37))/2)^(n+1) - ((1-sqrt(37))/2)^(n+1))/sqrt(37).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*9^k. - Paul Barry, Jul 20 2004
a(n) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*3^(n-k)/2. - Paul Barry, Aug 28 2005
a(n) = Sum_{k=0..n} A109466(n,k)*(-9)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = (-703*(1/2-sqrt(37)/2)^n + 199*sqrt(37)*(1/2-sqrt(37)/2)^n-333*(1/2+sqrt(37)/2)^n + 171*sqrt(37)*(1/2+sqrt(37)/2)^n)/(74*(5*sqrt(37)-14)). - Alexander R. Povolotsky, Oct 13 2010
a(n) = Sum_{k=1..n+1, k odd} C(n+1,k)*37^((k-1)/2)/2^n. - Vladimir Shevelev, Feb 05 2014
G.f.: 1/(1-x-9*x^2). - Philippe Deléham, Feb 19 2020
a(n) = J(n, 9/2), where J(n,x) are the Jacobsthal polynomials. - G. C. Greubel, Feb 18 2020
E.g.f.: exp(x/2)*(sqrt(37)*cosh(sqrt(37)*x/2) + sinh(sqrt(37)*x/2))/sqrt(37). - Stefano Spezia, Feb 19 2020

Edited by N. J. A. Sloane, Oct 11 2010

A099087 Expansion of 1/(1 - 2x + 2x^2).

Original entry on oeis.org

1, 2, 2, 0, -4, -8, -8, 0, 16, 32, 32, 0, -64, -128, -128, 0, 256, 512, 512, 0, -1024, -2048, -2048, 0, 4096, 8192, 8192, 0, -16384, -32768, -32768, 0, 65536, 131072, 131072, 0, -262144, -524288, -524288, 0, 1048576, 2097152, 2097152, 0, -4194304, -8388608, -8388608, 0, 16777216

Offset: 0

Paul Barry, Sep 24 2004

Yet another variation on A009545.
Row sums of Krawtchouk triangle A098593. Partial sums of e.g.f. exp(x)cos(x), or 2^(n/2)cos(Pi*n/2). See A009116.
Binomial transform of A057077. - R. J. Mathar, Nov 04 2008
Partial sums of A146559. - Philippe Deléham, Dec 01 2008
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012
Also the inverse Catalan transform of A000079. - Arkadiusz Wesolowski, Oct 26 2012

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Karl Dilcher and Maciej Ulas, Divisibility and Arithmetic Properties of a Class of Sparse Polynomials, arXiv:2008.13475 [math.NT], 2020. See Table 1, 1st column, p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-2).

Cf. A009545, A146559.

a:=[1,2];; for n in [3..50] do a[n]:=2*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Mar 16 2019

I:=[1,2]; [n le 2 select I[n] else 2*(Self(n-1) - Self(n-2)): n in [1..50]]; // G. C. Greubel, Mar 16 2019

CoefficientList[Series[1/(1 -2x +2x^2), {x, 0, 50}], x] (* Michael De Vlieger, Dec 24 2015 *)

x='x+O('x^50); Vec(1/(1-2*x+2*x^2)) \\ Altug Alkan, Dec 24 2015

[lucas_number1(n,2,2) for n in range(1, 50)] # Zerinvary Lajos, Apr 23 2009

E.g.f.: exp(x)*(cos(x) + sin(x)).
a(n) = 2^(n/2)*(cos(Pi*n/4) + sin(Pi*n/4)).
a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(n-k, k-i)*binomial(n, i) *(-1)^(k-i).
a(n) = 2*(a(n-1) - a(n-2)).
From R. J. Mathar, Apr 18 2008: (Start)
a(n) = (1-i)^(n-1) + (1+i)^(n-1) where i=sqrt(-1).
a(n) = 2 Sum_{k=0..(n-1)/2} (-1)^k*binomial(n-1,2k) if n>0. (End)
a(n) = Sum_{k=0..n} A109466(n,k)*2^k. - Philippe Deléham, Oct 28 2008
E.g.f.: (cos(x)+sin(x))*exp(x) = G(0); G(k)=1+2*x/(4*k+1-x*(4*k+1)/(2*(2*k+1)+x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
G.f.: U(0) where U(k)= 1 + x*(k+3) - x*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
a(n) = Re((1+i)^n) + Im((1+i)^n) where i = sqrt(-1) = A146559(n) + A009545(n). - Philippe Deléham, Feb 13 2013
a(n) = Sum_{j=0..n} binomial(n, j)*(-1)^binomial(j, 2); this is the case m=2 and z=-1 of f(m,n)(z) = Sum_{j=0..n} binomial(n, j)*z^binomial(j, m). See Dilcher and Ulas. - Michel Marcus, Sep 01 2020

Signs added by N. J. A. Sloane, Nov 14 2006

A106852 Expansion of 1/(1-x(1-3x)).

Original entry on oeis.org

1, 1, -2, -5, 1, 16, 13, -35, -74, 31, 253, 160, -599, -1079, 718, 3955, 1801, -10064, -15467, 14725, 61126, 16951, -166427, -217280, 282001, 933841, 87838, -2713685, -2977199, 5163856, 14095453, -1396115, -43682474, -39494129, 91553293, 210035680, -64624199, -694731239, -500858642

Offset: 0

Paul Barry, May 08 2005

Row sums of Riordan array (1, x*(1-3*x)). In general, Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*r^(n-k) yields the row sums of the Riordan array (1, x(1-r*x)).
Row sums of Riordan array (1/(1+3*x^2), x/(1+3*x^2)). - Paul Barry, Sep 10 2005
See A214733 for a differently signed version of this sequence. - Peter Bala, Nov 21 2016

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

Cf. A214733.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

CoefficientList[Series[1/(1 - x (1 - 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 07 2013 *)
LinearRecurrence[{1,-3},{1,1},40] (* Harvey P. Dale, Apr 02 2016 *)

a(n)=([0,1; -3,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Nov 21 2016

x='x+O('x^30); Vec(1/(1-x+3*x^2)) \\ G. C. Greubel, Jan 14 2018

[lucas_number1(n,1,+3) for n in range(1, 40)] # Zerinvary Lajos, Apr 22 2009

From Paul Barry, Sep 10 2005: (Start)
G.f.: 1/(1-x+3*x^2).
a(n) = 2*sqrt(33)*3^(n/2)*cos((n+1)*arctan(sqrt(11)/11)-pi*n/2)/11.
a(n) = 3^(n/2)(cos(-n*arccot(sqrt(11)/11))-sqrt(11)*sin(-n*arccot(sqrt(11)/11))/11).
a(n) = ((1+sqrt(-11))^(n+1)-(1-sqrt(-11))^(n+1))/(2^(n+1)sqrt(-11)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k, n-k)*3^(n-k) = Sum_{k=0..n} A109466(n,k)*3^(n-k).
a(n) = Sum_{k=0..n} C((n+k)/2, k)*(-3)^((n-k)/2)*(1+(-1)^(n-k))/2.
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)(-3)^k. (End)
a(n) = a(n-1) - 3*a(n-2), a(0)=1, a(1)=1. - Philippe Deléham, Oct 21 2008
G.f.: Q(0)/x -1/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
G.f.: Sum_{n >= 0} x^n * Product_{k = 1..n} (k - 3*x)/(1 + k*x). - Peter Bala, Jul 06 2025

A015442 a(n) = a(n-1) + 7*a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 8, 15, 71, 176, 673, 1905, 6616, 19951, 66263, 205920, 669761, 2111201, 6799528, 21577935, 69174631, 220220176, 704442593, 2245983825, 7177081976, 22898968751, 73138542583, 233431323840, 745401121921, 2379420388801

Offset: 0

Olivier Gérard

One obtains A015523 through a binomial transform, and A197189 by shifting one place left (starting 1,1,8 with offset 0) followed by a binomial transform. - R. J. Mathar, Oct 11 2011
The compositions of n in which each positive integer is colored by one of p different colors are called p-colored compositions of n. For n>=2, 8*a(n-1) equals the number of 8-colored compositions of n, with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011
a(n+1) is the number of compositions (ordered partitions) of n into parts 1 and 2, where there are 7 sorts of part 2. - Joerg Arndt, Jan 16 2024
Pisano period lengths: 1, 3, 8, 6, 4, 24, 1, 6, 24, 12, 60, 24, 12, 3, 8, 6, 288, 24, 120, 12, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook), section 14.8 "Strings with no two consecutive nonzero digits", pp.317-318
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (1,7).

Cf. A015440, A015441, A049310.

I:=[0, 1]; [n le 2 select I[n] else Self(n-1) + 7*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 17 2012

LinearRecurrence[{1, 7}, {0, 1}, 30] (* Vincenzo Librandi, Oct 17 2012 *)
nxt[{a_,b_}]:={b,7a+b}; NestList[nxt,{0,1},30][[All,1]] (* Harvey P. Dale, Feb 25 2022 *)

concat(0,Vec(1/(1-x-7*x^2)+O(x^99))) \\ Charles R Greathouse IV, Mar 12 2014

[lucas_number1(n,1,-7) for n in range(0, 27)] # Zerinvary Lajos, Apr 22 2009

O.g.f.: x/(1-x-7x^2). - R. J. Mathar, May 06 2008
a(n) = ( ((1+sqrt(29))/2)^(n+1) - ((1-sqrt(29))/2)^(n+1) )/sqrt(29).
a(n) = 8*a(n-2) + 7*a(n-3) with characteristic polynomial x^3 - 8*x - 7. - Roger L. Bagula, May 30 2007
a(n+1) = Sum_{k=0..n} A109466(n,k)*(-7)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = (Sum_{1<=k<=n, k odd} C(n,k)*29^((k-1)/2))/2^(n-1). - Vladimir Shevelev, Feb 05 2014
a(n) = sqrt(-7)^(n-1)*S(n-1, 1/sqrt(-7)), with the Chebyshev polynomial S(n, x), and S(-1, x) = 1 (see A049310). - Wolfdieter Lang, Nov 26 2023

A015446 Generalized Fibonacci numbers: a(n) = a(n-1) + 10*a(n-2).

Original entry on oeis.org

1, 1, 11, 21, 131, 341, 1651, 5061, 21571, 72181, 287891, 1009701, 3888611, 13985621, 52871731, 192727941, 721445251, 2648724661, 9863177171, 36350423781, 134982195491, 498486433301, 1848308388211, 6833172721221, 25316256603331, 93647983815541, 346810549848851

Offset: 0

Olivier Gérard

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, 11*a(n-2) equals the number of 11-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

For a(n) = ((1+(4*m+1)^(1/2))^n - (1-(4*m+1)^(1/2))^n)/(2^n*(4*m+1)^(1/2)), a(n)/a(n-1) appears to converge to (1+sqrt(4*m+1))/2. Here with m = 10, the numbers in the sequence are congruent with those of the Fibonacci sequence modulo m-1 = 9. For example, F(8) = 21 (Fibonacci) corresponds to a(8) = 5061 (here) because 2+1 and 5+0+1+6 are congruent. - Maleval Francis, Nov 12 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 10).

Cf. A015447, A015443.

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+10*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

Table[MatrixPower[{{1,2},{5,0}},n][[1]][[1]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
CoefficientList[Series[1/(1-x-10*x^2), {x,0,50}], x] (* G. C. Greubel, Apr 30 2017 *)
LinearRecurrence[{1,10},{1,1},30] (* Harvey P. Dale, Dec 12 2018 *)

a(n)=([1,2;5,0]^n)[1,1] \\ Charles R Greathouse IV, Mar 09 2014

[lucas_number1(n,1,-10) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

a(n) = (((1+sqrt(41))/2)^(n+1) - ((1-sqrt(41))/2)^(n+1))/sqrt(41).
From Paul Barry, Sep 10 2005: (Start)
a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(1+(-1)^(n-k))*10^((n-k)/2)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*10^k. (End)
a(n) is the entry (M^n)1,1 where the matrix M = [1,2;5,0]. - _Simone Severini, Jun 22 2006
a(n) = Sum_{k=0..n} A109466(n,k)*(-10)^(n-k). - Philippe Deléham, Oct 26 2008
G.f.: 1/(1-x-10*x^2). - Colin Barker, Feb 03 2012
a(n) = (Sum_{k=1..n+1, k odd} C(n+1,k)*41^((k-1)/2))/2^n. - Vladimir Shevelev, Feb 05 2014

A106853 Expansion of 1/(1 - x + 4*x^2).

Original entry on oeis.org

1, 1, -3, -7, 5, 33, 13, -119, -171, 305, 989, -231, -4187, -3263, 13485, 26537, -27403, -133551, -23939, 510265, 606021, -1435039, -3859123, 1881033, 17317525, 9793393, -59476707, -98650279, 139256549, 533857665, -23168531, -2158599191, -2065925067

Offset: 0

Paul Barry, May 08 2005

Row sums of Riordan array (1,x(1-4x)). In general, a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*r^(n-k) yields the row sums of the Riordan array (1,x(1-kx)).
For n >= 1, a(n) equals the determinant of the n X n matrix with 2's along the superdiagonal and the subdiagonal, and 1's along the main diagonal, and 0's everywhere else. - John M. Campbell, Jun 04 2011
For n >= 1, |a(n-1)| is the unique odd positive solution x to 4^(n+1) = 15*x^2 + y^2. The value of y is |A272931(n)|. - Jianing Song, Jan 22 2019
Define the sequence u(n) = (u(n-1) + u(n-2))/u(n-3) with u(1) = 1, u(2) = -1, u(3) = 2. Then u(4*n) = 2*(a(n-1)+4*a(n-2))*a(n-1)/(a(n)+a(n-1))/a(n), u(4*n+1) = a(n+1)/a(n), u(4*n+2) = -1, u(4*n+3) = 4*(a(n)+a(n-1))/(a(n)+a(n+1)). For example, a(2) = -3, a(3) = -7 and u(8) = 5/3, u(9) = 7/3, u(10) = -1. - Michael Somos, Oct 24 2023

			G.f. = 1 + x - 3*x^2 - 7*x^3 + 5*x^4 + 33*x^5 + 13*x^6 - 119*x^7 - 171*x^8 + ... - _Michael Somos_, Oct 24 2023

T. D. Noe, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 15.
Index entries for linear recurrences with constant coefficients, signature (1,-4).

Cf. A106852, A272931.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

f:= gfun:-rectoproc({a(n)=a(n-1)-4*a(n-2), a(0)=1,a(1)=1},a(n),remember):
map(f, [$0..100]); # Robert Israel, Jan 15 2018

Join[{a=1,b=1},Table[c=b-4*a;a=b;b=c,{n,80}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
CoefficientList[Series[1/(1-x*(1-4x)),{x,0,40}],x] (* or *) LinearRecurrence[ {1,-4},{1,1},40] (* Harvey P. Dale, May 26 2013 *)
a[ n_] := 2^n * ChebyshevU[n, 1/4]; (* Michael Somos, Oct 24 2023 *)

x='x+O('x^30); Vec(1/(1-x+4*x^2)) \\ G. C. Greubel, Jan 14 2018

{a(n) = 2^n*polchebyshev(n, 2, 1/4)}; /* Michael Somos, Oct 24 2023 */

[lucas_number1(n,1,4) for n in range(1, 36)] # Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1 - x + 4*x^2).
a(n) = 2^n*(cos(2*n*arctan(sqrt(15)/5))+sqrt(15)*sin(2*n*arctan(sqrt(15)/5))/15).
a(n) = ((1 + sqrt(-15))^(n+1) - (1 - sqrt(-15))^(n+1))/(2^(n+1)*sqrt(-15)).
a(n) = Sum_{k=0..n} ((-1)^(n-k)*binomial(k, n-k)*4^(n-k)).
a(n) = a(n-1) - 4*a(n-2), a(0) = 1, a(1) = 1. - Philippe Deléham, Oct 21 2008
a(n) = Sum_{k=0..n} A109466(n,k)*4^(n-k). - Philippe Deléham, Oct 25 2008
G.f.: 1/(1 - 2*x)^2/(1 + 3*x*G(0)/2), where G(k) = 1 + 1/(1 - x/(x + (k + 1)/(2*k + 4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
For n >= 1, 15*A272931(n)^2 + a(n-1)^2 = 4^(n+1). - Jianing Song, Jan 22 2019
a(n) = Product_{k=1..n} (1 + 4*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
a(n) = 2^n * U(n, 1/4), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Mar 28 2022

A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1

Offset: 0

Paul Barry, Apr 29 2005

A skew version of Sierpinski’s triangle A047999. - Johannes W. Meijer, Jun 05 2011
Row sums are A002487(n+1). Diagonal sums are A106345. Inverse is A106346.
Triangle formed by reading T triangle mod 2 with T := A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167. - Philippe Deléham, Dec 18 2008

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 0, 1;

G. C. Greubel, Rows n = 0..50, flattened
Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (1.6) p. 2.

Cf. A047999, A002487.
Cf. A106345 (diagonal sums), A106346 (inverse).
Cf. A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167.

Flat(List([0..15], n-> List([0..n], k-> (Binomial(k,n-k) mod 2) ))); # G. C. Greubel, Feb 07 2020

[ Binomial(k,n-k) mod 2: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 07 2020

seq(seq(`mod`(binomial(k, n-k), 2), k = 0..n), n = 0..15); # G. C. Greubel, Feb 07 2020

Table[Mod[Binomial[k, n-k], 2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 18 2017 *)

T(n,k) = binomial(k,n-k)%2;
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 07 2020

[[ mod(binomial(k,n-k), 2) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Feb 07 2020

A030240 Scaled Chebyshev U-polynomials evaluated at sqrt(7)/2.

Original entry on oeis.org

1, 7, 42, 245, 1421, 8232, 47677, 276115, 1599066, 9260657, 53631137, 310593360, 1798735561, 10416995407, 60327818922, 349375764605, 2023335619781, 11717718986232, 67860683565157, 393000752052475, 2275980479411226, 13180858091511257, 76334143284700217

Offset: 0

Wolfdieter Lang

Binomial transform of A030221. - Philippe Deléham, Nov 19 2009

Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=7, q=-7.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=7.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (7,-7).

Join[{a=1,b=7},Table[c=7*b-7*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)

Vec(1/(1-7*x+7*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

[lucas_number1(n,7,7) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

a(n) = 7*a(n-1)-7*a(n-2), a(-1)=0, a(0)=1; a(n)=sqrt(7)^n*U(n, sqrt(7)/2); G.f.: 1/(1-7*x+7*x^2); a(2*k)=7^k*A030221(k); a(2*k-1)=7^k*A004254(k)

a(n) = Sum_{k=0..n} A109466(n,k)*7^k. - Philippe Deléham, Oct 28 2008

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

Original entry on oeis.org

0, 1, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Labos Elemer, Apr 28 2003

a(n) is the least number x such that gcd(2^x, x-phi(x)) = 2^n. If cototient is replaced by totient, analogous values are different: A053576.

			G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000010, A007283, A009195, A050339, A134059.

Essentially the same as A003945 (and perhaps also A058764).

[0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // Klaus Brockhaus, Dec 02 2009

0,1,seq(3*2^(n-1), n=2..40); # G. C. Greubel, Apr 27 2021

{0}~Join~Map[Total, {{1}}~Join~Table[3 Binomial[n, k], {n, 30}, {k, 0, n}]] (* Michael De Vlieger, Jul 03 2016, after Harvey P. Dale at A134059 *)
Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* G. C. Greubel, Apr 27 2021 *)
Join[{0,1},NestList[2#&,6,30]] (* Harvey P. Dale, Jan 22 2024 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-6*k + 16) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n)=if(n<2,n,3<<(n-1)) \\ Charles R Greathouse IV, Jun 16 2012

[0,1]+[3*2^(n-1) for n in (2..40)] # G. C. Greubel, Apr 27 2021

a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.
G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - Michael Somos, Jun 15 2012
Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - Gary W. Adamson, Nov 18 2007
a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - Philippe Deléham, Oct 30 2008
a(n) = (-6*n + 16) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - Ilya Gutkovskiy, Jul 04 2016
a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - G. C. Greubel, Apr 27 2021

More terms from Klaus Brockhaus, Dec 02 2009

cp's OEIS Frontend

A057084 Scaled Chebyshev U-polynomials evaluated at sqrt(2).

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A015445 Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).

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A099087 Expansion of 1/(1 - 2*x + 2*x^2).

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A106852 Expansion of 1/(1-x*(1-3*x)).

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A015442 a(n) = a(n-1) + 7*a(n-2), a(0)=0, a(1)=1.

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A015446 Generalized Fibonacci numbers: a(n) = a(n-1) + 10*a(n-2).

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A099087 Expansion of 1/(1 - 2x + 2x^2).

A106852 Expansion of 1/(1-x(1-3x)).