A052947 -id:A052947 - cp's OEIS frontend

A113953 A Jacobsthal triangle.

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 4, 6, 1, 0, 0, 0, 12, 8, 1, 0, 0, 0, 8, 24, 10, 1, 0, 0, 0, 0, 32, 40, 12, 1, 0, 0, 0, 0, 16, 80, 60, 14, 1, 0, 0, 0, 0, 0, 80, 160, 84, 16, 1, 0, 0, 0, 0, 0, 32, 240, 280, 112, 18, 1, 0, 0, 0, 0, 0, 0, 192, 560, 448, 144, 20, 1, 0, 0, 0, 0, 0, 0, 64, 672, 1120, 672, 180, 22, 1

Offset: 0

Paul Barry, Nov 09 2005

Rows sums are the Jacobsthal numbers A001045(n+1).
Antidiagonal sums are the Padovan-Jacobsthal numbers A052947.
Inverse is (1,xc(-2x)), c(x) the g.f. of A000108, with general term k*C(2n-k-1,n-k)(-2)^(n - k)/n.
Triangle read by rows given by (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2013

			Rows begin
  1;
  0,  1;
  0,  2,  1;
  0,  0,  4,  1;
  0,  0,  4,  6,  1;
  0,  0,  0, 12,  8,  1;
  0,  0,  0,  8, 24, 10,  1;

D. Merlini, R. Sprugnoli, M. C. Verri, Strip tiling and regular grammars, Theor. Comp. Sci 242 (1-2) (2000) 109-124, Table 1, p=2.

A signed version is A110509.

G.f.: 1/(1-xy(1+2x)).
Riordan array (1, x(1+2x)).
T(n,k) = 2^(n-k)*binomial(k, n-k).
T(n,k) = A026729(n,k)*2^(n-k). - Philippe Deléham, Nov 22 2006
T(n,k) = T(n-1,k-1) + 2*T(n-2,k-1), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 01 2013

A106855 Expansion of 1/(1-x^2(1-3x)).

Original entry on oeis.org

1, 0, 1, -3, 1, -6, 10, -9, 28, -39, 55, -123, 172, -288, 541, -804, 1405, -2427, 3817, -6642, 11098, -18093, 31024, -51387, 85303, -144459, 239464, -400368, 672841, -1118760, 1873945, -3137283, 5230225, -8759118, 14642074, -24449793, 40919428, -68376015, 114268807, -191134299, 319396852

Offset: 0

Paul Barry, May 08 2005

Diagonal sums of Riordan array (1,x(1-3x)).

Index entries for linear recurrences with constant coefficients, signature (0,1,-3).

Cf. A052947, A106852.

G.f.: 1/(1-x^2+3x^3); a(n)=a(n-2)-3a(n-3); a(n)=sum{k=0..floor(n/2), (-1)^n*binomial(k, n-2k)*3^(n-2k)}.

A110291 Riordan array (1/(1-x), x(1+2x)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 9, 7, 1, 1, 3, 9, 19, 9, 1, 1, 3, 9, 27, 33, 11, 1, 1, 3, 9, 27, 65, 51, 13, 1, 1, 3, 9, 27, 81, 131, 73, 15, 1, 1, 3, 9, 27, 81, 211, 233, 99, 17, 1, 1, 3, 9, 27, 81, 243, 473, 379, 129, 19, 1, 1, 3, 9, 27, 81, 243, 665, 939, 577, 163, 21, 1

Offset: 0

Paul Barry, Jul 18 2005

Inverse is A110292.

			Rows begin
  1;
  1, 1;
  1, 3, 1;
  1, 3, 5,  1;
  1, 3, 9,  7,  1;
  1, 3, 9, 19,  9,   1;
  1, 3, 9, 27, 33,  11,  1;
  1, 3, 9, 27, 65,  51, 13,  1;
  1, 3, 9, 27, 81, 131, 73, 15, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000244, A001045, A001047, A005408, A014335.
Cf. A066810, A077890, A077912, A110292, A180146.
Cf. A000975 (row sums), A052947 (diagonal sums).

R:=PowerSeriesRing(Rationals(), 30);
F:= func< k | Coefficients(R!( x^k*(1+2*x)^k/(1-x) )) >;
A110291:= func< n,k | F(k)[n-k+1] >;
[A110291(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 05 2023

F[k_]:= CoefficientList[Series[x^k*(1+2*x)^k/(1-x), {x,0,40}], x];
A110291[n_, k_]:= F[k][[n+1]];
Table[A110291[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2023 *)

def p(k,x): return x^k*(1+2*x)^k/(1-x)
def A110291(n,k): return ( p(k,x) ).series(x, 30).list()[n]
flatten([[A110291(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 05 2023

T(n, k) = [x^n]( x^k*(1+2*x)^k/(1-x) ).
Sum_{k=0..n} T(n, k) = A000975(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A052947(n+1).
From G. C. Greubel, Jan 05 2023: (Start)
T(n, 0) = T(n, n) = 1.
T(n, n-1) = A005408(n-1).
T(2*n, n) = T(2*n+1, n) = A000244(n).
T(2*n, n+1) = A066810(n+1).
T(2*n, n-1) = A000244(n-1).
T(2*n+1, n+1) = A001047(n+1).
Sum_{k=0..n} (-1)^k * T(n, k) = A077912(n).
Sum_{k=0..n} 2^k * T(n, k) = A014335(n+2).
Sum_{k=0..n} 3^k * T(n, k) = A180146(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A077890(n). (End)

a(30) and following corrected by Georg Fischer, Oct 11 2022

A167434 Diagonal sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

Original entry on oeis.org

1, -4, 5, -6, 13, -16, 25, -42, 57, -92, 141, -206, 325, -488, 737, -1138, 1713, -2612, 3989, -6038, 9213, -14016, 21289, -32442, 49321, -75020, 114205, -173662, 264245, -402072, 611569, -930562, 1415713, -2153700, 3276837, -4985126, 7584237

Offset: 0

Paul Barry, Nov 03 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,-2)

Cf. A052947, A159287.

I:=[1,-4,5]; [n le 3 select I[n] else Self(n-2) - 2*Self(n-3): n in [1..40]]; // G. C. Greubel, Jun 27 2018

LinearRecurrence[{0, 1, -2}, {1, -4, 5}, 100] (* G. C. Greubel, Jun 13 2016 *)
CoefficientList[Series[(1-4x+4x^2)/(1-x^2+2x^3),{x,0,40}],x] (* Harvey P. Dale, Nov 08 2022 *)

x='x+O('x^40); Vec((1-4*x+4*x^2)/(1-x^2+2*x^3)) \\ G. C. Greubel, Jun 27 2018

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

a(n) = (-1)^n*A052947(n+4). - R. J. Mathar, Jun 24 2024

A373080 a(n) is the number of binary strings of length n not containing the substrings 0000, 0001, 0011, 0111, 1111.

Original entry on oeis.org

1, 2, 4, 8, 11, 18, 28, 40, 64, 96, 144, 224, 336, 512, 784, 1184, 1808, 2752, 4176, 6368, 9680, 14720, 22416, 34080, 51856, 78912, 120016, 182624, 277840, 422656, 643088, 978336, 1488400, 2264512, 3445072, 5241312, 7974096, 12131456, 18456720, 28079648

Offset: 0

Miquel A. Fiol, Jun 03 2024

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A052947, A164395, A164408, A368430.

LinearRecurrence[{0, 1, 2}, {1, 2, 4, 8, 11, 18, 28}, 50] (* Paolo Xausa, Jun 24 2024 *)

a(n) = a(n-2) + 2*a(n-3) for n >= 7.

G.f.: -(x+1)^2*(x^2+1)^2/(2*x^3+x^2-1). - Alois P. Heinz, Jun 03 2024

A099492 A Chebyshev transform of the Padovan-Jacobsthal numbers.

Original entry on oeis.org

1, 0, 0, 2, -1, -4, 5, 2, -16, 12, 27, -56, -3, 140, -144, -186, 547, -140, -1175, 1606, 1120, -5096, 2775, 9360, -16807, -4584, 45664, -38070, -69657, 167276, -11347, -393142, 450896, 467108, -1595725, 586584, 3235221, -4905692, -2556720, 14641550, -9572661, -25171740, 50306641, 6820750

Offset: 0

Paul Barry, Oct 19 2004

A Chebyshev transform of A052947, which has g.f. 1/(1-x^2-2x^3). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Index entries for linear recurrences with constant coefficients, signature (0,-2,2,-2,0,-1).

LinearRecurrence[{0,-2,2,-2,0,-1},{1,0,0,2,-1,-4},50] (* Harvey P. Dale, Dec 20 2015 *)

G.f.: (1+x^2)^2/(1+2x^2-2x^3+2x^4+x^6); a(n)=-2a(n-2)+2a(n-3)-2a(n-4)-a(n-6); a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*sum{j=0..floor((n-2k)/2), C(j, n-2k-2j)2^(n-2k-2j)}}.

A122439 Expansion of 1/(1-2x-x^2+4x^4).

Original entry on oeis.org

1, 2, 5, 12, 25, 54, 113, 232, 477, 970, 1965, 3972, 8001, 16094, 32329, 64864, 130053, 260594, 521925, 1044988, 2091689, 4185990, 8375969, 16757976, 33525165, 67064346, 134149981, 268332404, 536714129, 1073503278, 2147120761

Offset: 0

Paul Barry, Sep 05 2006

Diagonal sums of number triangle A122438. Convolution of A052947 and 2^n.

Index entries for linear recurrences with constant coefficients, signature (2,1,0,-4).

CoefficientList[Series[1/(1 - 2x - x^2 + 4x^4), {x, 0, 31}], x] (* Robert G. Wilson v, Sep 14 2006 *)
LinearRecurrence[{2,1,0,-4},{1,2,5,12},40] (* Harvey P. Dale, Jul 30 2024 *)

More terms from Robert G. Wilson v, Sep 14 2006

A077882 Expansion of x/((1-x)(1-x^2-2x^3)).

Original entry on oeis.org

0, 1, 1, 2, 4, 5, 9, 14, 20, 33, 49, 74, 116, 173, 265, 406, 612, 937, 1425, 2162, 3300, 5013, 7625, 11614, 17652, 26865, 40881, 62170, 94612, 143933, 218953, 333158, 506820, 771065, 1173137, 1784706, 2715268, 4130981, 6284681, 9561518, 14546644, 22130881, 33669681

Offset: 0

N. J. A. Sloane, Nov 17 2002

a(n+1) gives diagonal sums of Riordan array (1/(1-x),x*(1+2*x)) and partial sums of A052947. - Paul Barry, Jul 18 2005

Index entries for linear recurrences with constant coefficients, signature (1,1,1,-2).

a[0] = 0; a[1] = 1; a[2] = 1; a[3] = 2; a[n_Integer?Positive] := a[n] = a[n - 1] + a[n - 2] + a[n - 3] - 2a[n - 4]; aa = Table[a[n], {n, 0, 42}] (* Roger L. Bagula, Mar 25 2005 *)
CoefficientList[Series[x/((1-x)(1-x^2-2x^3)),{x,0,50}],x] (* or *) LinearRecurrence[{1,1,1,-2},{0,1,1,2},50] (* Harvey P. Dale, Aug 17 2017 *)

a(n) = a(n-1)+a(n-2)+a(n-3)-2*a(n-4). - Roger L. Bagula, Mar 25 2005

a(n+1) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} C(j, k-2*j)*2^(k-2*j). - Paul Barry, Jul 18 2005

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A104579 A Padovan-Jacobsthal convolution triangle.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 1, 4, 3, 0, 1, 4, 3, 6, 4, 0, 1, 5, 12, 6, 8, 5, 0, 1, 6, 16, 24, 10, 10, 6, 0, 1, 13, 24, 34, 40, 15, 12, 7, 0, 1, 16, 53, 60, 60, 60, 21, 14, 8, 0, 1, 25, 72, 135, 120, 95, 84, 28, 16, 9, 0, 1, 42, 126, 200, 275, 210, 140, 112, 36, 18, 10, 0, 1, 57, 220, 381

Offset: 0

Paul Barry, Mar 16 2005

First column is A052947. Row sums are A077947. Diagonal sums are A052907.

			Rows begin {1},{0,1},{1,0,1},{2,2,0,1},{1,4,3,0,1},{4,3,6,4,0,1},..

Riordan array (1/(1-x^2-2x^3), x/(1-x^2-2x^3))

T(n,k) = T(n-1,k-1)+T(n-2,k)+2*T(n-3,k), T(0,0)=1, T(n,k)=0 if k>n or if k<0. - Philippe Deléham, Jan 08 2014

A134271 a(n) = a(n-2) + 2*a(n-3), n > 3; with a(0)=0, a(1)=1, a(2)=2, a(3)=0.

Original entry on oeis.org

0, 1, 2, 0, 4, 4, 4, 12, 12, 20, 36, 44, 76, 116, 164, 268, 396, 596, 932, 1388, 2124, 3252, 4900, 7500, 11404, 17300, 26404, 40108, 61004, 92916, 141220, 214924, 327052, 497364, 756900, 1151468, 1751628, 2665268, 4054564, 6168524, 9385100

Offset: 0

Paul Curtz, Jan 30 2008

Recurrence in A052947.

Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Cf. A078050.

From R. J. Mathar, Feb 01 2008: (Start)
O.g.f.: 1/2 + (1/2)*(-2*x-5*x^2+1)/(-1+x^2+2*x^3).
a(n) = A052947(n-1) + 2*A052947(n-2) - A052947(n-3). (End)

cp's OEIS Frontend

A113953 A Jacobsthal triangle.

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A106855 Expansion of 1/(1-x^2(1-3x)).

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A110291 Riordan array (1/(1-x), x*(1+2*x)).

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A167434 Diagonal sums of the Riordan array (1-4*x+4*x^2, x*(1-2*x)) (A167431).

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A373080 a(n) is the number of binary strings of length n not containing the substrings 0000, 0001, 0011, 0111, 1111.

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A099492 A Chebyshev transform of the Padovan-Jacobsthal numbers.

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A122439 Expansion of 1/(1-2x-x^2+4x^4).

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A077882 Expansion of x/((1-x)*(1-x^2-2*x^3)).

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A110291 Riordan array (1/(1-x), x(1+2x)).

A167434 Diagonal sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

A077882 Expansion of x/((1-x)(1-x^2-2x^3)).