A110164 -id:A110164 - cp's OEIS frontend

A003953 Expansion of g.f.: (1+x)/(1-10*x).

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 110000000000, 1100000000000, 11000000000000, 110000000000000, 1100000000000000, 11000000000000000

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 11.
a(n) is sequence A003945(n-1) written in base 2: a(0)=1, a(n) for n >= 1: 2 times 1, (n-1) times 0. a(n) is also A007283(n-1) and A042950(n) for n >= 1 written in base 2. a(n) is also A098011(n+3) and A101229(n+10) for n >= 1 written in base 2. a(n) is also abs(A110164(n+1)) for n >= 1 written in base 2. - Jaroslav Krizek, Aug 17 2009
a(n) equals the numbers of words of length n on alphabet {0,1,...,10} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, Jun 02 2017]

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 312
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (10).
Index entries for sequences related to trees

Cf. A003945, A003952.

k:=11;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=11; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

k:=11; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 11*10^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[10#&,11,20]] (* Harvey P. Dale, Jul 04 2025 *)

a(n)=11*10^n\10 \\ Charles R Greathouse IV, Aug 14 2015

k=11; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 9. - Philippe Deléham, Jul 10 2005
G.f.: (1+x)/(1-10*x). - Paul Barry, Mar 22 2006
a(0) = 1, a(n) = 10^n + 10^(n-1) = 11*10^(n-1) for n >= 1. - Jaroslav Krizek, Aug 17 2009
E.g.f.: (11*exp(10*x) - 1)/10. - G. C. Greubel, Sep 24 2019

Edited by N. J. A. Sloane, Dec 04 2009

A110162 Riordan array ((1-x)/(1+x), x/(1+x)^2).

Original entry on oeis.org

1, -2, 1, 2, -4, 1, -2, 9, -6, 1, 2, -16, 20, -8, 1, -2, 25, -50, 35, -10, 1, 2, -36, 105, -112, 54, -12, 1, -2, 49, -196, 294, -210, 77, -14, 1, 2, -64, 336, -672, 660, -352, 104, -16, 1, -2, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 2, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 0

Paul Barry, Jul 14 2005

Inverse of Riordan array A094527. Rows sums are A099837. Diagonal sums are A110164. Product of Riordan array A102587 and inverse binomial transform (1/(1+x), x/(1+x)).
Coefficients of polynomials related to Cartan matrices of types C_n and B_n: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2), with p(x,0) = 1; p(x,1) = 2-x; p(x,2) = x^2-4*x-2. - Roger L. Bagula, Apr 12 2008
From Wolfdieter Lang, Nov 16 2012: (Start)
The alternating row sums are given in A219233.
For n >= 1 the row polynomials in the variable x^2 are R(2*n,x):=2*T(2*n,x/2) with Chebyshev's T-polynomials. See A127672 and also the triangle A127677.
(End)
From Peter Bala, Jun 29 2015: (Start)
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x/(1 + x)^2 and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = (1 - 2*x + sqrt(1 - 4*x))/2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

			Triangle T(n,k) begins:
m\k  0    1    2     3     4     5     6    7    8   9 10 ...
0:   1
1:  -2    1
2:   2   -4    1
3:  -2    9   -6     1
4:   2  -16   20    -8     1
5:  -2   25  -50    35   -10     1
6:   2  -36  105  -112    54   -12     1
7:  -2   49 -196   294  -210    77   -14    1
8:   2  -64  336  -672   660  -352   104  -16    1
9:  -2   81 -540  1386 -1782  1287  -546  135  -18   1
10:  2 -100  825 -2640  4290 -4004  2275 -800  170 -20  1
... Reformatted and extended by _Wolfdieter Lang_, Nov 16 2012
Row polynomial n=2: P(2,x) = 2 - 4*x + x^2. R(4,x):= 2*T(4,x/2) = 2 - 4*x^2 + x^4. For P and R see a comment above. - _Wolfdieter Lang_, Nov 16 2012.

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
T. M. Richardson, The Reciprocal Pascal Matrix, arXiv:1405.6315 [math.CO], 2014.

Cf. A128411. See A127677 for an almost identical triangle.

Cf. A136674, A053122.

/* As triangle */ [[(-1)^(n-k)*(Binomial(n+k,n-k) + Binomial(n+k-1,n-k-1)): k in [0..n]]: n in [0.. 12]]; // Vincenzo Librandi, Jun 30 2015

Table[If[n==0 && k==0, 1, (-1)^(n-k)*(Binomial[n+k, n-k] + Binomial[n+k-1, n-k-1])], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 16 2018 *)

{T(n,k) = (-1)^(n-k)*(binomial(n+k,n-k) + binomial(n+k-1,n-k-1))};
for(n=0, 12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 16 2018

[[(-1)^(n-k)*(binomial(n+k,n-k) + binomial(n+k-1,n-k-1)) for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 16 2018

T(n,k) = (-1)^(n-k)*(C(n+k,n-k) + C(n+k-1,n-k-1)), with T(0,0) = 1. - Paul Barry, Mar 22 2007
From Wolfdieter Lang, Nov 16 2012: (Start)
O.g.f. row polynomials P(n,x) := Sum(T(n,k)*x^k, k=0..n): (1-z^2)/(1+(x-2)*z+z^2) (from the Riordan property).
O.g.f. column No. k: ((1-x)/(1+x))*(x/(1+x)^2)^k, k >= 0.
T(0,0) = 1, T(n,k) = (-1)^(n-k)*(2*n/(n+k))*binomial(n+k,n-k), n>=1, and T(n,k) = 0 if n < k. (From the Chebyshev T-polynomial formula due to Waring's formula.)
(End)
T(n,k) = -2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 29 2013

A111286 Number of permutations avoiding the patterns {1342, 1432, 2341, 2431, 3142, 3241, 3412, 3421, 4132, 4231, 4312, 4321}; number of strong sorting class based on 1342.

Original entry on oeis.org

1, 1, 2, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Len Smiley, Nov 01 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb., Vol. 12 (2005), R31.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A003945, A007283, A042950, A098011, A110164 - differs from each by one initial term.

Table[If[n == 1, 1, If[n == 2, 2, 3*2^(n - 2)]], {n, 32}] (* Robert G. Wilson v *)
LinearRecurrence[{2},{1,2,6},40] (* Harvey P. Dale, Jul 14 2019 *)

a(n) = 3*2^(n-2), n>=3.

a(n) = 2*a(n-1) for n=3. G.f.: (1-x+2*x^3)/(1-2*x). - Colin Barker, Nov 29 2012

a(0)=1 prepended by Alois P. Heinz, Mar 12 2024

A010719 Period 2: repeat {5,8}.

Original entry on oeis.org

5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5, 8, 5

Offset: 0

N. J. A. Sloane, Dec 11 1996

The inverse binomial transform is 5, 3, -6, 12, -24, 48, -96, 192, -384, 768, ... essentially A110164. - R. J. Mathar, Feb 25 2023

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

Cf. A176323.

[ 3*(n mod 2) + 5 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 14 2014

A010719:=n->3*(n mod 2) + 5; seq(A010719(n), n=0..50); # Wesley Ivan Hurt, Jun 14 2014

Table[3 Mod[n, 2] + 5, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 14 2014 *)

a(n) = a(n-2) for n > 1; a(0) = 5, a(1) = 8. G.f.: (5+8*x)/((1-x)*(1+x)). - Vincenzo Librandi, Aug 01 2010

a(n) = 3*(n mod 2) + 5. - Wesley Ivan Hurt, Jun 14 2014

A173457 Number of cell turned "ON" at n-th stage of cellular automaton of A173456.

Original entry on oeis.org

0, 1, 8, 12, 4, 28, 36, 4, 28, 36, 12, 84, 108, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324, 12, 84, 108, 36, 252, 324, 36, 252, 324, 108, 756, 972, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324

Offset: 0

Omar E. Pol, Feb 18 2010

Essentially the first differences of A173456.

It appears that row lengths give A098011. After the initial zero, it appears that row lengths give the absolute values of A110164. - Omar E. Pol, Apr 22 2013

			From Omar E. Pol, Apr 22 2013 (Start):
When written as an irregular triangle begins:
0;
1;
8,12;
4,28,36;
4,28,36,12,84,108;
4,28,36,12,84,108,12,84,108,36,252,324;
4,28,36,12,84,108,12,84,108,36,252,324,12,84,108,36,252,324,36,252,324,108,756,972;
4,28,36,12,84,108,12,84,108,36,252,324,...
(End)

Lars Blomberg, Table of n, a(n) for n = 0..6000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A173456, A173458, A173459, A173461.

a(0)=0, a(1)=1, a(2)=8, a(3)=12, for n>=4 when (n MOD 3)=0,1,2 let m=36,4,28 then a(n)=m*A147610((n + 2) / 3). (Found empirically) [Lars Blomberg, Apr 22 2013]

a(41)-a(60) from Lars Blomberg, Apr 22 2013

A173461 Number of cells turned "ON" at n-th stage of cellular automaton of A173460.

Original entry on oeis.org

0, 1, 8, 12, 8, 52, 12, 12, 84, 36, 28, 188, 12, 12, 84, 36, 36, 252, 36, 36, 252, 108, 92, 628, 12, 12, 84, 36, 36, 252, 36, 36, 252, 108, 108, 756, 36, 36, 252, 108, 108, 756, 108, 108, 756, 324, 292, 2012, 12, 12, 84, 36, 36, 252, 36, 36, 252, 108, 108

Offset: 0

Omar E. Pol, Feb 18 2010

Essentially the first differences of A173460.

It appears that row lengths give the absolute values of A110164. - Omar E. Pol, Apr 25 2013

			From _Omar E. Pol_, Apr 25 2013: (Start)
When written as an irregular triangle begins:
0;
1,8;
12,8,52;
12,12,84,36,28,188;
12,12,84,36,36,252,36,36,252,108,92,628;
12,12,84,36,36,252,36,36,252,108,108,756,36,36,252,108,108,756,108,108,756,324,292,2012;
12,12,84,36,36,252,36,36,252,108,108,...
(End)

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A173457, A173460, A173462, A173463.

a(0)=0, a(1)=1, a(2)=8, for n>=3 let i=n/3+1, j=A147610(i), if 2^r==i for some r then let c1=2^(r+1), c2=2^(r+4) else let c1=c2=0, finally when (n MOD 3)=0,1,2 let a(n)=12*j, 12*j-c1, 84*j-c2. (Found empirically) [Lars Blomberg, Apr 23 2013]

More terms a(14)-a(17) from Omar E. Pol, Sep 25 2011

a(18)-a(58) from Lars Blomberg, Apr 23 2013

A176414 Expansion of (7+8x)/(1+2x).

Original entry on oeis.org

7, -6, 12, -24, 48, -96, 192, -384, 768, -1536, 3072, -6144, 12288, -24576, 49152, -98304, 196608, -393216, 786432, -1572864, 3145728, -6291456, 12582912, -25165824, 50331648, -100663296, 201326592, -402653184, 805306368

Offset: 0

Klaus Brockhaus, Apr 17 2010

Inverse binomial transform of A176415.

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

Cf. A176415, A110164 (essentially the same), A122803.

Join[{7},NestList[-2#&,-6,40]] (* Harvey P. Dale, Jun 20 2020 *)

{for(n=0, 29, print1(polcoeff((7+8*x)/(1+2*x)+x*O(x^n), n), ", "))}

A176414(n)=3*(-2)^n+!n*4 \\ M. F. Hasler, Apr 19 2015

a(n) = A110164(n+2) for n > 0.
a(n) = 3*(-2)^n = 3*A122803(n+1) for n > 0; a(0) = 7.
a(n) = -2*a(n-1) for n > 1; a(0) = 7, a(1) = -6.
a(n) = (-1)^n*A132477(n) = (-1)^n*A122391(n+3), n>1.
a(n) = (-1)^n*A111286(n+2) = (-1)^n*A098011(n+4) = (-1)^n*A091629(n) = (-1)^n*A087009(n+3) = (-1)^n*A082505(n+1) = (-1)^n*A042950(n+1) = (-1)^n*A007283(n) = (-1)^n*A003945(n+1), n>0. - R. J. Mathar, Dec 10 2010
E.g.f.: 4 + 3*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited by M. F. Hasler, Apr 19 2015

A114958 a(n) = 62^(n+1) - 5(n+1) - 4.

Original entry on oeis.org

3, 10, 29, 72, 163, 350, 729, 1492, 3023, 6090, 12229, 24512, 49083, 98230, 196529, 393132, 786343, 1572770, 3145629, 6291352, 12582803, 25165710, 50331529, 100663172, 201326463, 402653050, 805306229, 1610612592, 3221225323

Offset: 0

Creighton Dement, Feb 21 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A098011, A042950, A110164, A101229, A058764, A087009, A111286, A007283.

[6*2^(n+1) - 5*(n+1) - 4: n in [0..30] ]; // Vincenzo Librandi, May 18 2011

Vec((3 - 2*x + 4*x^2) / ((1 - x)^2*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Apr 30 2019

From Colin Barker, Apr 30 2019: (Start)
G.f.: (3 - 2*x + 4*x^2) / ((1 - x)^2*(1 - 2*x)).
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n>2.
(End)

cp's OEIS Frontend

A003953 Expansion of g.f.: (1+x)/(1-10*x).

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A110162 Riordan array ((1-x)/(1+x), x/(1+x)^2).

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A111286 Number of permutations avoiding the patterns {1342, 1432, 2341, 2431, 3142, 3241, 3412, 3421, 4132, 4231, 4312, 4321}; number of strong sorting class based on 1342.

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A010719 Period 2: repeat {5,8}.

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A173457 Number of cell turned "ON" at n-th stage of cellular automaton of A173456.

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A173461 Number of cells turned "ON" at n-th stage of cellular automaton of A173460.

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A176414 Expansion of (7+8*x)/(1+2*x).

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A176414 Expansion of (7+8x)/(1+2x).

A114958 a(n) = 62^(n+1) - 5(n+1) - 4.