A001590 -id:A001590 - cp's OEIS frontend

A108153 a(n) = 3a(n-1) + 2a(n-2) + a(n-3).

0, 1, 3, 11, 40, 145, 526, 1908, 6921, 25105, 91065, 330326, 1198213, 4346356, 15765820, 57188385, 207443151, 752472043, 2729490816, 9900859685, 35914032730, 130273308376, 472548850273, 1714107200301, 6217692609825, 22553841080350, 81811015661001

Offset: 0

Roger L. Bagula, Jun 06 2005

Index entries for linear recurrences with constant coefficients, signature (3,2,1).

Cf. A000073, A001590. Essentially the same as A010911.

M = {{0, 1, 0}, {0, 0, 1}, {1, 2, 3}} a3 = Table[MatrixPower[M, i][[1, 3]], {i, 1, 50}]
LinearRecurrence[{3,2,1},{0,1,3},40] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)
nxt[{a_,b_,c_}]:={b,c,a+2b+3c}; NestList[nxt,{0,1,3},30][[;;,1]] (* Harvey P. Dale, Nov 22 2024 *)

G.f.: x/(1-3*x-2*x^2-x^3). [R. J. Mathar, Mar 19 2009]

a(n) = 11*a(n-2) +7*a(n-3)+3*a(n-4). [Gary Detlefs, Sep 13 2010]

Definition replaced by the Adamson recurrence - the Associate Editors of the OEIS, Sep 28 2009

A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

Original entry on oeis.org

1, 1, 3, 7, 49, 321, 2131, 19783, 195777, 2101249, 25721731, 340358151, 4902173233, 75688032577, 1253701725459, 22347046050631, 418439924732161, 8318748086461953, 175769214730290307, 3871849719998940679, 89734800330818444721, 2187944831367633226561

Offset: 0

Vladeta Jovovic, Jan 19 2006

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A000726, A003724, A115276, A000246, A102736, A088009, A001590.

nmax := 30: B := x*(1+x)/(1-x^3) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(0=
      irem(j, 3), 0, a(n-j)*j!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

CoefficientList[Series[E^(x*(1+x)/(1-x^3)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)

E.g.f.: exp(x*(1+x)/(1-x^3)).
a(n) = a(n-1) + 2*(n-1)*a(n-2) + 2*(n-3)*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 6^(-1/4) * n^(n-1/4) * exp(2/3*sqrt(6*n)-n) * (1 - 43/(48*sqrt(6*n))). - Vaclav Kotesovec, Sep 25 2013

2 more terms from R. J. Mathar, Feb 06 2008

A198295 Riordan array (1, x*(1+x)/(1-x^3)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 3, 1, 0, 1, 2, 3, 4, 1, 0, 0, 4, 4, 6, 5, 1, 0, 1, 2, 9, 8, 10, 6, 1, 0, 1, 3, 9, 17, 15, 15, 7, 1, 0, 0, 6, 9, 24, 30, 26, 21, 8, 1, 0, 1, 3, 18, 26, 51, 51, 42, 28, 9, 1

Offset: 0

Philippe Deléham, Jan 26 2012

Triangle T(n,k), read by rows, given by (0, 1, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Antidiagonals sums: see A159284.

			Triangle begins:
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 1, 1, 3, 1
0, 1, 2, 3, 4, 1
0, 0, 4, 4, 6, 5, 1
0, 1, 2, 9, 8, 10, 6, 1
0, 1, 3, 9, 17, 15, 15, 7, 1

A. Luzón, D. Merlini, M. A. Morón, R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Cf. Columns: A000007, A011655, A186731, A185395, A185292.

Cf. Diagonals: A000012, A001477, A161680, A000125.

Sum_{k, 0<=k<=n} T(n,k) = A001590(n+2), n>0.
Sum_{k, 0<=k<=n}T(n,k)*(-1)^(n-k) = A078056(n-1), n>0.
T(n,n) = A000012(n), T(n+1,n) = A001477(n) = n, T(n+2,n) = A161680(n) = A000217(n-1); T(n+3,n) = A000125(n-1), n>=1.
G.f.: (-1+x)*(1+x+x^2)/(-1+x^3+x*y+x^2*y). - R. J. Mathar, Aug 11 2015

A233324 Triangle read by rows: T(n,k) = number of palindromic compositions of n in which no part exceeds k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 3, 4, 1, 2, 3, 3, 4, 1, 5, 6, 7, 7, 8, 1, 3, 6, 6, 7, 7, 8, 1, 8, 11, 14, 14, 15, 15, 16, 1, 5, 11, 12, 14, 14, 15, 15, 16, 1, 13, 20, 27, 28, 30, 30, 31, 31, 32, 1, 8, 20, 23, 28, 28, 30, 30, 31, 31, 32, 1, 21, 37, 52, 55, 60, 60, 62, 62, 63, 63, 64

Offset: 1

L. Edson Jeffery, Dec 11 2013

A palindromic composition of a natural number m is an ordered partition of m into N+1 natural numbers (or parts), p_0, p_1, ..., p_N, of the form m = p_0 + p_1 + ... + p_N such that p_j = p_{N-j}, for each j in {0,...,N}. Two palindromic compositions, sum_{j=0..N} p_j and sum_{j=0..N} q_j (say), are identical if and only if p_j = q_j, j = 0,...,N; otherwise they are taken to be distinct.
Partial sums of rows of A233323.
T(n,k) is defined for n,k >= 0.  T(n,k) = T(n,n) = A016116(n) for k>= 0. - Alois P. Heinz, Dec 11 2013

			Triangle T(n,k) begins:
1;
1,  2;
1,  1,  2;
1,  3,  3,  4;
1,  2,  3,  3,  4;
1,  5,  6,  7,  7,  8;
1,  3,  6,  6,  7,  7,  8;
1,  8, 11, 14, 14, 15, 15, 16;
1,  5, 11, 12, 14, 14, 15, 15, 16;
1, 13, 20, 27, 28, 30, 30, 31, 31, 32;

Alois P. Heinz, Rows n = 1..141, flattened
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A233323.

T(n,2) = A053602(n+1) = A123231(n). T(2n,3) = A001590(n+3). T(2n,4) = A001631(n+4). - Alois P. Heinz, Dec 11 2013

T:= proc(n, k) option remember; `if`(n<=k, 1, 0)+
      add(T(n-2*j, k), j=1..min(k, iquo(n, 2)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Dec 11 2013

T[n_, k_] := T[n, k] = If[n <= k, 1, 0] + Sum[T[n-2*j, k], {j, 1, Min[k, Quotient[ n, 2]]}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

T(n,k)=if(n<1,return(n==0));sum(i=1,k,T(n-2*i,k))+(n<=k) \\ Charles R Greathouse IV, Dec 11 2013

A242572 Indices of primes in A214828.

Original entry on oeis.org

3, 7, 8, 16, 19, 36, 44, 151, 292, 448, 467, 896, 1148, 1607, 1711, 1956, 2020, 6635, 14228, 25519, 43140, 74984, 77696, 137975

Offset: 1

Robert Price, May 17 2014

a(25) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523,A214825, A235862, A214827, A214828.

a={1,6,6}; For[n=3, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[n]]; a=RotateLeft[a]; a[[3]]=sum]

A243572 Irregular triangular array generated as in Comments; contains every positive integer exactly once.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 7, 10, 12, 18, 27, 8, 11, 13, 15, 19, 21, 28, 30, 36, 54, 81, 14, 16, 20, 22, 24, 29, 31, 33, 37, 39, 45, 55, 57, 63, 82, 84, 90, 108, 162, 243, 17, 23, 25, 32, 34, 38, 40, 42, 46, 48, 56, 58, 60, 64, 66, 72, 83, 85, 87, 91, 93, 99, 109

Offset: 1

Clark Kimberling, Jun 07 2014

Decree that row 1 is (1), row 2 is (2, 3), and row 3 is (4, 6, 9). Let r(n) = A001590(n+2), so that r(r) = r(n-1) + r(n-2) + r(n-3) with r(1) =1, r(2) = 2, r(3) = 3. Row n of the array, for n >= 4, consists of the numbers, in increasing order, defined as follows: all 3*x from x in row n-1, together with all 1 + 3*x from x in row n-2, together with all 2 + 3*x from x in row n-3. Thus, the number of numbers in row n is r(n), a tribonacci number. Every positive integer occurs exactly once in the array, so that the resulting sequence is a permutation of the positive integers.

			First 5 rows of the array:
1
2 ... 3
4 ... 6 ... 9
5 ... 7 ... 10 .. 12 .. 18 .. 27
8 ... 11 .. 13 .. 15 .. 19 .. 21 .. 28 .. 30 .. 36 .. 54 .. 81

Clark Kimberling, Table of n, a(n) for n = 1..1500

Cf. A232561, A243571, A243573, A232561, A002590.

z = 10; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 3 x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
u = Table[g[n], {n, 1, z}]; v = Flatten[u]  (* A243572 *)

A253318 Indices of primes in the 7th-order Fibonacci number sequence, A060455.

Original entry on oeis.org

7, 8, 11, 12, 14, 15, 16, 17, 18, 19, 21, 23, 26, 32, 33, 36, 42, 44, 71, 72, 137, 180, 193, 285, 679, 955, 1018, 1155, 1176, 1191, 2149, 2590, 2757, 3364, 4233, 6243, 6364, 7443, 10194, 11254, 13318, 18995, 20478, 22647, 29711, 34769, 61815, 71993, 107494, 135942, 148831

Offset: 1

Robert Price, Dec 30 2014

nonn

a(52) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288, A000322, A000383, A249413, A060455.

a={1,1,1,1,1,1,1}; step=7; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[7]]=sum]; lst

A007800 From a problem in AI planning: a(n) = 4+a(n-1)+a(n-2)+a(n-3)+a(n-4)-a(n-5)-a(n-6)-a(n-7), n>7.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 59, 111, 207, 384, 710, 1310, 2414, 4445, 8181, 15053, 27693, 50942, 93704, 172356, 317020, 583099, 1072495, 1972635, 3628251, 6673404, 12274314, 22575994, 41523738, 76374073, 140473833, 258371673, 475219609, 874065146

Offset: 1

Peter Jonsson [ petej(AT)ida.liu.se ]

The number of length n binary words with fewer than 3 zeros between any pair of consecutive ones. - Jeffrey Liese, Dec 23 2010

Robert Israel, Table of n, a(n) for n = 1..3397
T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

Cf. A062544.

for n from 1 to 5 do a[n]:= [1,2,4,8,16][n] od:
for n from 6 to 100 do a[n]:= 3*a[n-1]-2*a[n-2]-a[n-4]+a[n-5] od:
seq(a[n],n=1..100); # Robert Israel, Aug 19 2014

LinearRecurrence[{3,-2,0,-1,1},{1,2,4,8,16},40] (* Harvey P. Dale, Apr 24 2013 *)

Vec(-x*(x^4-x+1)/((x-1)^2*(x^3+x^2+x-1)) + O(x^100)) \\ Colin Barker, Aug 18 2014

a(1)=1, a(2)=2, a(3)=4, a(4)=8, a(5)=16, a(n)=3*a(n-1)-2*a(n-2)+0*a(n-3)- a(n-4)+ a(n-5). - Harvey P. Dale, Apr 24 2013
G.f.: -x*(x^4-x+1) / ((x-1)^2*(x^3+x^2+x-1)). - Colin Barker, Aug 18 2014
2*a(n) = A001590(n+4)-n. - R. J. Mathar, Aug 16 2017

A103685 Consider the morphism 1->{1,2}, 2->{1,3}, 3->{1}; a(n) is the total number of '3' after n substitutions.

Original entry on oeis.org

0, 0, 1, 5, 17, 51, 147, 419, 1191, 3383, 9607, 27279, 77455, 219919, 624415, 1772895, 5033759, 14292287, 40579903, 115217983, 327136895, 928835455, 2637230207, 7487852799, 21260161279, 60363694335, 171389837823, 486624896511, 1381667623423, 3922950583295

Offset: 0

Roger L. Bagula, Mar 26 2005

Examples of the morphism starting with {1} are shown in A103684. Counting the total number of '1' in rows 1 to n of A103684 yields 1, 3, 8,... = A073357(n+1),
counting the total number of '2' in rows 1 to n yields 0, 1, 4,.. = A115390(n+1),
and counting the total number '3' in rows 1 to n yields a(n), the sequence here.
Inverse binomial transform yields 0, 0, 1, 2, 3, 6, 11, 20,..., a variant of A001590 [Nov 18 2010]

Index entries for linear recurrences with constant coefficients, signature (5,-8,6,-2).

Cf. A073058, A103684.

LinearRecurrence[{5,-8,6,-2},{0,0,1,5},30] (* Harvey P. Dale, Nov 10 2011 *)

a(n)= +5*a(n-1) -8*a(n-2) +6*a(n-3) -2*a(n-4) = a(n-1)+A115390(n). [Nov 18 2010]

G.f.: x^2 / ( (x-1)*(2*x^3-4*x^2+4*x-1) ). [Nov 18 2010]

Depleted by the information already in A073357 and A115390; corrected image of {2} in the defn. - The Assoc. Eds. of the OEIS, Nov 18 2010

A106522 A Pascal type matrix based on the tribonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 13, 15, 15, 11, 5, 1, 24, 28, 30, 26, 16, 6, 1, 44, 52, 58, 56, 42, 22, 7, 1, 81, 96, 110, 114, 98, 64, 29, 8, 1, 149, 177, 206, 224, 212, 162, 93, 37, 9, 1, 274, 326, 383, 430, 436, 374, 255, 130, 46, 10, 1, 504, 600, 709, 813, 866, 810, 629, 385, 176, 56, 11, 1

Offset: 0

Paul Barry, May 06 2005

Row sums of A106522 mod 2 are A106524.

			Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   7,  8,  7,  4, 1;
  13, 13, 15, 11, 5, 1;

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

Cf. A000073, A001590 (row sums), A106523 (diagonal sums).

b[n_]:= b[n]= If[n<2, 0, If[n==2, 1, b[n-1] +b[n-2] +b[n-3]]]; (* A000073 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, b[n+2], T[n-1, k-1] +T[n-1, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 06 2021 *)

@CachedFunction
def b(n): return 0 if (n<2) else 1 if (n==2) else b(n-1) +b(n-2) +b(n-3)
def T(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return b(n+2)
    else: return T(n-1, k) + T(n-1, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 06 2021

Riordan array (1/(1-x-x^2-x^3), x/(1-x)).
Number triangle T(n, 0) = A000073(n+2), T(n, k) = T(n-1, k-1) + T(n-1, k).
Sum_{k=0..n} T(n,k) = A001590(n+3).
Sum_{k=0..floor(n/2)} T(n-k, k) = A106523(n).

cp's OEIS Frontend

A108153 a(n) = 3*a(n-1) + 2*a(n-2) + a(n-3).

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A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

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A198295 Riordan array (1, x*(1+x)/(1-x^3)).

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A233324 Triangle read by rows: T(n,k) = number of palindromic compositions of n in which no part exceeds k, 1 <= k <= n.

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A242572 Indices of primes in A214828.

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A243572 Irregular triangular array generated as in Comments; contains every positive integer exactly once.

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Mathematica

A253318 Indices of primes in the 7th-order Fibonacci number sequence, A060455.

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Mathematica

A007800 From a problem in AI planning: a(n) = 4+a(n-1)+a(n-2)+a(n-3)+a(n-4)-a(n-5)-a(n-6)-a(n-7), n>7.

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A108153 a(n) = 3a(n-1) + 2a(n-2) + a(n-3).