A085810 -id:A085810 - cp's OEIS frontend

A181880 Expansion of 1/(1-4x-3x^2-x^3).

1, 4, 19, 89, 417, 1954, 9156, 42903, 201034, 942001, 4414009, 20683073, 96916320, 454128508, 2127946065, 9971086104, 46722311119, 218930448853, 1025859814873, 4806952917170, 22524321562152, 105544004814991, 494555936863590, 2317380083461485, 10858732149251701, 50881624784254849, 238420075668235984, 1117183909174960184, 5234877488488803537, 24529481757148330684

Offset: 0

Wolfdieter Lang, Nov 27 2010

B(n):=a(n-2)*(-1)^n, B(0):=0, B(1):=0, (o.g.f. x^2/(1 + 4*x + 3*x^2 -x^3))appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with C(n)= A085810(n)*(-1)^n, and A(n)= A116423(n+1)*(-1)^(n+1). For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (4, 3, 1).

CoefficientList[Series[1/(1-4*x-3*x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,3,1},{1,4,19},40] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

O.g.f.: 1/(1-4*x-3*x^2-x^3).

a(n) = 4*a(n) + 3*a(n-2) +a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=4.

A123876 Riordan array (1/(1+2x), x(1+x)/(1+2*x)^2).

Original entry on oeis.org

1, -2, 1, 4, -5, 1, -8, 18, -8, 1, 16, -56, 41, -11, 1, -32, 160, -170, 73, -14, 1, 64, -432, 620, -377, 114, -17, 1, -128, 1120, -2072, 1666, -704, 164, -20, 1, 256, -2816, 6496, -6608, 3649, -1178, 223, -23, 1, -512, 6912, -19392, 24192, -16722, 7001, -1826, 291, -26, 1

Offset: 0

Paul Barry, Oct 16 2006

Inverse of A116395.
Row sums are A123877.
Diagonal sums are (-1)^n*A085810(n).
Unsigned version is A114164.

			Triangle begins
    1;
   -2,   1;
    4,  -5,    1;
   -8,  18,   -8,   1;
   16, -56,   41, -11,   1;
  -32, 160, -170,  73, -14, 1;

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

Cf. A085810, A114164, A116395, A123877.

Flat(List([0..12], n-> List([0..n], k-> (-1)^(n-k)*Sum([0..n], j-> 2^(n-j)*Binomial(k,j-k)*Binomial(n,j) ))));

[(-1)^(n-k)*(&+[2^(n-j)*Binomial(k,j-k)*Binomial(n,j): j in [0..n]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 08 2019

Table[(-1)^(n-k)*Sum[2^(n-j)*Binomial[k,j-k]*Binomial[n,j], {j,0,n}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 08 2019 *)

T(n,k) = b=binomial; (-1)^(n-k)*sum(j=0,n, 2^(n-j)*b(k,j-k)* b(n,j)); \\ G. C. Greubel, Aug 08 2019

b=binomial;
[[(-1)^(n-k)*sum(2^(n-j)*b(k,j-k)*b(n,j) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 08 2019

Number triangle T(n,k) = (-1)^(n-k)*Sum_{j=0..n} C(k,j-k)*C(n,j)*2^(n-j).

T(n,k) = T(n-1,k-1) - 4*T(n-1,k) + T(n-2,k-1) - 4*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = -2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 18 2014

More terms added by G. C. Greubel, Aug 08 2019

A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 18, 8, 1, 16, 56, 41, 11, 1, 32, 160, 170, 73, 14, 1, 64, 432, 620, 377, 114, 17, 1, 128, 1120, 2072, 1666, 704, 164, 20, 1, 256, 2816, 6496, 6608, 3649, 1178, 223, 23, 1, 512, 6912, 19392, 24192, 16722, 7001, 1826, 291, 26, 1, 1024, 16640, 55680, 83232, 69876, 36365, 12235, 2675, 368, 29, 1

Offset: 0

Paul Barry, Nov 15 2005

Row sums are A081567. Diagonal sums are A085810. Product of Pascal triangle A007318 and Morgan-Voyce triangle A085478.

Unsigned version of A123876. - Philippe Deléham, Oct 25 2007

			Triangle begins:
   1;
   2,   1;
   4,   5,   1;
   8,  18,   8,  1;
  16,  56,  41, 11,  1;
  32, 160, 170, 73, 14, 1;
  ...

Cf. A081567, A085810, A007318, A085478, A123876.

T(2n,n) gives A026000.

Number triangle T(n,k) = Sum_{j=0..n} C(n, j)*C(j+k, 2k);
T(n,k) = Sum_{j=0..n} C(n, k+j)*C(k, k-j)*2^(n-k-j);
T(n,k) = Sum_{j=0..n-k} C(n+k-j, n-k-j)*C(k, j)*(-1)^j*2^(n-k-j).
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 4*T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014

More terms from Michel Marcus, Sep 09 2024

A215492 a(n) = 21a(n-2) + 7a(n-3), with a(0)=0, a(1)=3, and a(2)=6.

Original entry on oeis.org

0, 3, 6, 63, 147, 1365, 3528, 29694, 83643, 648270, 1964361, 14199171, 45789471, 311933118, 1060973088, 6871121775, 24463966674, 151720368891, 561841152579, 3357375513429, 12860706786396, 74437773850062, 293576471108319, 1653218198356074, 6686170310225133

Offset: 0

Roman Witula, Aug 13 2012

We have a(n)=B(n;3), where B(n;d), n=1,2,..., d \in C, denote one of the quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. Its conjugate sequences A(n;3) and C(n;3) are discussed in A121458 and A215484 respectively. Similarly as in A121458 we deduce that each of the following elements a(3*n), a(3*n+1), a(3*n+2) is divided by 3*7^n for every n=0,1,... .

G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (0, 21, 7).

Cf. A121458, A215484, A121449, A085810, A215404, A077998, A006054, A033304, A052975, A094789, A005021, A121442, A121458.

I:=[0,3,6]; [n le 3 select I[n] else 21*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

LinearRecurrence[{0,21,7}, {0,3,6}, 50]
CoefficientList[Series[(3 x + 6 x^2)/(1 - 21 x^2 - 7 x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 18 2015 *)

concat(0,Vec((3+6*x)/(1-21*x^2-7*x^3)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012

a(n) = (1/7)*((c(1)-c(4))*(1+3*c(1))^n + (c(2)-c(1))*(1+3*c(2))^n + (c(4)-c(2))*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) (for the proof see Witula-Slota-Warzynski paper).

G.f.: (3*x+6*x^2)/(1-21*x^2-7*x^3).

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A181880 Expansion of 1/(1-4*x-3*x^2-x^3).

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A123876 Riordan array (1/(1+2*x), x*(1+x)/(1+2*x)^2).

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A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

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A215492 a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=0, a(1)=3, and a(2)=6.

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A181880 Expansion of 1/(1-4x-3x^2-x^3).

A123876 Riordan array (1/(1+2x), x(1+x)/(1+2*x)^2).

A215492 a(n) = 21a(n-2) + 7a(n-3), with a(0)=0, a(1)=3, and a(2)=6.