A135929 -id:A135929 - cp's OEIS frontend

A167375 a(n)=3*a(n-1)-a(n-2) with a(0)=1, a(1)=3, a(2)=11.

1, 3, 11, 30, 79, 207, 542, 1419, 3715, 9726, 25463, 66663, 174526, 456915, 1196219, 3131742, 8199007, 21465279, 56196830, 147125211, 385178803, 1008411198, 2640054791, 6911753175, 18095204734, 47373861027, 124026378347, 324705274014, 850089443695

Offset: 0

Jamel Ghanouchi, Nov 02 2009

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

Cf. A135929, A138034.

I:=[1,3,11]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jun 26 2014

Join[{1},LinearRecurrence[{3,-1},{3,11},30]] (* Harvey P. Dale, Jun 25 2014 *)
CoefficientList[Series[(3 x^2 + 1)/(1 - 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 26 2014 *)
Table[3LucasL[2n+1]-Fibonacci[2n], {n,0,20}] (* Rigoberto Florez, Dec 24 2018 *)

a(n) = (-1)^n*A098150(n-1), n>0.
G.f.: (3*x^2+1)/(1-3*x+x^2).
a(n) = 3*L(2n+1)-F(2n), where F(n) is the n-th Fibonacci number and L(n) is the n-th Lucas number. - Rigoberto Florez, Dec 24 2018

Edited by R. J. Mathar, Nov 03 2009

A167387 a(n) = (-1)^(n+1) * n(n-1)(n-4)*(n+1)/12.

Original entry on oeis.org

1, -2, 0, 10, -35, 84, -168, 300, -495, 770, -1144, 1638, -2275, 3080, -4080, 5304, -6783, 8550, -10640, 13090, -15939, 19228, -23000, 27300, -32175, 37674, -43848, 50750, -58435, 66960, -76384, 86768, -98175, 110670, -124320, 139194, -155363, 172900

Offset: 2

Jamel Ghanouchi, Nov 02 2009

The coefficient of [x^4] of the Polynomial B_{2n}(x) defined in A137276.

Essentially the same as A052472.

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (-5,-10,-10,-5,-1).

Cf. A052472, A137276, A135929, A138034.

List([2..40], n-> (-1)^(n+1)*(n-4)*Binomial(n+1,3)/2); # G. C. Greubel, May 19 2019

[(-1)^(n+1)*n*(n-1)*(n-4)*(n+1)/12: n in [2..40]]; // Vincenzo Librandi, Jun 13 2016

Table[(-1)^(n+1)*(n+1)*n*(n-1)*(n-4)/12, {n, 2, 40}] (* G. C. Greubel, Jun 12 2016 *)
LinearRecurrence[{-5, -10, -10, -5, -1}, {1, -2, 0, 10, -35}, 40] (* Vincenzo Librandi, Jun 13 2016 *)

vector(40, n, n++; (-1)^(n+1)*(n-4)*binomial(n+1,3)/2) \\ G. C. Greubel, May 19 2019

[(-1)^(n+1)*(n-4)*binomial(n+1,3)/2 for n in (2..40)] # G. C. Greubel, May 19 2019

a(n) = -5*a(n-1) -10*a(n-2) -10*a(n-3) -5*a(n-4) -a(n-5).
G.f.: x^2*(1+3*x)/(1+x)^5.
E.g.f.: x^2*(6 + 2*x - x^2)*exp(-x)/12. - G. C. Greubel, May 19 2019

A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.

Original entry on oeis.org

-2, -7, -15, -25, -35, -42, -42, -30, 0, 55, 143, 273, 455, 700, 1020, 1428, 1938, 2565, 3325, 4235, 5313, 6578, 8050, 9750, 11700, 13923, 16443, 19285, 22475, 26040, 30008, 34408, 39270, 44625, 50505, 56943, 63973, 71630, 79950, 88970, 98728, 109263, 120615

Offset: 8

Jamel Ghanouchi, Nov 06 2009

G. C. Greubel, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A111373, A135929, A137276.

[Binomial(n-5,3)*(n-16)/4: n in [8..60]]; // G. C. Greubel, Jul 30 2022

Table[(n-5)*(n-6)*(n-7)*(n-16)/24, {n,8,60}] (* G. C. Greubel, Jun 15 2016 *)

[binomial(n-5,3)*(n-16)/4 for n in (8..60)] # G. C. Greubel, Jul 30 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^8*(-2+3*x)/(1-x)^5.
E.g.f.: (1/24)*( -3360 - 1800*x - 420*x^2 - 52*x^3 - 3*x^4 + (3360 - 1560*x + 300*x^2 - 28*x^3 + x^4)*exp(x) ). - G. C. Greubel, Jul 30 2022

Definition simplified, sequence extended by R. J. Mathar, Nov 12 2009

A136255 Triangle T(n,k) read by rows: T(n,k) = (k+1) * A137276(n,k+1).

Original entry on oeis.org

1, 0, 2, 1, 0, 3, 0, 0, 0, 4, -3, 0, -3, 0, 5, 0, -6, 0, -8, 0, 6, 5, 0, -6, 0, -15, 0, 7, 0, 16, 0, 0, 0, -24, 0, 8, -7, 0, 30, 0, 15, 0, -35, 0, 9, 0, -30, 0, 40, 0, 42, 0, -48, 0, 10, 9, 0, -75, 0, 35, 0, 84, 0, -63, 0, 11

Offset: 1

Roger L. Bagula, Mar 17 2008

Row sums are 1, 2, 4, 4, -1, -8, -9, 0, 12, 14, 1, ... with g.f. x*(1+3*x^2) / (x^2-x+1)^2.

			Triangle starts:
{1},
{0, 2},
{1, 0, 3},
{0, 0, 0, 4},
{-3, 0, -3, 0, 5},
{0, -6, 0, -8, 0, 6},
{5, 0, -6, 0, -15, 0, 7},
{0, 16, 0, 0, 0, -24, 0, 8},
{-7, 0, 30, 0, 15, 0, -35, 0, 9},
{0, -30, 0, 40, 0,42, 0, -48, 0, 10},
{9, 0, -75, 0, 35, 0, 84, 0, -63, 0, 11},
...

Cf. A138034, A135929, A135936, A137276, A137277, A137289.

B := proc(n,x) if n = 0 then 1; else add( (-1)^j*binomial(n-j,j)*(n-4*j)/(n-j)*x^(n-2*j),j=0..n/2) ; fi; end:
A136255 := proc(n,k) diff( B(n,x),x) ; coeftayl(%,x=0,k) ; end: seq( seq(A136255(n,k),k=0..n-1),n=1..15) ;

B[x, 0] = 1; B[x, 1] = x; B[x, 2] = 2 + x^2; B[x, 3] = x + x^3; B[x, 4] = -2 + x^4; B[x_, n_] := B[x, n] = x*B[x, n-1] - B[x, n-2]; P[x_, n_] := D[B[x, n + 1], x]; Flatten @ Table[CoefficientList[P[x, n], x], {n, 0, 10}]

T(n,k) = (k+1) * A137276(n,k+1) .

Edited by the Associate Editors of the OEIS, Aug 27 2009

Edited by and new name from Joerg Arndt, May 15 2016

A195662 Triangle T(n,k) read by rows: T(0,0)= -3, T(1,0)= 2, T(1,1) = 0 and T(n,k) = T(n-1,k) -T(n-2,k-2) otherwise.

Original entry on oeis.org

-3, 2, 0, 2, 0, 3, 2, 0, 1, 0, 2, 0, -1, 0, -3, 2, 0, -3, 0, -4, 0, 2, 0, -5, 0, -3, 0, 3, 2, 0, -7, 0, 0, 0, 7, 0, 2, 0, -9, 0, 5, 0, 10, 0, -3, 2, 0, -11, 0, 12, 0, 10, 0, -10, 0, 2, 0, -13, 0, 21, 0, 5, 0, -20, 0, 3, 2, 0, -15, 0, 32, 0, -7, 0, -30, 0, 13, 0

Offset: 0

Paul Curtz, Sep 22 2011

In the notation of A195673, this defines polynomials P(n,x,p=-3,q=2), where p and q are the values of the constant and linear order for n=0 and 1.
Row sums -- the value P(n,1,-3,2) of the polynomial -- are A130848(n+5).
For general seed values in the two top rows of the triangle, the recurrence T(n,k) = T(n-1,k) - T(n-2,k-2) defines the triangle
p;
q,   0;
q,   0,      -p;
q,   0,    -p-q, 0;
q,   0,   -p-2q, 0,     p;
q,   0,   -p-3q, 0,  2p+q,  0;
and a companion triangle by adding 1 to both seed values:
p+1;
q+1, 0;
q+1, 0,    -p-1;
q+1, 0,  -p-q-2, 0;
q+1, 0, -p-2q-3, 0,    p+1;
q+1, 0, -p-3q-4, 0, 2p+q+3, 0;
The point-by-point difference between two companions is P(n,x,p+1,q+1) - P(n,x,p,q) = S(n,x) as represented (with increasing exponents) by A053119.
Examples of such triangles are A053119 (p=q=1), A192575 (p=1, q=2),
A162514 (p=2, q=1, up to a sign factor), A192011 (p=-1, q=2), A135929 (p=-2, q=1, apart from a irregular leading T(0,0)).

			The first few rows are
-3;
2, 0;
2, 0,   3;
2, 0,   1, 0;
2, 0,  -1, 0, -3;
2, 0,  -3, 0, -4, 0;
2, 0,  -5, 0, -3, 0,  3;
2, 0,  -7, 0,  0, 0,  7, 0;
2, 0,  -9, 0,  5, 0, 10, 0,  -3;
2, 0, -11, 0, 12, 0, 10, 0, -10, 0;
2, 0, -13, 0, 21, 0,  5, 0, -20, 0, 3;

Cf. A135929, A192011.

p = -3; q = 2; t[0, 0] = p; t[, 0] = q; t[, ?OddQ] = 0; t[n, k_] /; k > n = 0; t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = t[n-1, k] - t[n-2, k-2]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2012 *)

T(n,0) = 2 (n>0).
T(n,2) = -A060747(n-3), n>2.
T(n,4) = A028347(n-5), n>6.
T(2n,2n) = -3*(-1)^n ; T(n, 2k-1) = 0 ; T(2n+1,2n) = -(3n-2)*(-1)^n.  - M. F. Hasler, Sep 28 2011

A195673 Triangle T(n,k) read by rows: T(0,0)=-2, T(1,0)=3, T(1,1)=0 and T(n,k) = T(n-1,k)-T(n-2,k-2) otherwise.

Original entry on oeis.org

-2, 3, 0, 3, 0, 2, 3, 0, -1, 0, 3, 0, -4, 0, -2, 3, 0, -7, 0, -1, 0, 3, 0, -10, 0, 3, 0, 2, 3, 0, -13, 0, 10, 0, 3, 0, 3, 0, -16, 0, 20, 0, 0, 0, -2, 3, 0, -19, 0, 33, 0, -10, 0, -5, 0, 3, 0, -22, 0, 49, 0, -30, 0, -5, 0, 2, 3, 0, -25, 0, 68

Offset: 0

Paul Curtz, Sep 23 2011

Obviously T(n,k) = 0 for all odd k.

Conjecture: The polynomials p(n,x) = sum_{k=0..n} T(n,k)*x^(n-k) based on this simple recurrence for other initial constant values of T(0,0)=p and T(1,0)=q are related to the S-polynomials of A053119: p(n,x,p+1,q+1)-p(n,x,p,q) = S(n,x).

			-2;
3, 0;
3, 0,   2;
3, 0,  -1, 0;
3, 0,  -4, 0, -2;
3, 0,  -7, 0, -1, 0;
3, 0, -10, 0,  3, 0, 2;
3, 0, -13, 0, 10, 0, 3, 0.

Cf. A195662, A192011 (p=-1, q=2), A135929 (p=-2, q=1).

A136160 Triangle T(n,k) = k*A053120(n,k).

Original entry on oeis.org

1, 0, 4, -3, 0, 12, 0, -16, 0, 32, 5, 0, -60, 0, 80, 0, 36, 0, -192, 0, 192, -7, 0, 168, 0, -560, 0, 448, 0, -64, 0, 640, 0, -1536, 0, 1024, 9, 0, -360, 0, 2160, 0, -4032, 0, 2304, 0, 100, 0, -1600, 0, 6720, 0, -10240, 0, 5120, -11, 0, 660, 0, -6160, 0, 19712, 0, -25344, 0, 11264

Offset: 1

Roger L. Bagula, Mar 16 2008

The definition is equivalent to building the derivatives of the Chebyshev polynomials T(n,x) and listing the coefficients [x^k] dT/dx in row n.
Row sums are the squares A000079(n-1).
Obtained from A136265 by sign flips and nulling each second diagonal. - R. J. Mathar, Sep 04 2011

			1;
0, 4;
-3, 0, 12;
0, -16, 0, 32;
5, 0, -60, 0, 80;
0, 36, 0, -192, 0, 192;
-7, 0, 168, 0, -560, 0, 448;
0, -64, 0, 640, 0, -1536,0, 1024;
9, 0, -360, 0, 2160,0, -4032, 0, 2304;
0, 100, 0, -1600, 0, 6720, 0, -10240, 0, 5120;
-11, 0, 660, 0, -6160, 0, 19712, 0, -25344, 0, 11264;

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 8 and pages 42 - 43

Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.

Cf. A053120, A135929.

P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Q[x_, n_] := D[P[x, n + 1], x]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]

A138476 Triangle read by rows: expansion of (1+3t^2)/(1-t(2*x-t)).

Original entry on oeis.org

1, 0, 2, 2, 0, 4, 0, 2, 0, 8, -2, 0, 0, 16, 0, 6, 0, -8, 0, 32, 2, 0, -12, 0, -32, 64, 0, 10, 0, -16, 0, -96, 0, 128, -2, 0, 32, 0, 0, 0, -256, 0, 256, 0, -14, 0, 80, 0, 96, 0, -640, 0, 512

Offset: 1

A. Bannour (managing_office069(AT)yahoo.fr), Mar 19 2008

See A135929 and A138034 for further information.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972; see Chapter 22.

G.f.: (1+3*t^2)/(1-t*(2*x-t)).

New name from Joerg Arndt, May 15 2016

A160242 Triangle A(n,m) read by rows: a quarter of the Fourier coefficient [cos(mt)] of the shifted Boubaker polynomial B_n(2cos t)-2cos(nt).

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 2

Haydar Rahmanov, May 05 2009

Starting from the polynomials B_n(x) defined in A137276 and A135929, we insert x=2*cos(t), and define the Fourier coefficients A(n,m) by B_n(2*cos t)-2*cos(n*t) = 4*sum(m=0,..,n-2) A(n,m)*cos(m*t).
A(n,m) is not an integer for n=0, so the table starts at n=1. Furthermore, A(n,m)=0 if n-m is odd, these regular zeros are skipped as usual, so effectively the first table entry appears at n=2.
Simpler definition from R. J. Mathar, Apr 15 2010: a(n)=1 if n =0 or n in A002061, otherwise a(n)=2. So this is a kind of characteristic function of the central polygonal numbers A002061.

			Using T^m =cos(m*t) as a notational shortcut, the expansions start
; B_1(2 cos t)-2*cos 1 t = 0
1 ; B_2(2 cos t)-2*cos 2 t = 1
0 2 ; B_3(2 cos t)-2*cos 3 t = 2*T
1 0 2 ; B_4(2 cos t)-2*cos 4 t = 1+2*T^2
0 2 0 2 ; B_5(2 cos t)-2*cos 5 t = 2*T+2*T^3
1 0 2 0 2 ; B_6(2 cos t)-2*cos 6 t = 1+2*T^2+2*T^4
0 2 0 2 0 2 ; B_7(2 cos t)-2*cos 7 t = 2*T+2*T^3+2*T^5
1 0 2 0 2 0 2 ; B_8(2 cos t)-2*cos 8 t = 1+2*T^2+2*T^4+2*T^6
0 2 0 2 0 2 0 2 ; B_9(2 cos t)-2*cos 9 t = 2*T+2*T^3+2*T^5+2*T^7
1 0 2 0 2 0 2 0 2 ; B_10(2 cos t)-2*cos 10 t = 1+2*T^2+2*T^4+2*T^6+2*T^8
0 2 0 2 0 2 0 2 0 2 ; B_11(2 cos t)-2*cos 11 t = 2*T^3+2*T^5+2*T^7+2*T^9+2*T

A. Luzon and M. A. Morón, Recurrence relations for polynomial sequences via Riordan matrices, arXiv:0904.2672 [math.CO]

centralPolygonalQ[n_] := Resolve[Exists[k, k>0, n == k^2-k+1], Integers];
b[n_] := If[n == 0 || centralPolygonalQ[n], 1, 2];
a[n_] := b[n-1];
Table[a[n], {n, 2, 106}] (* Jean-François Alcover, Oct 31 2018, after R. J. Mathar *)

Definition clarified, publication title corrected, sequence extended by R. J. Mathar, Dec 07 2009

cp's OEIS Frontend

A167375 a(n)=3*a(n-1)-a(n-2) with a(0)=1, a(1)=3, a(2)=11.

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A167387 a(n) = (-1)^(n+1) * n*(n-1)*(n-4)*(n+1)/12.

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A167543 a(n) = (n-5)*(n-6)*(n-7)*(n-16)/24.

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A136255 Triangle T(n,k) read by rows: T(n,k) = (k+1) * A137276(n,k+1).

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A195662 Triangle T(n,k) read by rows: T(0,0)= -3, T(1,0)= 2, T(1,1) = 0 and T(n,k) = T(n-1,k) -T(n-2,k-2) otherwise.

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A195673 Triangle T(n,k) read by rows: T(0,0)=-2, T(1,0)=3, T(1,1)=0 and T(n,k) = T(n-1,k)-T(n-2,k-2) otherwise.

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A136160 Triangle T(n,k) = k*A053120(n,k).

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A138476 Triangle read by rows: expansion of (1+3*t^2)/(1-t*(2*x-t)).

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A167387 a(n) = (-1)^(n+1) * n(n-1)(n-4)*(n+1)/12.

A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.

A138476 Triangle read by rows: expansion of (1+3t^2)/(1-t(2*x-t)).

A160242 Triangle A(n,m) read by rows: a quarter of the Fourier coefficient [cos(mt)] of the shifted Boubaker polynomial B_n(2cos t)-2cos(nt).