A052992 -id:A052992 - cp's OEIS frontend

A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.

1, 1, 3, 5, 9, 14, 24, 35, 55, 81, 120, 171, 248, 345, 486, 669, 920, 1246, 1690, 2256, 3014, 3984, 5253, 6870, 8970, 11618, 15022, 19306, 24745, 31557, 40154, 50845, 64244, 80850, 101501, 126982, 158514, 197218, 244865, 303143, 374497, 461435

Offset: 0

James Sellers, Apr 04 2000

Sum of products of multiplicities of odd parts in all partitions of n (if there are no odd parts in a partition then product of multiplicities is considered to be 1). - Vladeta Jovovic, Feb 16 2005

The sequence A077285 is the same but with multiplicities of all parts.

			1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 14*x^5 + 24*x^6 + 35*x^7 + 55*x^8 + ...
q^-1 + q^11 + 3*q^23 + 5*q^35 + 9*q^47 + 14*q^59 + 24*q^71 + 35*q^83 + ...
a(6) = 24 since the 5 partitions 6 = 5+1 = 4+2 = 3+2+1 = 2+2+2 each contribute 1, the 3 partitions 4+1+1 = 3+3 = 2+2+1+1 each contribute 2, the partition 3+1+1+1 contributes 3, the partition 2+1+1+1+1 contributes 4, and the partition 1+1+1+1+1+1 contributes 6 to give total 24 = 5*1 + 3*2 + 1*3 + 1*4 + 1*6. - _Michael Somos_, Mar 09 2011

George E. Andrews, Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See formula (18) on page 3944.

Cf. A077285, A052992, A097242, A247661, A247662.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
      +add(b(n-i*j, i-1)*`if`(irem(i, 2)=1, j, 1), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 16 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1] * If[Mod[i, 2] == 1, j, 1], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
QP = QPochhammer; s = QP[q^4] * (QP[q^6]^2 / (QP[q] * QP[q^2] * QP[q^3] * QP[q^12])) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))} /* Michael Somos, Mar 09 2011 */

Expansion of q^(1/12) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - Michael Somos, Mar 09 2011
Euler transform of period 12 sequence [ 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, ...]. - Michael Somos, Mar 09 2011
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(12*k - 10)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 2)))^(-1). [Andrews, p. 10, equation (5.9)]
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*sqrt(2)*n). - Vaclav Kotesovec, Nov 28 2015

A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

0, 1, 0, -1, 2, 0, 3, -2, 4, 0, -5, 6, -4, 8, 0, 11, -10, 12, -8, 16, 0, -21, 22, -20, 24, -16, 32, 0, 43, -42, 44, -40, 48, -32, 64, 0, -85, 86, -84, 88, -80, 96, -64, 128, 0, 171, -170, 172, -168, 176, -160, 192, -128, 256, 0, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 0

Offset: 0

Paul Curtz, Jul 24 2008

A variant of the triangle A140503, now including the diagonal.

Since the diagonal contains zeros, rows sums are those of A140503.

			Triangle begins as:
    0;
    1,   0;
   -1,   2,   0;
    3,  -2,   4,  0;
   -5,   6,  -4,  8,   0;
   11, -10,  12, -8,  16,  0;
  -21,  22, -20, 24, -16, 32,  0;

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

Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.

Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.

[2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 18 2023

A001045:= n -> (2^n-(-1)^n)/3;
A140944:= proc(n,k) if n = 0 then A001045(k); else procname(n-1,k+1)-procname(n-1,k) ; fi; end:
seq(seq(A140944(n,k),k=0..n),n=0..10); # R. J. Mathar, Sep 07 2009

T[0, 0]=0; T[1, 0]= T[0, 1]= 1; T[0, k_]:= T[0, k]= T[0, k-1] + 2*T[0, k-2]; T[n_, n_]=0; T[n_, k_]:= T[n, k] = T[n-1, k+1] - T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2014 *)
Table[2^k*(1-(-2)^(n-k))/3, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2023 *)

T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ Jianing Song, Aug 11 2022

def A140944(n,k): return 2^k*(1 - (-2)^(n-k))/3
flatten([[A140944(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 18 2023

T(n, k) = T(n-1, k+1) - T(n-1, k). T(0, k) = A001045(k).
T(n, k) = (2^k - 2^n*(-1)^(n+k))/3, for n >= k >= 0. - Jianing Song, Aug 11 2022
From G. C. Greubel, Feb 18 2023: (Start)
T(n, n-1) = A000079(n).
T(2*n, n) = (-1)^(n+1)*A192382(n+1).
T(2*n, n-1) = (-1)^n*A246036(n-1).
T(2*n, n+1) = A083086(n).
T(3*n, n) = -A115489(n).
Sum_{k=0..n} T(n, k) = A052992(n)*[n>0] + 0*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A045883(n).
Sum_{k=0..n} 2^k*T(n, k) = A084175(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^(n+1)*A109765(n).
Sum_{k=0..n} 3^k*T(n, k) = A091056(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*A097038(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^(n+1)*A138495(n). (End)

Edited and extended by R. J. Mathar, Sep 07 2009

A281146 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 515", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 100, 1000, 10100, 101000, 1010100, 10101000, 101010100, 1010101000, 10101010100, 101010101000, 1010101010100, 10101010101000, 101010101010100, 1010101010101000, 10101010101010100, 101010101010101000, 1010101010101010100, 10101010101010101000

Offset: 0

Robert Price, Feb 22 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A052992, A153772, A279053.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 515; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Conjectures from Colin Barker, Feb 22 2017: (Start)
a(n) = 100*(10^n - 1) / 99 for n>1 and even.
a(n) = 100*(10^n - 10) / 99 for n>1 and odd.
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n>2.
G.f.: (1 - x^2 + 100*x^4) / ((1 - x)*(1 + x)*(1 - 10*x)).
(End)

A339694 Triangle read by rows: A(n, k) = Sum_{i=0..n-1} x(i, k)*2^i, where x(i, k) = A014682^(i)(k) (mod 2) using the i-th iteration of A014682.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 0, 5, 2, 3, 4, 1, 6, 7, 0, 5, 10, 3, 4, 1, 6, 7, 8, 13, 2, 11, 12, 9, 14, 15, 0, 21, 10, 3, 20, 17, 6, 23, 8, 29, 2, 11, 12, 9, 14, 15, 16, 5, 26, 19, 4, 1, 22, 7, 24, 13, 18, 27, 28, 25, 30, 31, 0, 21, 42, 35, 20, 17, 6, 23, 40, 29, 34, 11

Offset: 1

Sebastian Karlsson, Dec 13 2020

A(n, k) is periodic with period 2^n, i.e., A(n, k) = A(n, k + 2^n). Each row in the triangle is therefore [A(n, 0), A(n, 1), ..., A(n, 2^n-1)].
The binary modular Collatz graph C(n) is the graph representing the dynamics of the Collatz function (A014682) modulo 2^n. For example, in C(3), there is an arrow from 3 to 5 and from 3 to 1 because any number that is 3 modulo 8 either gets mapped to 5 modulo 8 or 1 modulo 8. The vertices of the de Bruijn graph B(2,n) are words of length n consisting of the two symbols 0 and 1. If one represents these vertices as integers, b_0 b_1 ... b_{n-1} -> Sum_{i=0..n-1} b_i*2^i, then A(n) : C(n) -> B(2,n) is a graph isomorphism [Laarhoven, de Weger].
The n-th row is a permutation on the set {0..2^n-1}. For n > 5, the order of this permutation is 2^(n-4) [Bernstein, Lagarias]. - Sebastian Karlsson, Jan 17 2021

			Triangle begins:
n=1 : 0 1;
n=2 : 0 1  2 3;
n=3 : 0 5  2 3 4 1 6 7;
n=4 : 0 5 10 3 4 1 6 7 8 13 2 11 12 9 14 15;
...
A(3, 4) = Sum_{i=0..2} x(i, 4)*2^i = 0*2^0 + 0*2^1 + 1*2^2 = 4.
A(4, 1) = Sum_{i=0..3} x(i, 1)*2^i = 1*2^0 + 0*2^1 + 1*2^2 + 0*2^3 = 5.

Sebastian Karlsson, Rows n = 1..13, flattened
D. J. Bernstein and J. C. Lagarias, The 3x+1 conjugacy map, Canadian Journal of Mathematics, Vol. 48, 1996, pp. 1154-1169.
Thijs Laarhoven and Benne de Weger, The Collatz conjecture and De Bruijn graphs, Indagationes Mathematicae. New Series, 24(4) (2013), 971-983. arXiv version, arXiv:1209.3495 [math.NT], 2012.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A014682, A304715.

Cf. A000004 (column 0), A052992 (column 1), A263053 (column 2).

A339694row[n_]:=Table[Sum[Mod[Nest[If[OddQ[#],(3#+1)/2,#/2]&,k,i],2]2^i,{i,0,n-1}],{k,0,2^n-1}];Array[A339694row,6] (* Paolo Xausa, Aug 08 2023 *)

f(n) = if(n%2, 3*n+1, n)/2 \\ A014682
x(i, n) = my(x=n); for (k=1, i, x = f(x)); x % 2;
A(n, k) = sum(i=0, k-1, x(i, n)*2^i);
row(n) = vector(2^n, i, A(i-1, n));
tabf(nn) = for (n=1, nn, print(row(n))); \\ Michel Marcus, Dec 21 2020

def A014682(k):
    if k % 2 == 0:
        return k // 2
    else:
        return (3*k + 1) // 2
def x(i, k):
    while i > 0:
        k = A014682(k)
        i = i - 1
    return k % 2
def A(n, k):
    L = [x(i, k) * 2**i for i in range(0, n)]
    return sum(L)

A000120( T(n, (m + 1) mod 2^n) ) = log_3( A014682^n(m + 1 + 2^n) - A014682^n(m + 1) ), m = 0..2^n-1. (A000120 is the binary weight.) - Thomas Scheuerle, Aug 23 2021

A093833 3^n-Jacobsthal(n).

Original entry on oeis.org

1, 2, 8, 24, 76, 232, 708, 2144, 6476, 19512, 58708, 176464, 530076, 1591592, 4777508, 14337984, 43024876, 129096472, 387333108, 1162086704, 3486434876, 10459654152, 31379661508, 94140382624, 282423944076, 847277424632

Offset: 0

Paul Barry, Apr 17 2004

Binomial transform of A052992. Binomial transform is A093834. Partial sums are A004054. Sums of consecutive pairs yield A053581.
Contribution from Johannes W. Meijer, Aug 15 2010: (Start)
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 343, 349, 373 and 469, lead to this sequence. For the central square these vectors lead to the companion sequence A175659.
(End)

Index entries for linear recurrences with constant coefficients, signature (4,-1,-6)

G.f.: (1-x)^2/((1+x)(1-2x)(1-3x)); a(n)=3^n-2^n/3+(-1)^n/3; a(n)=3^n-A001045(n).

A117413 Expansion of (1-x^2)/(1-2x^2+4x^3+x^4).

Original entry on oeis.org

1, 0, 1, -4, 1, -12, 17, -24, 81, -104, 241, -508, 817, -1876, 3425, -6512, 13537, -24848, 49697, -97332, 185249, -368604, 710129, -1380872, 2709425, -5233656, 10232209, -19924140, 38689617, -75543460, 146843585, -285921248, 557171393, -1083673376, 2111184193, -4110111076

Offset: 0

Paul Barry, Mar 14 2006

Diagonal sums of number triangle A117411.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,-4,-1).

Cf. A117411.

I:=[1,0,1,-4]; [n le 4 select I[n] else 2*Self(n-1) -4*Self(n-2) -Self(n-3): n in [1..41]]; // G. C. Greubel, Sep 07 2022

CoefficientList[Series[(1-x^2)/(1-2x^2+4x^3+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,-4,-1},{1,0,1,-4},40] (* Harvey P. Dale, Jul 12 2017 *)

@CachedFunction
def a(n): # a = A117413
    if(n<4): return (1, 0, 1, -4)[n]
    else: return 2*a(n-2) - 4*a(n-3) - a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Sep 07 2022

a(n) = 2*a(n-2) - 4*a(n-3) - a(n-4).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2*k} C(n-k, k-j)*C(j, n-2*k)*(-4)^(n-2*k).
a(n) = (-)^n*A052992(2*n). - R. J. Mathar, Nov 22 2024

A164925 Array, binomial(j-i,j), read by rising antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 0, 1, 1, -2, 0, 0, 1, 1, -3, 1, 0, 0, 1, 1, -4, 3, 0, 0, 0, 1, 1, -5, 6, -1, 0, 0, 0, 1, 1, -6, 10, -4, 0, 0, 0, 0, 1, 1, -7, 15, -10, 1, 0, 0, 0, 0, 1, 1, -8, 21, -20, 5, 0, 0, 0, 0, 0, 1, 1, -9, 28, -35, 15, -1, 0, 0, 0, 0, 0, 1, 1, -10, 36, -56, 35, -6, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Mark Dols, Aug 31 2009

Inverse of A052509, or A004070???

			Array, A(n, k), begins as:
  1,  1,  1,   1,  1,   1,  1,  1,  1, ...
  1,  0,  0,   0,  0,   0,  0,  0,  0, ...
  1, -1,  0,   0,  0,   0,  0,  0,  0, ...
  1, -2,  1,   0,  0,   0,  0,  0,  0, ...
  1, -3,  3,  -1,  0,   0,  0,  0,  0, ...
  1, -4,  6,  -4,  1,   0,  0,  0,  0, ...
  1, -5, 10, -10,  5,  -1,  0,  0,  0, ...
  1, -6, 15, -20, 15,  -6,  1,  0,  0, ...
  1, -7, 21, -35, 35, -21,  7, -1,  0, ...
Antidiagonal triangle, T(n, k), begins as:
  1;
  1,  1;
  1,  0,  1;
  1, -1,  0,  1;
  1, -2,  0,  0,  1;
  1, -3,  1,  0,  0,  1;
  1, -4,  3,  0,  0,  0,  1;
  1, -5,  6, -1,  0,  0,  0,  1;
  1, -6, 10, -4,  0,  0,  0,  0,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A004070, A008346, A010892, A052509, A052992.

Cf. A109466, A130595, A164899, A164915, A164965.

A164925:= func< n,k | k eq 0 or k eq n select 1 else Binomial(2*k-n,k) >;
[A164925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2023

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[2*k-n, k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 10 2023 *)

{A(i, j) = if( i<0, 0, if(i==0 || j==0, 1, binomial(j-i, j)))}; /* Michael Somos, Jan 25 2012 */

def A164925(n,k): return 1 if (k==0 or k==n) else binomial(2*k-n, k)
flatten([[A164925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 10 2023

Sum_{k=0..n} T(n, k) = A164965(n). - Mark Dols, Sep 02 2009
From G. C. Greubel, Feb 10 2023: (Start)
A(n, k) = binomial(k-n, k), with A(0, k) = A(n, 0) = 1 (array).
T(n, k) = binomial(2*k-n, k), with T(n, 0) = T(n, n) = 1 (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A008346(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^n*A052992(n). (End)

Edited by Michael Somos, Jan 26 2012

Offset changed by G. C. Greubel, Feb 10 2023

A263053 Number of (n+1) X 2 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

Original entry on oeis.org

2, 2, 10, 10, 42, 42, 170, 170, 682, 682, 2730, 2730, 10922, 10922, 43690, 43690, 174762, 174762, 699050, 699050, 2796202, 2796202, 11184810, 11184810, 44739242, 44739242, 178956970, 178956970, 715827882, 715827882, 2863311530, 2863311530

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

Each row must be either 01 or 10. The two columns are therefore binary complements with sum 2^k-1, where k = n + 1 is the number of rows. If k is even then 2^k-1 is divisible by 3 and the number of solutions is 2*(2^k-1)/3. If k is odd then 2^k-1 == 1 (mod 3) and the number of solutions is (2^k-2)/3. - Andrew Howroyd, Feb 03 2022

			All solutions for n=4:
  0 1   0 1   1 0   1 0   1 0   0 1   1 0   1 0   0 1   0 1
  0 1   0 1   0 1   1 0   0 1   1 0   1 0   0 1   1 0   1 0
  1 0   0 1   0 1   1 0   1 0   1 0   0 1   1 0   0 1   0 1
  0 1   1 0   0 1   0 1   1 0   1 0   1 0   0 1   1 0   0 1
  1 0   0 1   1 0   1 0   1 0   0 1   0 1   0 1   1 0   0 1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 1 of A263060.

Cf. A052992.

[int(2**n - 2/3 -((-2)**n)/3) for n in range(1,40)] # Pascal Bisson, Feb 03 2022

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
From Colin Barker, Jan 01 2019: (Start)
G.f.: 2*x / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 2^n - 2/3 - (-2)^n/3.
(End)
a(n) = 2*A052992(n). - Pascal Bisson, Feb 03 2022

A247661 The number phi_4(n) of Frobenius partitions that allow up to 4 repetitions of an integer in a row.

Original entry on oeis.org

1, 1, 3, 6, 12, 20, 35, 56, 92, 142, 221, 330, 496, 724, 1056, 1512, 2155, 3028, 4240, 5866, 8085

Offset: 0

N. J. A. Sloane, Oct 05 2014

George E. Andrews, Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.

Cf. A000041, A053993, A052992, A247662.

A247662 The number phi_5(n) of Frobenius partitions that allow up to 5 repetitions of an integer in a row.

Original entry on oeis.org

1, 1, 3, 6, 12, 21, 37, 60, 99, 156, 244, 371, 561, 829, 1218, 1763, 2532, 3594, 5068, 7075, 9819

Offset: 0

N. J. A. Sloane, Oct 05 2014

George E. Andrews, Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.

Cf. A000041, A053993, A052992, A247661.

cp's OEIS Frontend

A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.

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A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

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A281146 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 515", based on the 5-celled von Neumann neighborhood.

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A339694 Triangle read by rows: A(n, k) = Sum_{i=0..n-1} x(i, k)*2^i, where x(i, k) = A014682^(i)(k) (mod 2) using the i-th iteration of A014682.

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A093833 3^n-Jacobsthal(n).

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A117413 Expansion of (1-x^2)/(1-2*x^2+4*x^3+x^4).

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A164925 Array, binomial(j-i,j), read by rising antidiagonals.

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A117413 Expansion of (1-x^2)/(1-2x^2+4x^3+x^4).