A077957 -id:A077957 - cp's OEIS frontend

A135247 a(n) = 3a(n-1) + 2a(n-2) - 6*a(n-3).

1, 3, 11, 33, 103, 309, 935, 2805, 8431, 25293, 75911, 227733, 683263, 2049789, 6149495, 18448485, 55345711, 166037133, 498111911, 1494335733, 4483008223, 13449024669, 40347076055, 121041228165, 363123688591, 1089371065773, 3268113205511, 9804339616533

Offset: 0

Paul Curtz, Feb 15 2008

This sequence interleaves A016133 and 3*A016133, see formulas. - Mathew Englander, Jan 08 2024

a(n) is the number of partitions of n into parts 1 (in three colors) and 2 (in two colors) where the order of colors matters. For example, the a(2)=11 such partitions (using parts 1, 1', 1'', 2, and 2') are 2, 2', 1+1, 1+1', 1+1'', 1'+1, 1'+1', 1'+1'', 1''+1, 1''+1', 1''+1''. For such partitions where the order of colors does not matter see A002624. - Joerg Arndt, Jan 18 2024

Sean A. Irvine, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,-6).

Cf. A016133.

a:=[1,3,11];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-6*a[n-3]; od; a; # G. C. Greubel, Nov 20 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-3*x-2*x^2+6*x^3) )); // G. C. Greubel, Nov 20 2019

seq(coeff(series(1/(1-3*x-2*x^2+6*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 20 2019

LinearRecurrence[{3,2,-6},{1,3,11},30] (* Harvey P. Dale, Jun 27 2015 *)

my(x='x+O('x^30)); Vec(1/(1-3*x-2*x^2+6*x^3)) \\ G. C. Greubel, Nov 20 2019

def A135247_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-3*x-2*x^2+6*x^3) ).list()
A135247_list(30) # G. C. Greubel, Nov 20 2019

G.f.: 1/((1-3*x)*(1-2*x^2)). - G. C. Greubel, Oct 04 2016
From Mathew Englander, Jan 08 2024: (Start)
a(n) = A010684(n) * A016133(floor(n/2)).
a(n) = 3*a(n-1) + A077957(n) for n >= 1.
a(n) = (A000244(n+2) - A164073(n+3))/7.
(End)

More terms from Harvey P. Dale, Jun 27 2015

Dropped two leading terms = 0. - Joerg Arndt, Jan 18 2024

A162666 a(n) = 20a(n-1) - 98a(n-2) for n > 1; a(0) = 1, a(1) = 10.

Original entry on oeis.org

1, 10, 102, 1060, 11204, 120200, 1306008, 14340560, 158822416, 1771073440, 19856872032, 223572243520, 2525471411264, 28599348360320, 324490768902528, 3687079238739200, 41941489422336256, 477496023050283520

Offset: 0

Klaus Brockhaus, Jul 20 2009

Binomial transform of A147960. Tenth binomial transform of A077957.

G. C. Greubel, Table of n, a(n) for n = 0..940
Index entries for linear recurrences with constant coefficients, signature (20,-98).

Cf. A147960, A077957.

a:=[1,10];; for n in [3..20] do a[n]:=20*a[n-1]-98*a[n-2]; od; a; # G. C. Greubel, Aug 27 2019

[ n le 2 select 9*n-8 else 20*Self(n-1)-98*Self(n-2): n in [1..18] ];

seq(coeff(series((1-10*x)/(1-20*x+98*x^2), x, n+1), x, n), n = 0..20); # G. C. Greubel, Aug 27 2019

Union[Flatten[NestList[{#[[2]],20#[[2]]-98#[[1]]}&,{1,10},20]]]  (* Harvey P. Dale, Feb 25 2011 *)
LinearRecurrence[{20,-98}, {1,10}, 20] (* G. C. Greubel, Aug 27 2019 *)

my(x='x+O('x^20)); Vec((1-10*x)/(1-20*x+98*x^2)) \\ G. C. Greubel, Aug 27 2019

def A162666_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-10*x)/(1-20*x+98*x^2)).list()
A162666_list(20) # G. C. Greubel, Aug 27 2019

a(n) = ((10+sqrt(2))^n + (10-sqrt(2))^n)/2.
G.f.: (1-10*x)/(1-20*x+98*x^2).
E.g.f.: exp(10*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

A178945 Expansion of x(1-x)^2/( (1-2x^2)(1-2x)^2).

Original entry on oeis.org

1, 2, 7, 16, 42, 96, 228, 512, 1160, 2560, 5648, 12288, 26656, 57344, 122944, 262144, 557184, 1179648, 2490624, 5242880, 11010560, 23068672, 48235520, 100663296, 209717248, 436207616, 905973760, 1879048192, 3892322304, 8053063680, 16643014656

Offset: 1

Gary W. Adamson, Dec 30 2010

			(1, 4, 12, 32, 80, 192, 448, 1024,...) +
..(1, 0,..2,..0,..4,...0,...8,....0...) =
..(2, 4, 14, 32, 84, 192, 456, 1024,...). Then dividing the sum by 2 we obtain:
..(1, 2, 7, 16, 42, 96, 228,...).

Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).

Cf. A000079, A001787, A077957, column k=2 of A290222.

CoefficientList[Series[x (1-x)^2/((1-2x^2)(1-2x)^2),{x,0,50}],x] (* or *) LinearRecurrence[{4,-2,-8,8},{0,1,2,7},50] (* Harvey P. Dale, Dec 29 2023 *)

a(2n+1) = ( A001787(2n+1)+A077957(2n))/2.
a(2n) = A001787(2n)/2.
a(n) = 2^(n-2)*n + 2^(n/2-5/2)*(1-(-1)^n).
a(n) = +4*a(n-1) -2*a(n-2) -8*a(n-3) +8*a(n-4).
G.f.: x*(S(x)^2 + S(x^2))/2 where S(x) is the g.f. for A000079.

A206800 Riordan array (1/(1-3x+x^2), x(1-x)/(1-3*x+x^2)).

Original entry on oeis.org

1, 3, 1, 8, 5, 1, 21, 19, 7, 1, 55, 65, 34, 9, 1, 144, 210, 141, 53, 11, 1, 377, 654, 534, 257, 76, 13, 1, 987, 1985, 1905, 1111, 421, 103, 15, 1, 2584, 5911, 6512, 4447, 2041, 641, 134, 17, 1, 6765, 17345, 21557, 16837, 9038, 3440, 925, 169, 19, 1

Offset: 0

Philippe Deléham, Feb 12 2012

			Triangle begins :
1
3, 1
8, 5, 1
21, 19, 7, 1
55, 65, 34, 9, 1
144, 210, 141, 53, 11, 1
377, 654, 534, 257, 76, 13, 1
987, 1985, 1905, 1111, 421, 103, 15, 1
2584, 5911, 6512, 4447, 2041, 641, 134, 17, 1
6765, 17345, 21557, 16837, 9038, 3440, 925, 169, 19, 1
Triangle (0,3,-1/3,1/3,0,0,0,0,0,...) DELTA (1,0,-1/3,1/3,0,0,0,0,...) begins :
1
0, 1
0, 3, 1
0, 8, 5, 1
0, 21, 19, 7, 1
0, 55, 65, 34, 9, 1...

Subtriangle of the triangle given by (0, 3, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Antidiagonal sums are A072264(n).

Cf. A000045, A001519, A001906, A001870,

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1).
G.f.: 1/(1-(y+3)*x+(y+1)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n* A015587(n+1), (-1)^n*A190953(n+1), (-1)^n*A015566(n+1), (-1)*A189800(n+1), (-1)^n*A015541(n+1), (-1)^n*A085939(n+1), (-1)^n*A015523(n+1), (-1)^n*A063727(n), (-1)^n*A006130(n), A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n) for x = -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A108085 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) - T(n-1,k-1).

Original entry on oeis.org

1, 2, -1, 8, -6, 1, 64, -56, 14, -1, 1024, -960, 280, -30, 1, 32768, -31744, 9920, -1240, 62, -1, 2097152, -2064384, 666624, -89280, 5208, -126, 1, 268435456, -266338304, 87392256, -12094464, 755904, -21336, 254, -1, 68719476736, -68451041280, 22638755840, -3183575040, 205605888, -6217920

Offset: 0

Gerald McGarvey, Jun 05 2005

For n > 0, n-th row sum = Product_{i=1..n} (2^i - 1), i.e., A005329(n).

Triangle T(n,k), 0 <= k <= n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [-1, 0, -2, 0, -4, 0, -8, 0, -16, 0, -32, 0, ...] = A014236(first zero omitted)DELTA -A077957 where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006

			Triangle begins
     1;
     2,   -1;
     8,   -6,   1;
    64,  -56,  14,  -1;
  1024, -960, 280, -30, 1;

t(n, k) = {if (k < 0, return (0)); if (n < k, return (0)); if (n == 0, return (1)); return (2^n*t(n-1,k) - t(n-1,k-1));}  \\ Michel Marcus, Apr 11 2013

A122742 Numbers of polypentagons with two connected internal vertices (see Cyvin et al. for precise definition).

Original entry on oeis.org

0, 0, 1, 3, 6, 16, 34, 80, 172, 384, 824, 1792, 3824, 8192, 17376, 36864, 77760, 163840, 343936, 720896, 1507072, 3145728, 6553088, 13631488, 28310528, 58720256, 121632768, 251658240, 520089600, 1073741824, 2214584320, 4563402752, 9395224576, 19327352832, 39728414720, 81604378624

Offset: 4

N. J. A. Sloane, Sep 24 2006

S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).

G.f.: x^6*(x-1)*(2*x^3-4*x^2+1) / ((2*x-1)^2*(2*x^2-1)). - Colin Barker, Aug 29 2013

a(n) = A001787(n+1)/256 - 2^(n-7) - A077957(n)/16, n>6. - R. J. Mathar, Jul 26 2019

A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = xZ(n-1,x) - 2Z(n-2,x), n >= 2.

Original entry on oeis.org

1, 1, 0, 1, 0, -2, 1, 0, -4, 0, 1, 0, -6, 0, 4, 1, 0, -8, 0, 12, 0, 1, 0, -10, 0, 24, 0, -8, 1, 0, -12, 0, 40, 0, -32, 0, 1, 0, -14, 0, 60, 0, -80, 0, 16, 1, 0, -16, 0, 84, 0, -160, 0, 80, 0, 1, 0, -18, 0, 112, 0, -280, 0, 240, 0, -32

Offset: 0

Paul Curtz, Jun 19 2011

The coefficients of the Z(n,x) polynomials by decreasing exponents, see the formulas, define this triangle.

			The first few rows of the coefficients of the Z(n,x) are
  1;
  1,    0;
  1,    0,   -2;
  1,    0,   -4,    0;
  1,    0,   -6,    0,    4;
  1,    0,   -8,    0,   12,    0;
  1,    0,  -10,    0,   24,    0,   -8;
  1,    0,  -12,    0,   40,    0,  -32,    0;
  1,    0,  -14,    0,   60,    0,  -80,    0,   16;
  1,    0,  -16,    0,   84,    0, -160,    0,   80,    0;

Row sums: A107920(n+1). Main diagonal: A077966(n).
Z(n,x=1) = A107920(n+1), Z(n,x=2)  = A009545(n+1),
Z(n,x=3) = A000225(n+1), Z(n,x=4)  = A007070(n),
Z(n,x=5) = A107839(n),   Z(n,x=6)  = A154244(n),
Z(n,x=7) = A186446(n),   Z(n,x=8)  = A190975(n+1),
Z(n,x=9) = A190979(n+1), Z(n,x=10) = A190869(n+1).
Row sum without sign: A113405(n+1).
Cf. A128099, A013609.

nmax:=10: Z(0, x):=1 : Z(1, x):=x: for n from 2 to nmax do Z(n, x) := x*Z(n-1, x) - 2*Z(n-2, x) od: for n from 0 to nmax do for k from 0 to n do T(n, k) := coeff(Z(n, x), x, n-k) od: od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 27 2011, revised Nov 29 2012

a[n_, k_] := If[OddQ[k], 0, 2^(k/2)*Coefficient[ ChebyshevU[n, x/2], x, n-k]]; Flatten[ Table[ a[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Aug 02 2012, from 2nd formula *)

Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.

a(n,k) = A077957(k) * A053119(n,k). - Paul Curtz, Sep 30 2011

Edited and information added by Johannes W. Meijer, Jun 27 2011

A208459 Triangle T_x = T(n,k) given by (0, 1/x, 1-1/x, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (x, 1/x-1, -1/x, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938, for x = 0.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, -3, 0, 1, 0, 3, -1, 0, 5, 0, 1, 0, 4, -2, 3, 2, -8, 0, 1, 0, 5, -3, 7, -2, -5, 13, 0, 1, 0, 6, -4, 12, -8, 2, 12, -21, 0, 1, 0, 7, -5, 18, -16, 15, 3, -25, 34

Offset: 0

Philippe Deléham, Feb 27 2012

Triangle T_x : T_1 = A103631, T_2 = A208343, T_3 = A208345.

			Triangle begins :
1
0, 0
0, 1, 1
0, 1, 0, -1
0, 1, 0, 1, 2
0, 1, 0, 2, 0, -3
0, 1, 0, 3, -1, 0, 5
0, 1, 0, 4, -2, 3, 2, -8
0, 1, 0, 5, -3, 7, -2, -5, 13
0, 1, 0, 6, -4, 12, -8, 2, 12, -21
0, 1, 0, 7, -5, 18, -16, 15, 3, -25, 34

Cf. A103631, A208343, A208345, A000045 (Fibonacci)

T(n,k) = T(n-1,k) - T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) with T(0,0) = 1 T(1,0) = 0, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-x+y*x)/(1-x+y*x- y^2*x^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = 12*A015548(n-1), 6*A085939(n-1), A106434(n), A000007(n), A000007(n), A077957(n), (-1)^n*A102901(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sm_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A034834(n-1), A077957(n), A052533(n), (-1)^n*A086344(n) for x = -1, 0, 1, 2, 3 respectively.

A235501 Riordan array (1/(1-2*x^2), x/(1-x)).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 4, 3, 5, 3, 1, 0, 7, 8, 8, 4, 1, 8, 7, 15, 16, 12, 5, 1, 0, 15, 22, 31, 28, 17, 6, 1, 16, 15, 37, 53, 59, 45, 23, 7, 1, 0, 31, 52, 90, 112, 104, 68, 30, 8, 1, 32, 31, 83, 142, 202, 216, 172, 98, 38, 9, 1, 0, 63, 114, 225

Offset: 0

Philippe Deléham, Jan 11 2014

Row sums are A007179(n+1).

			Triangle begins (0<=k<=n):
1
0, 1
2, 1, 1
0, 3, 2, 1
4, 3, 5, 3, 1
0, 7, 8, 8, 4, 1
8, 7, 15, 16, 12, 5, 1
0, 15, 22, 31, 28, 17, 6, 1

Cf. Columns: A077957, A052551, A077866.
Diagonals: A000012, A001477, A022856.
Cf. Similar sequences: A059260, A191582.

T(n,n)=1, T(2n,0)=2^n, T(2n+1,0)=0, T(n,k)=T(n-1,k-1)+T(n-1,k) for 0

T(n,k)=T(n-1,k)+T(n-1,k-1)+2*T(n-2,k)-T(n-3,k)-2*T(n-3,k-1), T(0,0)=1, T(1,0)=0, T(1,1)=1, T(n,k)=0 if k<0 or if k>n.

T(n,n)=1, T(n+1,n)=n, T(n+2,n)=n*(n+1)/2 + 2.

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(3*x + 2*x^2/2! + x^3/3!) = 3*x + 8*x^2/2! + 16*x^3/3! + 28*x^4/4! + 45*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

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A135247 a(n) = 3*a(n-1) + 2*a(n-2) - 6*a(n-3).

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A162666 a(n) = 20*a(n-1) - 98*a(n-2) for n > 1; a(0) = 1, a(1) = 10.

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A178945 Expansion of x*(1-x)^2/( (1-2*x^2)*(1-2*x)^2).

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A206800 Riordan array (1/(1-3*x+x^2), x*(1-x)/(1-3*x+x^2)).

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A108085 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) - T(n-1,k-1).

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A122742 Numbers of polypentagons with two connected internal vertices (see Cyvin et al. for precise definition).

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A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.

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A208459 Triangle T_x = T(n,k) given by (0, 1/x, 1-1/x, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (x, 1/x-1, -1/x, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938, for x = 0.

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A135247 a(n) = 3a(n-1) + 2a(n-2) - 6*a(n-3).

A162666 a(n) = 20a(n-1) - 98a(n-2) for n > 1; a(0) = 1, a(1) = 10.

A178945 Expansion of x(1-x)^2/( (1-2x^2)(1-2x)^2).

A206800 Riordan array (1/(1-3x+x^2), x(1-x)/(1-3*x+x^2)).

A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = xZ(n-1,x) - 2Z(n-2,x), n >= 2.