A102900 -id:A102900 - cp's OEIS frontend

A102901 a(n) = a(n-1) + 6*a(n-2), a(0)=1, a(1)=0.

1, 0, 6, 6, 42, 78, 330, 798, 2778, 7566, 24234, 69630, 215034, 632814, 1923018, 5719902, 17258010, 51577422, 155125482, 464590014, 1395342906, 4182882990, 12554940426, 37652238366, 112981880922, 338895311118, 1016786596650

Offset: 0

Paul Barry, Jan 17 2005

Binomial transform is A102900.

Hankel transform is = 1,6,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

			a(6) = 330; (2*3^6 + 3*(-2)^6)/5 = (1458 + 192)/5 = 330.

Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,6).

Cf. A015441, A102900.

[(2*3^n+3*(-2)^n)/5: n in [0..50]]; // Vincenzo Librandi, Jul 20 2013

A102901:=n->(2*3^n+3*(-2)^n)/5; seq(A102901(k), k=0..60); # Wesley Ivan Hurt, Nov 05 2013

CoefficientList[Series[(1-x)/((1+2x)(1-3x)), {x,0,50}], x] (* Vincenzo Librandi, Jul 20 2013 *)

a(n)=([0,1; 6,1]^n*[1;0])[1,1] \\ Charles R Greathouse IV, Mar 28 2016

A102901=BinaryRecurrenceSequence(1,6,1,0)
[A102901(n) for n in range(51)] # G. C. Greubel, Dec 09 2022

G.f.: (1-x)/((1+2*x)*(1-3*x)).
a(n) = (2*3^n + 3*(-2)^n)/5.
a(n) = 6*A015441(n-1), for n>0.

A247666 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 7-celled neighborhood.

Original entry on oeis.org

1, 7, 7, 25, 7, 49, 25, 103, 7, 49, 49, 175, 25, 175, 103, 409, 7, 49, 49, 175, 49, 343, 175, 721, 25, 175, 175, 625, 103, 721, 409, 1639, 7, 49, 49, 175, 49, 343, 175, 721, 49, 343, 343, 1225, 175, 1225, 721, 2863, 25, 175, 175, 625

Offset: 0

N. J. A. Sloane, Sep 22 2014

nonn

The neighborhood of a cell consists of the cell itself together with its six surrounding cells. A cell is ON at generation n iff an odd number of its neighbors were ON at the previous generation. We start with one ON cell.
This is the Run Length Transform of the sequence 1,7,25,103,409,1639,26215,... (almost certainly A102900).
This appears to be the same as the number of ON cells in a certain 2-D CA on the square grid in which the neighborhood of a cell is defined by f = 1/(x*y)+1/x+1/x*y+1/y+x/y+x+x*y, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation. Here is the neighborhood:
[X, 0, X]
[X, 0, X]
[X, X, X]
which contains a(1) = 7 ON cells.
This is the odd-rule cellular automaton defined by OddRule 557 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Furthermore, this is also the number of ON cells in the 2-D CA on the square grid in which the neighborhood of a cell is defined by f = 1/(x*y)+1/x+1/y+1+y+x+x*y, with the same rule. Here is the neighborhood:
[0, X, X]
[X, X, X]
[X, X, 0]
- N. J. A. Sloane, Feb 19 2015
This is the odd-rule cellular automaton defined by OddRule 376 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
The partial sums are in A253767 in which the structure looks like an irregular stepped pyramid, apparently with a like-hexagonal base. - Omar E. Pol, Jan 29 2015

			From _Omar E. Pol_, Jan 29 2015: (Start)
May be arranged into blocks of sizes A011782:
1;
7;
7, 25;
7, 49, 25, 103;
7, 49, 49, 175, 25, 175, 103, 409;
7, 49, 49, 175, 49, 343, 175, 721, 25, 175, 175, 625, 103, 721, 409, 1639;
7, 49, 49, 175, 49, 343, 175, 721, 49, 343, 343, 1225, 175, 1225, 721, 2863, 25, 175, 175, 625, ...
It appears that right border gives A102900 without repetitions, see Comments section. [This is just a restatement of the fact that this sequence is the run length transform of what is presumably A102900. - _N. J. A. Sloane_, Feb 06 2015]
(End)
From _Omar E. Pol_, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
1;
..
7;
..
7;
25;
.........
7,    49;
25;
103;
...................
7,    49,  49, 175;
25,  175;
103;
409;
......................................
7,    49,  49, 175, 49, 343, 175, 721;
25,  175, 175, 625;
103, 721;
409;
1639;
...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
(End)

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, Illustrations of generations 0 to 4
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Cf. A102900, A071053, A160239, A247640.

C := f->`if`(type(f,`+`),nops(f),1);
f := 1+1/x+x+1/y+y+1/(x*y)+x*y;
g := n->expand(f^n) mod 2;
[seq(C(g(n)),n=0..100)];

A247666[n_] := Total[CellularAutomaton[{170, {2, {{1, 1, 0}, {1, 1, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A247666, 52, 0] (* JungHwan Min, Sep 01 2016 *)
A247666L[n_] := Total[#, 2] & /@ CellularAutomaton[{170, {2, {{1, 1, 0}, {1, 1, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A247666L[51] (* JungHwan Min, Sep 01 2016 *)

a(n) = number of terms in expansion of f^n mod 2, where f = 1+1/x+x+1/y+y+1/(x*y)+x*y (mod 2);

A086901 a(1) = a(2) = 1; for n>2, a(n) = 4a(n-1) + 3a(n-2).

Original entry on oeis.org

1, 1, 7, 31, 145, 673, 3127, 14527, 67489, 313537, 1456615, 6767071, 31438129, 146053729, 678529303, 3152278399, 14644701505, 68035641217, 316076669383, 1468413601183, 6821884412881, 31692778455073, 147236767058935

Offset: 1

Rick Powers (rick.powers(AT)mnsu.edu), Sep 18 2003

			a(3) = 4*1 + 3*1 = 7;
a(4) = 4*7 + 3*1 = 31.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Sela Fried and Toufik Mansour, Staircase graph words, arXiv:2312.08273 [math.CO], 2023.
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.

Cf. A015530, A102900, A122542.

a086901 n = a086901_list !! (n-1)
a086901_list = 1 : 1 : zipWith (+)
               (map (* 3) a086901_list) (map (* 4) $ tail a086901_list)
-- Reinhard Zumkeller, Feb 13 2015

[n le 2 select 1 else 4*Self(n-1) +3*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 28 2024

a[n_]:=(MatrixPower[{{3,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{3,4},#]}]&, {1,1},40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{4,3}, {1,1}, 41] (* G. C. Greubel, Oct 28 2024 *)

A086901(n)=if(n<3,1,4*A086901(n-1)+3*A086901(n-2)) \\ Michael B. Porter, Apr 04 2010

A086901=BinaryRecurrenceSequence(4,3,1,1)
[A086901(n) for n in range(41)] # G. C. Greubel, Oct 28 2024

a(n) = ((c + 5)*b^n - (b + 5)*c^n)/14, where b = 2 + sqrt(7), c = 2 - sqrt(7).
From Ralf Stephan, Feb 01 2004: (Start)
G.f.: x*(1-3*x)/(1 - 4*x - 3*x^2).
a(n) = A015530(n) - 3*A015530(n-1) = 1 + 6*Sum_{k=0..n-2} A015530(k). (End)
a(n+1) = Sum_{k=0..n} 3^(n-k)*A122542(n,k), n>=0. - Philippe Deléham, Oct 27 2006
a(n) = upper left term in the 2 X 2 matrix [1,2; 3,3]^(n-1). - Gary W. Adamson, Mar 02 2008
G.f.: G(0)*(1-3*x)/(2-4*x), where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013
E.g.f.: exp(2*x)*( cosh(sqrt(7)*x) - (1/sqrt(7))*sinh(sqrt(7)*x) ). - G. C. Greubel, Oct 28 2024

More terms from Ray Chandler, Sep 19 2003

A128625 Expansion of (1+3x)/(1-5x).

Original entry on oeis.org

1, 8, 40, 200, 1000, 5000, 25000, 125000, 625000, 3125000, 15625000, 78125000, 390625000, 1953125000, 9765625000, 48828125000, 244140625000, 1220703125000, 6103515625000, 30517578125000, 152587890625000, 762939453125000, 3814697265625000, 19073486328125000

Offset: 0

Paul Barry, Mar 14 2007

Binomial transform of A102900(n+1).

Hankel transform is := 1,-24,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A102900.

[n eq 0 select 1 else 8*5^(n-1): n in [0..50]]; // G. C. Greubel, Mar 12 2024

CoefficientList[Series[(1 + 3 x)/(1 - 5 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1},NestList[5#&,8,30]] (* or *) LinearRecurrence[{5},{1,8},30] (* Harvey P. Dale, Dec 23 2021 *)

[(8*5^n - 3*int(n==0))//5 for n in range(51)] # G. C. Greubel, Mar 12 2024

a(n) = (8/5)*5^n - (3/5)*0^n.
a(0)=1, a(n) = Sum_{k=0..n} ((n+k)/n)*binomial(n,k)*2^(n-k)*3^k, n > 0.
E.g.f.: (8*exp(5*x) - 3)/5. - G. C. Greubel, Mar 12 2024

A122016 Riordan array(1, x(1+2x)/(1-x)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 3, 6, 1, 0, 3, 15, 9, 1, 0, 3, 24, 36, 12, 1, 0, 3, 33, 90, 66, 15, 1, 0, 3, 42, 171, 228, 105, 18, 1, 0, 3, 51, 279, 579, 465, 153, 21, 1, 0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1, 0, 3, 69, 576, 2172, 3858, 3258, 1344, 276, 27, 1

Offset: 0

Philippe Deléham, Sep 24 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Rising and falling diagonals are A078010 and A122552.

			Triangle begins:
  1;
  0, 1;
  0, 3,  1;
  0, 3,  6,   1;
  0, 3, 15,   9,    1;
  0, 3, 24,  36,   12,    1;
  0, 3, 33,  90,   66,   15,   1;
  0, 3, 42, 171,  228,  105,  18,   1;
  0, 3, 51, 279,  579,  465, 153,  21,  1;
  0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1;

Huyile Liang, Jinyang Zhang, and Yu Wang, Some properties of the matrix related to q-coloured coordination number, Filomat (2024) Vol. 38, No. 4, 1465-1477. See p. 1466.

Cf. Diagonals A000012, A008585, A062741, columns A000007, A122553, A122709, row sums A026150.

Cf. A078010, A084938, A102900, A117575, A122117, A122552, A147518.

T[n_,k_]:=SeriesCoefficient[(1-x)/(1-(y+1)*x-2*y*x^2),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Dec 27 2023 *)

Sum_{k=0..n} T(n,k)*x^(n-k) = A026150(n), A102900(n) for x = 1, 2.
T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1). - Philippe Deléham, Sep 25 2006
G.f.: (1-x)/(1-(y+1)*x-2*y*x^2). - Philippe Deléham, Jan 31 2012
Sum_{k=0..n} T(n,k)*x^k = A117575(n+1), A000007(n), A026150(n), A122117(n), A147518(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 31 2012

More terms from Stefano Spezia, Dec 27 2023

A140323 First differences of A140322.

Original entry on oeis.org

1, 9, 31, 129, 511, 2049, 8191, 32769, 131071, 524289, 2097151, 8388609, 33554431, 134217729, 536870911, 2147483649, 8589934591, 34359738369, 137438953471, 549755813889, 2199023255551, 8796093022209, 35184372088831, 140737488355329, 562949953421311

Offset: 1

Paul Curtz, May 26 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A140322.

a[n_] := (2*4^n - (-1)^n); Array[a, 25, 0] (* Robert G. Wilson v, Aug 13 2011 *)

Mix A004171(2*n)-1, A004171(2*n+1)+1.
a(n+1)-4*a(n) = period 2: repeat [5, -5].
G.f.: x*(1 + 6*x)/(1 - 3*x - 4*x^2). - Ilya Gutkovskiy, May 11 2016
From Klaus Purath, Aug 03 2020: (Start)
a(n) = 8*A015521(n-1) + A102900(n-1).
a(n) = 30*4^(n-3) + a(n-2), n>2.
a(n+2) = 3*a(n+1) + 4*a(n). (End)

Extended and offset changed from 0 to 1 by Vincenzo Librandi, Aug 09 2011

A275667 Number of ON cells after n generations in a 2-dimensional "Odd-Rule" cellular automaton on triangular tiling.

Original entry on oeis.org

1, 3, 7, 9, 7, 21, 25, 27, 7, 21, 49, 63, 25, 75, 103, 81, 7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243, 7, 21, 49, 63, 49, 147, 175, 189, 49, 147, 343, 441, 175, 525, 721, 567, 25, 75, 175, 225, 175, 525, 625, 675, 103, 309, 721

Offset: 0

Kovba Alexey, Aug 04 2016

Each triangular tile has 3 neighbors. A cell is ON in a given generation if and only if there was an odd number of ON cells among the three nearest neighbors in the preceding generation.
At the initial moment there is a single ON cell.
Given pattern replicates after a number of generations which is a power of 2 when a(n) = 7.
Number of cells on each even step minus one is divisible by 6.
By analogy with the Ekhad, Sloane, Zeilberger link, one may suppose that using ternary expansion of n, recurrence relations for a(n) can be obtained and proved.
From Andrey Zabolotskiy, Aug 04 2016: (Start)
If the first conjecture from the Formula section is true then the fact that the right border of the triangle (see Example) gives A000244 follows directly from it.
If the second conjecture is true then the numbers just before the right border give A102900.
Since the 7 cells which are ON at the beginning of every row are farther and farther away from each other, the n-th term of a row (with offset 0) is a(n)*7 for not very large n.
See also comments to A247666.
(End)
This is ETA rule 170. See the Sadat-Benedek reference for proof of pattern replication. - Paul Cousin, Apr 22 2025

			From _Omar E. Pol_, Aug 04 2016: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
3;
7, 9;
7, 21, 25, 27;
7, 21, 49, 63, 25, 75, 103, 81;
7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243;
...
It appears that the right border gives A000244.
(End)

Paul Cousin, Table of n, a(n) for n = 0..16384
Paul Cousin, Illustration for n = 0..128
Paul Cousin, Elementary Triangular Automaton Rule 170
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015.
Alexey Kovba, Illustration for n = 0..5
MohammadReza Saadat and Benedek Nagy, Copy Machines - Self-reproduction with 2 States on Archimedean Tilings, Journal of Cellular Automata, vol. 17, pp. 221-249, 2023.
Index to sequences in the OEIS related to cellular automata

Cf. A160239 (square tiling analog), A247640, A247666 (hexagonal tiling analogs).
Cf. A000244, A011782, A102900, A382971.
Pattern replicating ETA rules: A383369 (rule 90).

a(0) = 1. Conjecture: a(2*t+1) = 3*a(t).

Conjectures: a(8*t+6) = 3*a(4*t+2) + 4*a(2*t), a(8*t+2) = 3*a(4*t) + 4*a(2*t), a(4*t) = a(2*t). These conjectured formulas together give recurrent relations for a(n) for any n. Also, obviously a(2*n) = A247666(n). - Andrey Zabolotskiy, Aug 04 2016

A335794 Number of ON cells at n-th generation in an "Ulam-Warburton and Friedkin Replicator" hybrid two-dimensional cellular automaton.

Original entry on oeis.org

0, 1, 7, 13, 31, 37, 67, 85, 139, 145, 175, 205, 283, 301, 391, 445, 595, 601, 631, 661, 739, 769, 895, 973, 1195, 1213, 1303, 1393, 1627, 1681, 1927, 2077, 2467, 2473, 2503, 2533, 2611, 2641, 2767, 2845, 3067, 3097, 3223, 3349, 3667, 3745, 4111, 4333, 4939, 4957

Offset: 0

Cody B Duncan, Jun 23 2020

This 2D CA uses the neighborhood:
[0 X X]
[X X X]
[X X 0]
If a cell has an even number of ON neighbors and it is currently OFF, stay OFF; otherwise turn ON.
The results are similar to those for A151723, but with a distorted grid.
(The "look" keyword refers to the animation. - N. J. A. Sloane, Jul 03 2020)

S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

Cody B. Duncan, Animation

Cf. A102900, A147562, A151723, A247666, A335795.

A251634 Numerators of inverse Riordan triangle of Riordan triangle A029635. Riordan (1/(1-x), x/(1+2*x)). Triangle read by rows for 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 3, -3, 1, 1, -5, 9, -5, 1, 1, 11, -23, 19, -7, 1, 1, -21, 57, -61, 33, -9, 1, 1, 43, -135, 179, -127, 51, -11, 1, 1, -85, 313, -493, 433, -229, 73, -13, 1, 1, 171, -711, 1299, -1359, 891, -375, 99, -15, 1, 1, -341, 1593, -3309, 4017, -3141, 1641, -573, 129, -17, 1

Offset: 0

Wolfdieter Lang, Jan 09 2015

The denominators are given by 2*A130321(n,m).
The rational lower triangular matrix with entries R(n,m) = T(n,m)/(2*A130321(n,m)) = T(n,m)/2^(n-m+1) for n >= m >= 0 and 0 otherwise is the inverse of the Riordan matrix A029635.
R is the rational Riordan triangle (1/(2-x), x/(1+x)).
The numerator triangle T is the Riordan array (1/(1-x), x/(1+2*x)). From the o.g.f. of the column sequences of R and T(n,m) = 2^(n-m+1)*R(n,m).
Row sums of the rational triangle R are [1/2, seq(3/2^(n+1), for n >= 1)].
Row sums of the present triangle T give [repeat(1,2,)].
Alternating row sums of the rational triangle R give (-1)^n*A102900(n)/2^(n+1), n >= 0: 1/2, -1/4, 7/8, -25/16, 103/32, -409/64, 1639/128, -6553/256, 26215/512, ... .
Alternating row sums of the present triangle T give A084567.
The inverse of the T Riordan matrix is ((1-3*x)/(1-2*x), x/(1-2*x)) = A251636.
Equals A248810 when the first column (m = 0) of ones is removed. - Georg Fischer, Jul 26 2023

			The triangle T(n,m) begins:
  n\m  0    1    2     3     4     5    6    7    8   9 ...
  0:   1
  1:   1    1
  2:   1   -1    1
  3:   1    3   -3     1
  4:   1   -5    9    -5     1
  5:   1   11  -23    19    -7     1
  6:   1  -21   57   -61    33    -9    1
  7:   1   43 -135   179  -127    51  -11    1
  8:   1  -85  313  -493   433  -229   73  -13    1
  9:   1  171 -711  1299 -1359   891 -375   99  -15   1
  ...
The rational Riordan triangle R(n,m) begins:
  n\m  0      1      2      3     4    5  ...
  0:  1/2
  1:  1/4    1/2
  2:  1/8   -1/4    1/2
  3:  1/16   3/8   -3/4    1/2
  4:  1/32  -5/16   9/8   -5/4   1/2
  5:  1/64  11/3  -23/1   19/8  -7/4  1/2
  ...
For more rows see the link.

Wolfdieter Lang, Eleven rows of the triangle, and rational triangle.

Cf. A029635, A084567, A102900, A130321, A251636, A248810, A331333.

A251634 := proc(n, k) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`if`(k > n, 0, S(n-1, k-1)/k - 2*S(n-1, k))) end: k!*S(n, k) end:
seq(seq(A251634(n, k), k=0..n)), n=0..9); # Peter Luschny, Jan 19 2020

O.g.f. of the row polynomials P(n,x) = Sum_{m=0..n} R(n,m)*x^m of the rational triangle R: G(z,x) = Sum_{n>=0} P(n,x)*z^n = (1+z)/((2-z)*(1+(1-x)*z)).
O.g.f. column m of the rational triangle R: (1/(2-x))*(x/(1+x))^m, m >= 0 (Riordan property of R).
O.g.f. column m of the numerator triangle T: (1/(1-x))*(x/(1+2*x))^m, m >= 0. (Riordan property of T).
T(n, k) = k!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else S(n-1, k-1)/k - 2*S(n-1, k). - Peter Luschny, Jan 19 2020

A274845 a(0)=1, a(1)=0, a(4n+2) = a(4n+3) = a(4n+5) = (4^(n+1) +(-1)^n)/5, a(4n+4) = (24^(n+1) -3(-1)^n)/5.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 3, 7, 3, 13, 13, 25, 13, 51, 51, 103, 51, 205, 205, 409, 205, 819, 819, 1639, 819, 3277, 3277, 6553, 3277, 13107, 13107, 26215, 13107, 52429, 52429, 104857, 52429, 209715, 209715, 419431, 209715, 838861, 838861, 1677721, 838861, 3355443

Offset: 0

Paul Curtz, Jul 08 2016

Antidiagonals of the array in A274613 written as a triangle:
1,
0, 1/2,
0, 1/2, 1/4,
0,   0, 1/2, 1/8,
0,   0, 1/4, 3/8, 1/16,
... .
a(n) is the numerators of the antidiagonal sums i.e. 1, 0, 1/2, 1/2, 1/4, 1/2, 3/8, 3/8, 7/16, 3/8, 13/32, 13/32, 25/64, 13/32, ... = a(n)/b(n).
The denominators b(n) are A173300(n).
a(0)+a(1) = 1, a(4n+2) +a(4n+3) +a(4n+4) +a(4n+5) = 4, 16, 64, 256, ... .

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 3, 0, 0, 0, 4).

Cf. A000302, A102900, A015521, A173300, A274613.

LinearRecurrence[{0,0,0,3,0,0,0,4}, {1,0,1,1,1,1,3,3}, 50] (* G. C. Greubel, Jul 08 2016 *)

Vec((1+x^2+x^3-2*x^4+x^5)/((1-2*x^2)*(1+2*x^2)*(1+x^4)) + O(x^60)) \\ Colin Barker, Jul 22 2016

a(4n) = A102900(n), a(4n+1) = A015521(n), a(4n+2) = a(4n+3) = A015521(n+1).
a(n) = 3*a(n-4) + 4*a(n-8). - G. C. Greubel, Jul 08 2016
G.f.: (1+x^2+x^3-2*x^4+x^5) / ((1-2*x^2)*(1+2*x^2)*(1+x^4)). - Colin Barker, Jul 22 2016

More terms from Colin Barker, Jul 22 2016

cp's OEIS Frontend

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A086901 a(1) = a(2) = 1; for n>2, a(n) = 4a(n-1) + 3a(n-2).

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A274845 a(0)=1, a(1)=0, a(4n+2) = a(4n+3) = a(4n+5) = (4^(n+1) +(-1)^n)/5, a(4n+4) = (24^(n+1) -3(-1)^n)/5.