A004254 -id:A004254 - cp's OEIS frontend

A158880 Number of spanning trees in C_6 X P_n.

6, 8100, 7741440, 7138643400, 6551815840350, 6009209192448000, 5511006731579419434, 5054037303588059379600, 4634949992739663836897280, 4250612670512943969574312500, 3898145031429828405122837863554

Offset: 1

Alois P. Heinz, Mar 28 2009

A linear divisibility sequence of order 18. - Peter Bala, May 02 2014

Alois P. Heinz, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Index to divisibility sequences

Column 6 of A173958.

Cf. A001353, A003690, A003753, A003733, A158898. A001109, A001906, A004254.

a:= n-> 6* (Matrix(1,18, (i,j)-> -sign(j-10) *[0, 1, 1350, 1290240, 1189773900, 1091969306725, 1001534865408000, 918501121929903239, 842339550598009896600, 772491665456610639482880][1+abs(j-10)]). Matrix(18, (i,j)-> if i=j-1 then 1 elif j=1 then [842608511100, -639641521152, 276457068288, -65829977967, 8292106368, -524839680, 16393554, -232704, 1152, -1][1+abs(i-9)] else 0 fi)^n) [1,10]: seq(a(n), n=1..15);

See program.

a(n) = 6*U(n-1,3/2)^2*U(n-1,5/2)^2*U(n-1,3) = 6*A001906(n)^2*A004254(n)^2*A001109(n), where U(n,x) is a Chebyshev polynomial of the second kind. - Peter Bala, May 02 2014

A164975 Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 8, 8, 8, 5, 15, 25, 20, 16, 8, 30, 55, 70, 48, 32, 13, 56, 125, 175, 184, 112, 64, 21, 104, 262, 440, 512, 464, 256, 128, 34, 189, 539, 1014, 1401, 1416, 1136, 576, 256, 55, 340, 1075, 2270, 3501, 4170, 3760, 2720, 1280, 512

Offset: 1

Mark Dols, Sep 03 2009

A164975 is jointly generated with A209125 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+ 2x*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 05 2012

			Triangle T(n,k), 0 <= k < n, n >= 1, begins:
   1;
   1,   2;
   2,   3,   4;
   3,   8,   8,   8;
   5,  15,  25,  20,  16;
   8,  30,  55,  70,  48,  32;
  13,  56, 125, 175, 184, 112,  64;
  21, 104, 262, 440, 512, 464, 256, 128;
  ...
T(7,1) = 30 + 2*8 + 15 - 5 = 56.
T(6,1) = 15 + 2*5 +  8 - 3 = 30.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A000045, A000079, A000244 (row sums).

A164975 := proc(n,k) option remember; if n <=0 or k > n or k< 1 then 0; elif k= 1 then combinat[fibonacci](n); else procname(n-1,k)+2*procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc: # R. J. Mathar, Jan 27 2011

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209125 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A164975 *)
(* Clark Kimberling, Mar 05 2012 *)
With[{nmax = 10}, Rest[CoefficientList[CoefficientList[Series[ x/(1 - 2*y*x-x-x^2+y*x^2), {x,0,nmax}, {y,0,nmax}], x], y]]//Flatten] (* G. C. Greubel, Jan 14 2018 *)

T(n,n-1) = A000079(n-1).
T(n,n-2) = A001792(n-2). - R. J. Mathar, Jan 27 2011
T(n,1) = A099920(n-1). - R. J. Mathar, Jan 27 2011
G.f.: x/(1-2*y*x-x-x^2+y*x^2). - Philippe Deléham, Mar 21 2012
Sum_{k=0..n-1, n>0} T(n,k)*x^k = A000045(n), A000244(n-1), A004254(n), A186446(n-1), A190980(n) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 21 2012

Corrected by Philippe Deléham, Mar 21 2012

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

Original entry on oeis.org

1, 4, 14, 54, 204, 774, 2934, 11124, 42174, 159894, 606204, 2298294, 8713494, 33035364, 125246574, 474845814, 1800277164, 6825368934, 25876938294, 98106921684, 371951579934, 1410175504854, 5346381254364, 20269670277654, 76848154596054, 291353474621124

Offset: 0

Johannes W. Meijer, Aug 13 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
The sequence above corresponds to 16 A[5] vectors with decimal values between 3 and 384. These vectors lead for the corner squares to A123620 and for the central square to A155116.
This sequence appears among the members of a family of sequences with g.f. (1 + x - k*x^2)/(1 - 3*x + (k-4)*x^2). Berserker sequences that are members of this family are 4*A007482 (k=2; with leading 1 added), A180142 (k=1; this sequence), A000302 (k=0), A180140 (k=-1) and 4*A154964 (k=-2; n>=1 and a(0)=1). Some other members of this family are 2*A180148 (k=3; with leading 1 added), 4*A025192 (k=4; with leading 1 added), 2*A005248 (k=5; with leading 1 added) and A123932 (k=6).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3).

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

with(LinearAlgebra): nmax:=23; m:=2; A[5]:=[0,0,0,0,0,0,0,1,1]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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);
# second Maple program:
a:= n-> ceil((<<0|1>, <3|3>>^n. <<2/3, 4>>)[1,1]):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 14 2021

LinearRecurrence[{3, 3}, {1, 4, 14}, 26] (* Jean-François Alcover, Jan 18 2025 *)

G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).
a(n) = 3*a(n-1) + 3*a(n-2) for n >= 2 with a(0)=1, a(1)=4 and a(2)=14.
a(n) = (6-2*A)*A^(-n-1)/21 + (6-2*B)*B^(-n-1)/21 with A=(-3+sqrt(21))/6 and B=(-3-sqrt(21))/6.
Lim_{k->infinity} a(2*n+k)/a(k) = 2*A000244(n)/(A003501(n) - A004254(n)*sqrt(21)) for n >= 1.
Lim_{k->infinity} a(2*n-1+k)/a(k) = 2*A000244(n)/(A004253(n)*sqrt(21) - 3*A030221(n-1)) for n >= 1.

A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 7, 27, 25, 21, 19, 66, 126, 90, 55, 40, 204, 392, 504, 300, 144, 97, 522, 1363, 1884, 1851, 954, 377, 217, 1425, 4065, 7281, 8011, 6435, 2939, 987, 508, 3642, 12332, 24606, 34044, 31446, 21524, 8850, 2584, 1159, 9441, 35236, 82020, 127830

Offset: 1

Clark Kimberling, Jan 20 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
    1
    1    3
    4    6    8
    7   27   25   21
   19   66  126   90   55
   40  204  392  504  300  144
   97  522 1363 1884 1851  954  377
  217 1425 4065 7281 8011 6435 2939 987
Row 4 represents the polynomial p(4,x) = 7 + 27*x + 25*x^2 + 21*x^3, so (T(4,k)) = (7,27,25,21), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), 1-28, Paper No. A14.

Cf. A006130 (column 1); A001906 (p(n,n-1)); A090017 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004187, (p(n,2)); A004254, (p(n,-2)); A190988, (p(n,3)); A190978 (unsigned), (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151.

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 3 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 3 - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(13 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).

A078368 A Chebyshev S-sequence with Diophantine property.

Original entry on oeis.org

1, 19, 360, 6821, 129239, 2448720, 46396441, 879083659, 16656193080, 315588584861, 5979526919279, 113295422881440, 2146633507828081, 40672741225852099, 770635449783361800, 14601400804658022101

Offset: 0

Wolfdieter Lang, Nov 29 2002

a(n) gives the general (positive integer) solution of the Pell equation b^2 - 357*a^2 =+4 with companion sequence b(n)=A078369(n+1), n>=0.
This is the m=21 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..20 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364, A077412, A078366 and A049660. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 19's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,18}. Milan Janjic, Jan 25 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=19, q=-1.
Tanya Khovanova, Recursive Sequences
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=21.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (19,-1).

a(n) = sqrt((A078369(n+1)^2 - 4)/357), n>=0, (Pell equation d=357, +4).

Cf. A077428, A078355 (Pell +4 equations).

Join[{a=1,b=19},Table[c=19*b-a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{19,-1},{1,19},20] (* Harvey P. Dale, Feb 10 2019 *)

[lucas_number1(n,19,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

a(n) = 19*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n) = (ap^(n+1)-am^(n+1))/(ap-am) with ap = (19+sqrt(357))/2 and am = (19-sqrt(357))/2.
a(n) = S(2*n+1, sqrt(21))/sqrt(21) = S(n, 19); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
G.f.: 1/(1-19*x+x^2).
a(n) = Sum_{k=0..n} A101950(n,k)*18^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/17*(17 + sqrt(357)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/38*(17 + sqrt(357)). - Peter Bala, Dec 23 2012

A083861 Square array T(n,k) of second binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 5, 0, 1, 5, 19, 0, 1, 5, 20, 65, 0, 1, 5, 21, 75, 211, 0, 1, 5, 22, 85, 275, 665, 0, 1, 5, 23, 95, 341, 1000, 2059, 0, 1, 5, 24, 105, 409, 1365, 3625, 6305, 0, 1, 5, 25, 115, 479, 1760, 5461, 13125, 19171, 0, 1, 5, 26, 125, 551, 2185, 7573, 21845, 47500, 58025

Offset: 0

Paul Barry, May 06 2003

Row n >= 0 of the array gives the solution to the recurrence b(k) = 5*b(k-1) + (n - 6)*b(k-2) for k >= 2 with b(0) = 0 and b(1) = 1. The rows are the binomial transforms of the rows of array A083857. The rows are the second binomial transforms of the generalized Fibonacci numbers in array A083856.

			Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  0, 1, 5, 19,  65, 211,  665,  2059,  6305,  19171, ...
  0, 1, 5, 20,  75, 275, 1000,  3625, 13125,  47500, ...
  0, 1, 5, 21,  85, 341, 1365,  5461, 21845,  87381, ...
  0, 1, 5, 22,  95, 409, 1760,  7573, 32585, 140206, ...
  0, 1, 5, 23, 105, 479, 2185,  9967, 45465, 207391, ...
  0, 1, 5, 24, 115, 551, 2640, 12649, 60605, 290376, ...
  0, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, ...
  ...

G. C. Greubel, Antidiagonals n = 0..100, flattened
OEIS, Transformations of integer sequences.

Rows include A001047 (n=0), A093131 (n=1), A002450 (n=2), A004254 (n=5), A000351 (n=6), A052918 (n=7), A015535 (n=8), A015536 (n=9), A015537 (n=10).

Cf. A083856 (second inverse binomial transform), A083856 (first inverse binomial transform), A082297 (main diagonal).

T:= func< n,k | Round( (((5+Sqrt(4*n+1))/2)^k - ((5-Sqrt(4*n+1))/2)^k)/Sqrt(4*n + 1) ) >;
[T(n-k,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 27 2019

seq(seq(round( (((5+sqrt(4*(n-k)+1))/2)^k - ((5-sqrt(4*(n-k)+1))/2)^k)/sqrt(4*(n-k)+1) ), k=0..n), n=0..10); # G. C. Greubel, Dec 27 2019

T[n_, k_]:= Round[(((5 +Sqrt[4*n+1])/2)^k - ((5 -Sqrt[4*n+1])/2)^k)/Sqrt[4*n+1]]; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 27 2019 *)

T(n, k) = round( (((5+sqrt(4*n+1))/2)^k - ((5-sqrt(4*n+1))/2)^k)/sqrt(4*n + 1) );
for(n=0,10, for(k=0,n, print1(T(n-k,k), ", "))) \\ G. C. Greubel, Dec 27 2019

[[round( (((5+sqrt(4*(n-k)+1))/2)^k - ((5-sqrt(4*(n-k)+1))/2)^k)/sqrt(4*(n-k)+1) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 27 2019

T(n, k) = (((5 + sqrt(4*n + 1))/2)^k - ((5 - sqrt(4*n + 1))/2)^k)/sqrt(4*n + 1).
O.g.f. for row n >= 0: -x/(-1 + 5*x + (n-6)*x^2) . - R. J. Mathar, Dec 02 2007
From Petros Hadjicostas, Dec 25 2019: (Start)
T(n,k) = 5*T(n,k-1) + (n - 6)*T(n,k-2) for k >= 2 with T(n,0) = 0 and T(n,1) = 1 for all n >= 0.
T(n,k) = Sum_{i = 0..k} binomial(k,i) * A083857(n,i).
T(n,k) = Sum_{i = 0..k} Sum_{j = 0..i} binomial(k,i) * binomial(i,j) * A083856(n,j). (End)

Name and various sections edited by Petros Hadjicostas, Dec 25 2019

A099025 Expansion of 1 / ((1+x) * (1-5*x+x^2)).

Original entry on oeis.org

1, 4, 20, 95, 456, 2184, 10465, 50140, 240236, 1151039, 5514960, 26423760, 126603841, 606595444, 2906373380, 13925271455, 66719983896, 319674648024, 1531653256225, 7338591633100, 35161304909276, 168467932913279, 807178359657120, 3867423865372320

Offset: 0

Ralf Stephan, Sep 26 2004

			1 + 4*x + 20*x^2 + 95*x^3 + 456*x^4 + 2184*x^5 + 10465*x^6 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Index entries for linear recurrences with constant coefficients, signature (4,4,-1).

First differences of A089927. First differences are in A003769 and A005386. Pairwise sums are in A004254.

I:=[1, 4, 20]; [n le 3 select I[n] else 4*Self(n-1) + 4*Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 31 2017

CoefficientList[Series[1/((1+x)*(1-5*x+x^2)), {x,0,50}], x] (* or *) LinearRecurrence[{4,4,-1}, {1,4,20}, 30] (* G. C. Greubel, Dec 31 2017 *)

Vec(1/(1+x)/(1-5*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = (3 * (-1)^n + 38 * subst( poltchebi(n), x, 5/2) - 8 * subst( poltchebi(n-1), x, 5/2)) / 21} /* Michael Somos, Jan 25 2013 */

a(n) = (1/7)*[A030221(n+2) - A003501(n+2) + (-1)^n].
a(n) = 5*a(n-1) -a(n-2) +(-1)^n, a(0)=1, a(1)=4. - Vincenzo Librandi, Mar 22 2011
G.f.: 1 / ((1 + x) * (1 - 5*x + x^2)).
a(-3-n) = -a(n). - Michael Somos, Jan 25 2013
a(n) = (2^(-n)*(3*(-2)^n+(9-2*sqrt(21))*(5-sqrt(21))^n+(5+sqrt(21))^n*(9+2*sqrt(21))))/21. - Colin Barker, Nov 02 2016

A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.

Original entry on oeis.org

1, 4, -1, 15, -8, 1, 56, -46, 12, -1, 209, -232, 93, -16, 1, 780, -1091, 592, -156, 20, -1, 2911, -4912, 3366, -1200, 235, -24, 1, 10864, -21468, 17784, -8010, 2120, -330, 28, -1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36, -1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 01 2006

The matrices are {4} if n=1, {{4,-1},{-1,4}} if n=2, {{4,-1,0},{-1,4,-1},{0,-1,4}} if n=3 etc. The empty matrix at n=0 has an empty product (determinant) with assigned value =1.

Riordan array (1/(1-4*x+x^2), -x/(1-4*x+x^2)). - Philippe Deléham, Mar 04 2016

			Triangle begins as:
      1;
      4,     -1;
     15,     -8,     1;
     56,    -46,    12,    -1;
    209,   -232,    93,   -16,    1;
    780,  -1091,   592,  -156,   20,   -1;
   2911,  -4912,  3366, -1200,  235,  -24,  1;
  10864, -21468, 17784, -8010, 2120, -330, 28, -1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Joanne Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334.

Cf. A000302, A001353, A001906, A004254, A007070, A123966.

Cf. A139278, A159764, A181983, A207823 (absolute values).

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A124029:= func< n,k | Coefficient(R!( Evaluate(ChebyshevU(n+1), (4-x)/2) ), k) >;
[A124029(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023

A123966x := proc(n,x)
    local A,r,c ;
    A := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
            A[r,c] :=0 ;
        if r = c then
            A[r,c] := A[r,c]+4 ;
        elif abs(r-c)= 1 then
            A[r,c] :=  A[r,c]-1 ;
        end if;
    end do:
    end do:
    (-1)^n*LinearAlgebra[CharacteristicPolynomial](A,x) ;
end proc;
A123966 := proc(n,k)
    coeftayl( A123966x(n,x),x=0,k) ;
end proc:
seq(seq(A123966(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 06 2011

(* Matrix version*)
k = 4;
T[n_, m_, d_]:= If[n==m, k, If[n==m-1 || n==m+1, -1, 0]];
M[d_]:= Table[T[n, m, d], {n,d}, {m,d}];
Table[M[d], {d,10}]
Table[Det[M[d]], {d,10}]
Table[Det[M[d] - x*IdentityMatrix[d]], {d,10}]
Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d,10}]]//Flatten
(* Recursive Polynomial form*)
p[0, x]= 1; p[1, x]= (4-x); p[k_, x_]:= p[k, x]= (4-x)*p[k-1, x] - p[k -2, x];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
(* Additional program *)
Table[CoefficientList[ChebyshevU[n, (4-x)/2], x], {n,0,12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)

def A124029(n,k): return ( chebyshev_U(n, (4-x)/2) ).series(x, n+2).list()[k]
flatten([[A124029(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (4-x)*p(n-1, x) - p(n-2, x), p(0, x) = 1, p(1, x) = 4-x.
From G. C. Greubel, Aug 20 2023: (Start)
T(n, k) = [x^k]( ChebyshevU(n, (4-x)/2) ).
Sum_{k=0..n} T(n, k) = A001906(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A004254(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007070(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000302(n).
T(n, n) = (-1)^n.
T(n, n-1) = 4*A181983(n), n >= 1.
T(n, n-2) = (-1)^n*A139278(n-1), n >= 2.
T(n, 0) = A001353(n+1). (End)

A144109 INVERT transform of the cubes A000578.

Original entry on oeis.org

1, 9, 44, 207, 991, 4752, 22769, 109089, 522676, 2504295, 11998799, 57489696, 275449681, 1319758713, 6323343884, 30296960703, 145161459631, 695510337456, 3332390227649, 15966440800785, 76499813776276, 366532628080599

Offset: 0

R. J. Mathar, Sep 11 2008

Analog of A033453 (INVERT squares) and A001906 (INVERT first powers).

For n>1, a(n-1) is the number of generalized compositions of n when there are i^3 different types of i, (i=1,2,...). [Milan Janjic, Sep 24 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -2, 5, -1).

Cf. A004254, A056594.

CoefficientList[Series[(1 + 4*x + x^2)/((1 + x^2)*(1 -5 *x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 14 2012 *)

G.f.: (1+4*x+x^2)/((1+x^2)*(1-5*x+x^2)).

a(n) = (9*A004254(n+1)-4*A056594(n))/5.

A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.

Original entry on oeis.org

0, 5, 120, 2760, 63365, 1454640, 33393360, 766592645, 17598237480, 403992869400, 9274237758725, 212903475581280, 4887505700610720, 112199727638465285, 2575706229984090840, 59129043561995624040, 1357392295695915262085, 31160893757444055403920

Offset: 1

Paul Weisenhorn, May 24 2009

The ansatz 3*a(n)+1=A^2, 7*a(n)+1=B^2 is equivalent to the Pell equation x^2-21*y^2=1 (see A077232 for d=21), with x=(21*a(n)+5)/2 and y=A*B/2.
The associated A are in A004253, the B in A030221.
Bisection of A089927. - R. J. Mathar, Jul 10 2009

Michael De Vlieger, Table of n, a(n) for n = 1..736
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A089927, A160682, A245031, A334673.

j:=0: for n from 0 to 1000000 do a:=sqrt(3*n+1): b:=sqrt(7*n+1):
if (trunc(a)=a) and (trunc(b)=b) then j:=j+1: print(j,n,a,b): end if:
end do:

LinearRecurrence[{24,-24,1},{0,5,120},30] (* Harvey P. Dale, Dec 17 2013 *)

a(n) = 24*a(n-1) - 24*a(n-2) + a(n-3).
a(n) = (A004253(n)^2 - 1)/3 = (A030221(n)^2 - 1)/7.
a(n) = ((5+w)/2*((23+5*w)/2)^(n-1) + (5-w)/2*((23-5*w)/2)^(n-1) - 5)/21; where w=sqrt(21). [Corrected by Kevin Ryde, Sep 11 2020]
G.f.: 5*x^2/((1-x)*(x^2-23*x+1)). - R. J. Mathar, Jul 10 2009
From Francesca Arici, Sep 12 2020: (Start)
a(n) = 23*a(n-1) - a(n-2) + 5.
a(n) = A004254(n)* A004254(n+1). (End)
a(n) = 5*A334673(n-1). - Hugo Pfoertner, Apr 07 2021

Edited and extended by R. J. Mathar, Jul 10 2009

Name edited by Michel Marcus, Sep 12 2020

cp's OEIS Frontend

A158880 Number of spanning trees in C_6 X P_n.

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A164975 Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.

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A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).

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A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

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A078368 A Chebyshev S-sequence with Diophantine property.

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A083861 Square array T(n,k) of second binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

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A099025 Expansion of 1 / ((1+x) * (1-5*x+x^2)).

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A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

A368152 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 3 - x^2.

A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.