A057083 -id:A057083 - cp's OEIS frontend

A015523 a(n) = 3a(n-1) + 5a(n-2), with a(0)=0, a(1)=1.

0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469, 5477472, 22964761, 96281643, 403668734, 1692414417, 7095586921, 29748832848, 124724433149, 522917463687, 2192374556806, 9191710988853, 38537005750589

Offset: 0

Olivier Gérard

From Johannes W. Meijer, Aug 01 2010: (Start)
a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 and 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
For n >= 1, the sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the side squares to A152187 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. 1/(1-3*x-k*x^2). Red king sequences that are members of this family are A007482 (k=2), A015521 (k=4), A015523 (k=5; this sequence), A083858 (k=6), A015524 (k=7) and A015525 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A049072 (k=-4), A057083 (k=-3), A000225 (k=-2), A001906 (k=-1), A000244 (k=0), A006190 (k=1), A030195 (k=3), A099012 (k=9), A015528 (k=10) and A015529 (k=11).
Inverse binomial transform of A052918 (with extra leading 0).
(End)
First differences in A197189. - Bruno Berselli, Oct 11 2011
Pisano period lengths:  1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12, ... - R. J. Mathar, Aug 10 2012
This is the Lucas U(P=3, Q=-5) sequence, and hence for n >= 0, a(n+2)/a(n+1) equals the continued fraction 3 + 5/(3 + 5/(3 + 5/(3 + ... + 5/3))) with n 5's. - Greg Dresden, Oct 06 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A072263, A072264, A152187, A179606, A197189.

[ n eq 1 select 0 else n eq 2 select 1 else 3*Self(n-1)+5*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

Join[{a = 0, b = 1}, Table[c = 3 * b + 5 * a; a = b; b = c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
a[0] := 0; a[1] := 1; a[n_] := a[n] = 3a[n - 1] + 5a[n - 2]; Table[a[n], {n, 0, 49}] (* Alonso del Arte, Jan 16 2011 *)

x='x+O('x^30); concat([0], Vec(x/(1-3*x-5*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,3,-5) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

a(n) = 3*a(n-1) + 5*a(n-2).
From Paul Barry, Jul 20 2004: (Start)
a(n) = ((3/2 + sqrt(29)/2)^n - (3/2 - sqrt(29)/2)^n)/sqrt(29).
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)*5^k*3^(n-2*k-1). (End)
G.f.: x/(1 - 3*x - 5*x^2). - R. J. Mathar, Nov 16 2007
From Johannes W. Meijer, Aug 01 2010: (Start)
Limit_{k->oo} a(n+k)/a(k) = (A072263(n) + a(n)*sqrt(29))/2.
Limit_{n->oo} A072263(n)/a(n) = sqrt(29). (End)
G.f.: G(0)*x/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013
E.g.f.: 2*exp(3*x/2)*sinh(sqrt(29)*x/2)/sqrt(29). - Stefano Spezia, Oct 06 2019

A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j+1).

Original entry on oeis.org

0, 1, 2, 3, 3, 0, -9, -27, -54, -81, -81, 0, 243, 729, 1458, 2187, 2187, 0, -6561, -19683, -39366, -59049, -59049, 0, 177147, 531441, 1062882, 1594323, 1594323, 0, -4782969, -14348907, -28697814, -43046721, -43046721, 0, 129140163, 387420489, 774840978

Offset: 0

N. J. A. Sloane, Oct 20 2000

Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1-M)^n = A057681(n)-a(n)*M+z(n)*M^2, where z(0)=z(1)=0 and, apparently, z(n+2)=A057083(n). - Stanislav Sykora, Jun 10 2012
From Tom Copeland, Nov 09 2014: (Start)
This array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interp. (here t=-2) o.g.f. G(x,t) = x(1-x)/[1+(t-1)x(1-x)] and inverse o.g.f. Ginv(x,t) =  [1-sqrt(1-4x/(1+(1-t)x))]/2 (Cf. A005773 and A091867 and A030528 for more info on this family). (End)
{A057681, A057682, A*}, where A* is A057083 prefixed by two 0's, is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x)} of order 3. For the definitions of {k_i(x)} and the difference analog {K_i (n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Jul 31 2017

			G.f. = x + 2*x^2 + 3*x^3 + 3*x^4 - 9*x^6 - 27*x^7 - 54*x^8 - 81*x^9 + ...
If M^3=1 then (1-M)^6 = A057681(6) - a(6)*M + A057083(4)*M^2 = -18 + 9*M + 9*M^2. - _Stanislav Sykora_, Jun 10 2012

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mark W. Coffey, Reductions of particular hypergeometric functions 3F2 (a, a+1/3, a+2/3; p/3, q/3; +-1), arXiv preprint arXiv:1506.09160 [math.CA], 2015.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3).

Alternating row sums of triangle A030523.

Cf. A000108, A000484, A000748, A005043, A005773, A057083, A057681, A091867.

I:=[0,1,2]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

A057682:=n->add((-1)^j*binomial(n,3*j+1), j=0..floor(n/3)):
seq(A057682(n), n=0..50); # Wesley Ivan Hurt, Nov 11 2014

A[n_] := Array[KroneckerDelta[#1, #2 + 1] - KroneckerDelta[#1, #2] + Sum[KroneckerDelta[#1, #2 -q], {q, n}] &, {n, n}];
Join[{0,1}, Table[(-1)^(n-1)*Total[CoefficientList[ CharacteristicPolynomial[A[(n-1)], x], x]], {n,2,30}]] (* John M. Campbell, Mar 16 2012 *)
Join[{0}, LinearRecurrence[{3,-3}, {1,2}, 40]] (* Jean-François Alcover, Jan 08 2019 *)

{a(n) = sum( j=0, n\3, (-1)^j * binomial(n, 3*j + 1))} /* Michael Somos, May 26 2004 */

{a(n) = if( n<2, n>0, n-=2; polsym(x^2 - 3*x + 3, n)[n + 1])} /* Michael Somos, May 26 2004 */

b=BinaryRecurrenceSequence(3,-3,1,2)
def A057682(n): return 0 if n==0 else b(n-1)
[A057682(n) for n in range(41)] # G. C. Greubel, Jul 14 2023

G.f.: (x - x^2) / (1 - 3*x + 3*x^2).
a(n) = 3*a(n-1) - 3*a(n-2), if n>1.
Starting at 1, the binomial transform of A000484. - Paul Barry, Jul 21 2003
It appears that abs(a(n)) = floor(abs(A000748(n))/3). - John W. Layman, Sep 05 2003
a(n) = ((3+i*sqrt(3))/2)^(n-2) + ((3-i*sqrt(3))/2)^(n-2). - Benoit Cloitre, Oct 27 2003
a(n) = n*3F2(1/3-n/3,2/3-n/3,1-n/3 ; 2/3,4/3 ; 1) for n>=1. - John M. Campbell, Jun 01 2011
Let A(n) be the n X n matrix with -1's along the main diagonal, 1's everywhere above the main diagonal, and 1's along the subdiagonal. Then a(n) equals (-1)^(n-1) times the sum of the coefficients of the characteristic polynomial of A(n-1), for all n>1 (see Mathematica code below). - John M. Campbell, Mar 16 2012
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) - z(n), y(n+1) = y(n) - x(n), z(n+1) = z(n) - y(n). Then a(n) = -y(n). But this recurrence falls into a repetitive cycle of length 6 and multiplicative factor -27, so that a(n) = -27*a(n-6) for any n>6. - Stanislav Sykora, Jun 10 2012
a(n) = A057083(n-1) - A057083(n-2). - R. J. Mathar, Oct 25 2012
G.f.: 3*x - 1/3 + 3*x/(G(0) - 1) where G(k)= 1 + 3*(2*k+3)*x/(2*k+1 - 3*x*(k+2)*(2*k+1)/(3*x*(k+2) + (k+1)/G(k+1)));(continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
G.f.: Q(0,u) -1, where u=x/(1-x), Q(k,u) = 1 - u^2 + (k+2)*u - u*(k+1 - u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
From Vladimir Shevelev, Jul 31 2017: (Start)
For n>=1, a(n) = 2*3^((n-2)/2)*cos(Pi*(n-2)/6);
For n>=2, a(n) = K_1(n) + K_3(n-2);
For m,n>=2, a(n+m) = a(n)*K_1(m) + K_1(n)*a(m) - K_3(n-2)*K_3(m-2), where
K_1 = A057681, K_3 = A057083. (End)

A129339 Main diagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 23, 37, 74, 175, 431, 1024, 2291, 4825, 9650, 18571, 34955, 65536, 124511, 242461, 484922, 989527, 2038103, 4194304, 8565755, 17308657, 34617314, 68703187, 135812051, 268435456, 532087943, 1059392917, 2118785834

Offset: 1

Paul Curtz, May 28 2007

			First seven rows of T are
[ 1 ]
[ 1,  2 ]
[ 1,  2,  4 ]
[ 0,  1,  3,  7 ]
[ 0,  0,  1,  4, 11 ]
[ 0,  0,  0,  1,  5, 16 ]
[ 1,  1,  1,  1,  2,  7, 23 ].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Paul Curtz, Comments on this sequence
Index entries for linear recurrences with constant coefficients, signature (5,-9,6).

Cf. A038504, A131022 (T read by rows), A131023 (first subdiagonal of T), A131024 (row sums of T), A131025 (antidiagonal sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

m:=33; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ M[n, n]: n in [1..m] ]; // Klaus Brockhaus, Jun 10 2007

m:=33; S:=[ [1, 1, 1, 0, 0, 0][(n-1) mod 6 + 1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // Klaus Brockhaus, Jun 17 2007

I:=[1,2,4,7]; [n le 4 select I[n] else 5*Self(n-1)-9*Self(n-2)+6*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 13 2018

a[n_] := 2^(n-2) + 2*3^((n-3)/2)*Sin[n*Pi/6]; a[1]=1; Table[a[n], {n, 1, 33}] (* Jean-François Alcover, Aug 13 2012 *)
CoefficientList[Series[(1 - x)^3 / ((1 - 2 x) (1 - 3 x + 3 x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 13 2018 *)

{m=33; v=concat([1, 2, 4, 7], vector(m-4)); for(n=5, m, v[n]=5*v[n-1]-9*v[n-2]+6*v[n-3]); v} \\ Klaus Brockhaus, Jun 10 2007

G.f.: x*(1-x)^3/((1-2*x)*(1-3*x+3*x^2)). [multiplied by x to match the offset by R. J. Mathar, Jul 22 2009]
a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 7; for n > 4, a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3).
Binomial transform of A088911. - Klaus Brockhaus, Jun 17 2007
a(n+1) = A057083(n)/3+2^(n-1), n > 1. - R. J. Mathar, Jul 22 2009

Edited and extended by Klaus Brockhaus, Jun 10 2007

A057681 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j).

Original entry on oeis.org

1, 1, 1, 0, -3, -9, -18, -27, -27, 0, 81, 243, 486, 729, 729, 0, -2187, -6561, -13122, -19683, -19683, 0, 59049, 177147, 354294, 531441, 531441, 0, -1594323, -4782969, -9565938, -14348907, -14348907, 0, 43046721, 129140163, 258280326, 387420489, 387420489

Offset: 0

N. J. A. Sloane, Oct 20 2000

Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1-M)^n = a(n)-A057682(n)*M+z(n)*M^2, where z(0)=z(1)=0 and, apparently, z(n+2)=A057083(n). - Stanislav Sykora, Jun 10 2012
Pisano period lengths: 1, 3, 1, 6, 24, 3, 6, 12, 1, 24, 60, 6, 12, 6, 24, 24, 96, 3, 18, 24, ... . - R. J. Mathar, Aug 10 2012
{A057681, A057682, A*}, where A* is A057083 prefixed by two 0's, is the difference analog of the trigonometric functions of order 3, {k_1(x), k_2(x), k_3(x)}. For a definition see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jun 25 2017

			If M^3=1 then (1-M)^6 = a(6)-A057682(6)*M+A057083(4)*M^2 = -18+9*M+9*M^2.

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
T. Alden Gassert, Discriminants of simplest 3^n-tic extensions, arXiv:1409.7829 [math.NT], 2014.
Mark W. Coffey, Reductions of particular hypergeometric functions 3F2 (a, a+1/3, a+2/3; p/3, q/3; +-1), arXiv:1506.09160 [math.CA], 2015.
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv:1610.09361 [math.NT], 2016.
Ira Gessel, The Smith College diploma problem.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3).

Cf. A009116, A009545, A057682, A057083, A103312.

a:=[1,1];; for n in [3..40] do a[n]:=3*a[n-1]-3*a[n-2]; od; Concatenation([1],a); # Muniru A Asiru, Oct 24 2018

I:=[1,1]; [1] cat [n le 2 select I[n] else 3*Self(n-1) - 3*Self(n-2): n in [1..40]]; // G. C. Greubel, Oct 23 2018

A057681 := n->add((-1)^j*binomial(n,3*j),j=0..floor(n/3)); seq(A057681(n), n=0..50);
A057681_list := proc(n) local i; series((1+2*exp(3*z/2)*cos(z*sqrt(3/4)))/3, z,n+2): seq(i!*coeff(%,z,i),i=0..n) end: A057681_list(38); # Peter Luschny, Jul 10 2012

Join[{1},LinearRecurrence[{3,-3},{1,1},40]] (* Harvey P. Dale, Aug 19 2014 *)

x='x+O('x^40); Vec((1-x)^2/((1-x)^3+x^3)) \\ G. C. Greubel, Oct 23 2018

From Paul Barry, Feb 26 2004: (Start)
G.f.: (1-x)^2/((1-x)^3+x^3).
a(n) = 0^n/3 + 2*3^((n-2)/2)*cos(Pi*n/6). (End)
From Paul Barry, Feb 27 2004: (Start)
Binomial transform of (1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, ...).
E.g.f.: 2*exp(3x/2)*cos(sqrt(3)*x/2)/3+1/3.
a(n) = (((3+sqrt(-3))/2)^n+((3-sqrt(-3))/2)^n)/3+0^n/3. (End)
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5). - Paul Curtz, Jan 02 2008
Start with x(0)=1,y(0)=0,z(0)=0 and set x(n+1)=x(n)-z(n), y(n+1)=y(n)-x(n),z(n+1)=z(n)-y(n). Then a(n)=x(n). But this recurrence falls into a repetitive cycle of length 6 and multiplicative factor -27, so that a(n) = -27*a(n-6) for any n>6. - Stanislav Sykora, Jun 10 2012
E.g.f.: (1+2*exp(3*z/2)*cos(z*sqrt(3/4)))/3. - Peter Luschny, Jul 10 2012
a(0)=a(1)=a(2)=1, a(n)=3*a(n-1)-3*a(n-2), n>=3. - Wesley Ivan Hurt, Nov 11 2014
For n>=1, a(n) = 2*3^((n-2)/2)*cos(Pi*n/6). - Vladimir Shevelev, Jun 25 2017
a(n+m) = a(n)*a(m)-A057682(n)*A*057083(m)-A*057083(n)*A057682(m), where A*057083 is A057083 prefixed by two 0's. - Vladimir Shevelev, Jun 25 2017

A131713 Period 3: repeat [1, -2, 1].

Original entry on oeis.org

1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1

Offset: 0

Paul Curtz, Sep 14 2007

Second differences of A131534. Binomial transform of 1, -3, 6, -9, 9, 0, ..., A057083 signed.

Nonsimple continued fraction expansion of sqrt(2)-1 = 0.414213562... - R. J. Mathar, Mar 08 2012

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

Cf. A057083, A061347, A131534.

&cat [[1, -2, 1]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([1, -2, 1]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

f[n_] := If[ Mod[n, 3] == 1, -2, 1]; Array[f, 105, 0]
CoefficientList[Series[(1 - x)/(1 + x + x^2), {x, 0, 104}], x]
PadRight[{}, 120, {1,-2,1}] (* Harvey P. Dale, Jan 25 2014 *)

a(n)=[1,-2,1][1+n%3] \\ Jaume Oliver Lafont, Mar 24 2009

a(n)=1-3*(n%3==1) \\ Jaume Oliver Lafont, Mar 24 2009

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^n) = 0^n, b(3^n) = 3 * 0^n - 2, b(p^n) = 1 if p > 3. - Michael Somos, Jan 02 2011
G.f.: (1-x)/(x^2+x+1). - R. J. Mathar, Nov 14 2007
a(n) = 2*cos((2n+1)*Pi/3). - Jaume Oliver Lafont, Nov 23 2008
a(n) = A117188(2*n). - R. J. Mathar, Jun 13 2011
a(n) + a(n-1) + a(n-2) = 0 for n>1, a(n) = a(n-3) for n>2. - Wesley Ivan Hurt, Jul 01 2016
a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*(-3)^k. - Peter Bala, Feb 06 2019
E.g.f.: 2*exp(-x/2)*cos(sqrt(3)*x/2 + Pi/3). - Fabian Pereyra, Oct 17 2024

Corrected and extended by Michael Somos, Jan 02 2011

A038221 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)3^j.

Original entry on oeis.org

1, 3, 3, 9, 18, 9, 27, 81, 81, 27, 81, 324, 486, 324, 81, 243, 1215, 2430, 2430, 1215, 243, 729, 4374, 10935, 14580, 10935, 4374, 729, 2187, 15309, 45927, 76545, 76545, 45927, 15309, 2187, 6561, 52488, 183708, 367416, 459270, 367416, 183708, 52488, 6561

Offset: 0

N. J. A. Sloane

Triangle of coefficients in expansion of (3 + 3x)^n = 3^n (1 +x)^n, where n is a nonnegative integer. (Coefficients in expansion of (1 +x)^n are given in A007318: Pascal's triangle). - Zagros Lalo, Jul 23 2018

			Triangle begins as:
     1;
     3,     3;
     9,    18,      9;
    27,    81,     81,     27;
    81,   324,    486,    324,     81;
   243,  1215,   2430,   2430,   1215,    243;
   729,  4374,  10935,  14580,  10935,   4374,    729;
  2187, 15309,  45927,  76545,  76545,  45927,  15309,  2187;
  6561, 52488, 183708, 367416, 459270, 367416, 183708, 52488, 6561;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Indranil Ghosh, Rows 0..100 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A007318, A304236, A304249.
Cf. A000007, A000400, A027465, A030195, A057083.
Columns k: A000244 (k=0), 3*A027471 (k=1), 3^2*A027472 (k=2), 3^3*A036216 (k=3), 3^4*A036217 (k=4), 3^5*A036219 (k=5), 3^6*A036220 (k=6), 3^7*A036221 (k=7), 3^8*A036222 (k=8), 3^9*A036223 (k=9), 3^10*A172362 (k=10).

Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*3^(i-j)*3^j))); # Muniru A Asiru, Jul 23 2018

a038221 n = a038221_list !! n
a038221_list = concat $ iterate ([3,3] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

[3^n*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 17 2022

(* programs from Zagros Lalo, Jul 23 2018 *)
t[0, 0]=1; t[n_, k_]:= t[n, k]= If[n<0 || k<0, 0, 3 t[n-1, k] + 3 t[n-1, k-1]]; Table[t[n, k], {n,0,10}, {k,0,n}]//Flatten
Table[CoefficientList[Expand[3^n *(1+x)^n], x], {n,0,10}]//Flatten
Table[3^n Binomial[n, k], {n,0,10}, {k,0,n}]//Flatten  (* End *)

def A038221(n,k): return 3^n*binomial(n,k)
flatten([[A038221(n,k) for k in range(n+1)] for n in range(10)]) # G. C. Greubel, Oct 17 2022

G.f.: 1/(1 - 3*x - 3*x*y). - Ilya Gutkovskiy, Apr 21 2017
T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 3 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 23 2018
From G. C. Greubel, Oct 17 2022: (Start)
T(n, k) = T(n, n-k).
T(n, n) = A000244(n).
T(n, n-1) = 3*A027471(n).
T(n, n-2) = 9*A027472(n+1).
T(n, n-3) = 27*A036216(n-3).
T(n, n-4) = 81*A036217(n-4).
T(n, n-5) = 243*A036219(n-5).
Sum_{k=0..n} T(n, k) = A000400(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A030195(n+1), n >= 0.
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A057083(n).
T(n, k) = 3^k * A027465(n, k). (End)

A057086 Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.

Original entry on oeis.org

1, 10, 90, 800, 7100, 63000, 559000, 4960000, 44010000, 390500000, 3464900000, 30744000000, 272791000000, 2420470000000, 21476790000000, 190563200000000, 1690864100000000, 15003009000000000, 133121449000000000, 1181184400000000000, 10480629510000000000

Offset: 0

Wolfdieter Lang, Aug 11 2000

This is the m=10 member of the m-family of sequences S(n,sqrt(m))*(sqrt(m))^n; for S(n,x) see Formula. The m=4..9 instances are A001787, A030191, A030192, A030240, A057084-5 and the m=1..3 signed sequences are A010892, A009545, A057083.

The characteristic roots are rp(m) := (m + sqrt(m*(m-4)))/2 and rm(m) := (m-sqrt(m*(m-4)))/2 and a(n,m)= (rp(m)^(n+1) - rm(m)^(n+1))/(rp(m) - rm(m)) is the Binet form of these m-sequences.

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=10, q=-10.
Wolfdieter 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=10.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,-10).

Cf. A001090, A001787, A030191, A030192, A030240, A049310.

Cf. A009545, A010892, A057080, A057083, A057084, A057085.

[(10)^n*Evaluate(DicksonSecond(n, 1/10), 1): n in [0..30]]; // G. C. Greubel, May 02 2022

Join[{a=1,b=10},Table[c=10*b-10*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 20 2011 *)

Vec(1/(1-10*x+10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

[lucas_number1(n,10,10) for n in range(1, 20)] # Zerinvary Lajos, Apr 26 2009

a(n) = 10*(a(n-1) - a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, sqrt(10))*(sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(2*k) = A057080(k)*10^k, a(2*k+1) = A001090(k)*10^(k+1).
G.f.: 1/(1-10*x+10*x^2).
a(n) = Sum_{k=0..n} A109466(n,k)*10^k. - Philippe Deléham, Oct 28 2008

A306914 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k+x^k).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 0, 0, 1, 5, 10, 9, -4, 0, 1, 6, 15, 20, 9, -8, 0, 1, 7, 21, 35, 34, 0, -8, 0, 1, 8, 28, 56, 70, 48, -27, 0, 0, 1, 9, 36, 84, 126, 125, 48, -81, 16, 0, 1, 10, 45, 120, 210, 252, 200, 0, -162, 32, 0

Offset: 0

Seiichi Manyama, Mar 16 2019

			Square array begins:
   1,  1,    1,    1,   1,    1,    1,    1, ...
   0,  2,    3,    4,   5,    6,    7,    8, ...
   0,  2,    6,   10,  15,   21,   28,   36, ...
   0,  0,    9,   20,  35,   56,   84,  120, ...
   0, -4,    9,   34,  70,  126,  210,  330, ...
   0, -8,    0,   48, 125,  252,  462,  792, ...
   0, -8,  -27,   48, 200,  461,  924, 1716, ...
   0,  0,  -81,    0, 275,  780, 1715, 3432, ...
   0, 16, -162, -164, 275, 1209, 2989, 6434, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-9 give A000007, A099087, A057083, A099589(n+3), A289389(n+4), A306940, (-1)^n * A049018(n), A306941, A306942.

Cf. A039912, A306913, A306915, A307039, A307079, A307394.

A[n_, k_] := SeriesCoefficient[1/((1-x)^k + x^k), {x, 0, n}];
Table[A[n-k+1, k], {n, 0, 11}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Mar 20 2019 *)

A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+k-1,k*j+k-1).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019

A192080 Expansion of 1/((1-x)^6 - x^6).

Original entry on oeis.org

1, 6, 21, 56, 126, 252, 463, 804, 1365, 2366, 4368, 8736, 18565, 40410, 87381, 184604, 379050, 758100, 1486675, 2884776, 5592405, 10919090, 21572460, 43144920, 87087001, 176565486, 357913941, 723002336, 1453179126, 2906358252

Offset: 0

Bruno Berselli, Jun 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Cf. A006090, A049016, A049017, A057079, A057083, A057681.

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), this sequence (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

m:=30; R:=PowerSeriesRing(Integers(),m); Coefficients(R!(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))));

CoefficientList[Series[1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), {x,0,50}], x] (* Vincenzo Librandi, Oct 15 2012 *)
LinearRecurrence[{6,-15,20,-15,6},{1,6,21,56,126},30] (* Harvey P. Dale, Feb 22 2017 *)

makelist(coeff(taylor(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)), x, 0, n), x, n), n, 0, 29);

Vec(1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2))+O(x^99)) \\ Charles R Greathouse IV, Jun 23 2011

def A192080_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^6-x^6) ).list()
A192080_list(51) # G. C. Greubel, Apr 11 2023

a(n) = abs(A006090(n)) = (-1)^n * A006090(n).
G.f.: 1/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)).
From G. C. Greubel, Apr 11 2023: (Start)
a(n) = (2^(n+5) + A010892(n) - 2*A010892(n-1) - 27*(A057083(n) - 2*A057083(n-1)))/6.
a(n) = (2^(n+5) + A057079(n+2) - 27*A057681(n+1))/6. (End)

A129267 Triangle with T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) and T(0,0)=1 .

Original entry on oeis.org

1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -1, -3, -2, 1, 1, 0, -2, -5, -3, 1, 1, 1, 2, -2, -7, -4, 1, 1, 1, 5, 7, -1, -9, -5, 1, 1, 0, 3, 12, 15, 1, -11, -6, 1, 1, -1, -3, 3, 21, 26, 4, -13, -7, 1, 1, -1, -7, -15, -3, 31, 40, 8, -15, -8, 1, 1

Offset: 0

Philippe Deléham, Jun 08 2007

Triangle T(n,k), 0<=k<=n, read by rows given by [1,-1,1,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . Riordan array (1/(1-x+x^2),(x*(1-x))/(1-x+x^2)); inverse array is (1/(1+x),(x/(1+x))*c(x/(1+x))) where c(x)is g.f. of A000108 .

Row sums are ( with the addition of a first row {0}): 0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32,... (see A009545). - Roger L. Bagula, Nov 15 2009

			Triangle begins:
   1;
   1,  1;
   0,  1,   1;
  -1, -1,   1,  1;
  -1, -3,  -2,  1,  1;
   0, -2,  -5, -3,  1,   1;
   1,  2,  -2, -7, -4,   1,   1;
   1,  5,   7, -1, -9,  -5,   1,   1;
   0,  3,  12, 15,  1, -11,  -6,   1,  1;
  -1, -3,   3, 21, 26,   4, -13,  -7,  1, 1;
  -1, -7, -15, -3, 31,  40,   8, -15, -8, 1, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A063967, A167925, A199324, A202503, A202551.

T:= proc(n, k) option remember;
      if k<0 or  k>n  then 0
    elif n=0 and k=0 then 1
    else T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 14 2020

m = {{a, 1}, {-1, 1}}; v[0]:= {0, 1}; v[n_]:= v[n] = m.v[n-1]; Table[CoefficientList[v[n][[1]], a], {n, 0, 10}]//Flatten (* Roger L. Bagula, Nov 15 2009 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0 && k==0, 1, T[n-1, k-1] + T[n-1, k] - T[n-2, k-1] - T[n-2, k] ]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 14 2020 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    else: return T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 14 2020

Sum{k=0..n} T(n,k)*x^k = { (-1)^n*A057093(n), (-1)^n*A057092(n), (-1)^n*A057091(n), (-1)^n*A057090(n), (-1)^n*A057089(n), (-1)^n*A057088(n), (-1)^n*A057087(n), (-1)^n*A030195(n+1), (-1)^n*A002605(n), A039834(n+1), A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n), A057086(n) } for x=-11, -10, ..., 8, 9, respectively .
Sum{k=0..n} T(n,k)*A000045(k) = A100334(n).
Sum{k=0..floor(n/2)} T(n-k,k) = A050935(n+2).
T(n,k)= Sum{j>=0} A109466(n,j)*binomial(j,k).
T(n,k) = (-1)^(n-k)*A199324(n,k) = (-1)^k*A202551(n,k) = A202503(n,n-k). - Philippe Deléham, Mar 26 2013
G.f.: 1/(1-x*y+x^2*y-x+x^2). - R. J. Mathar, Aug 11 2015

Riordan array definition corrected by Ralf Stephan, Jan 02 2014

cp's OEIS Frontend

A015523 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=0, a(1)=1.

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A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j+1).

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A129339 Main diagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.

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A057681 a(n) = Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j).

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A131713 Period 3: repeat [1, -2, 1].

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A038221 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.

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A015523 a(n) = 3a(n-1) + 5a(n-2), with a(0)=0, a(1)=1.

A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j+1).

A057681 a(n) = Sum_{j=0..floor(n/3)} (-1)^jbinomial(n,3j).

A038221 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)3^j.