A013979 -id:A013979 - cp's OEIS frontend

A078012 a(n) = a(n-1) + a(n-3) for n >= 3, with a(0) = 1, a(1) = a(2) = 0. This recurrence can also be used to define a(n) for n < 0.

1, 0, 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925

Offset: 0

N. J. A. Sloane, Nov 17 2002, Mar 08 2008

Number of compositions of n into parts >= 3. - Milan Janjic, Jun 28 2010
From Adi Dani, May 22 2011: (Start)
Number of compositions of number n into parts of the form 3*k+1, k >= 0.
For example, a(10)=19 and all compositions of 10 in parts 1,4,7 or 10 are
(1,1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,4), (1,1,1,1,1,4,1), (1,1,1,1,4,1,1), (1,1,1,4,1,1,1), (1,1,4,1,1,1,1), (1,4,1,1,1,1,1), (4,1,1,1,1,1,1), (1,1,4,4), (1,4,1,4), (1,4,4,1), (4,1,1,4),(4,1,4,1), (4,4,1,1), (1,1,1,7), (1,1,7,1), (1,7,1,1), (7,1,1,1), (10). (End)
For n >= 0 a(n+1) is the number of 00's in the Narayana word NW(n); equivalently the number of two neighboring 0's at level n of the Narayana tree. See A257234. This implies that if a(0) is put to 0 then a(n) is the number of -1's in the Narayana word NW(n), and also at level n of the Narayana tree. - Wolfdieter Lang, Apr 24 2015

			G.f. = 1 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 6*x^9 + 9*x^10 + 13*x^11 + ...

Taylor L. Booth, Sequential Machines and Automata Theory, John Wiley and Sons, Inc., 1967, page 331ff.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Christian Ballot, On Functions Expressible as Words on a Pair of Beatty Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.2.
C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f_n.]
Taras Goy, On identities with multinomial coefficients for Fibonacci-Narayana sequence, Annales Mathematicae et Informaticae, Vol. 49 (2018), 75-84.
Bahar Kuloğlu, Engin Özkan, and Marin Marin, On the period of Pell-Narayana sequence in some groups, arXiv:2305.04786 [math.CO], 2023.
J. D. Opdyke, A unified approach to algorithms generating unrestricted.., J. Math. Model. Algor. 9 (2010) 53-97.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000930, A065417, A068921, A077961.

Cf. also A135851, A257234.

a:=[1,0,0];; for n in [4..50] do a[n]:=a[n-1]+a[n-3]; od; a; # G. C. Greubel, Jun 28 2019

a078012 n = a078012_list !! n
a078012_list = 1 : 0 : 0 : 1 : zipWith (+) a078012_list
   (zipWith (+) (tail a078012_list) (drop 2 a078012_list))
-- Reinhard Zumkeller, Mar 23 2012

I:=[1,0,0]; [n le 3 select I[n] else Self(n-1) + Self(n-3): n in [1..50]]; // G. C. Greubel, Jan 19 2018

A078012 := proc(n): if n=0 then 1 else add(binomial(n-3-2*i,i),i=0..(n-3)/3) fi: end: seq(A078012(n), n=0..46); # Johannes W. Meijer, Aug 11 2011
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <1|0|1>>^n)[1, 1]:
seq(a(n), n=0..46);  # Alois P. Heinz, May 08 2025

CoefficientList[ Series[(1-x)/(1-x-x^3), {x,0,50}], x] (* Robert G. Wilson v, May 25 2011 *)
LinearRecurrence[{1,0,1}, {1,0,0}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
a[ n_]:= If[ n >= 0, SeriesCoefficient[ (1-x)/(1-x-x^3), {x, 0, n}], SeriesCoefficient[1/(1+x^2-x^3), {x, 0, -n}]]; (* Michael Somos, Feb 03 2018 *)

{a(n) = if( n<0, n = -n; polcoeff( 1 / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - x) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, May 03 2011 */

((1-x)/(1-x-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 28 2019

a(n) = Sum_{i=0..(n-3)/3} binomial(n-3-2*i, i), n >= 1, a(0) = 1.
From Michael Somos, May 03 2011: (Start)
Euler transform of A065417.
G.f.: (1 - x) / (1 - x - x^3).
a(-n) = A077961(n). a(n+3) = A000930(n).
a(n+5) = A068921(n). (End)
a(n+1) = A013979(n-3) + A135851(n) + A107458(n), n >= 3.
G.f.: 1/(1 - Sum_{k>=3} x^k). - Joerg Arndt, Aug 13 2012
G.f.: Q(0)*(1-x)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x^2)/( x*(4*k+3 + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
0 = -1 + a(n)*(a(n)*(a(n) + a(n+2)) + a(n+1)*(a(n+1) - 3*a(n+2))) + a(n+1)*(+a(n+1)*(+a(n+1) + a(n+2)) + a(n+2)*(-2*a(n+2))) + a(n+2)^3 for all n in Z. - Michael Somos, Feb 03 2018
a(-n) = a(n)*a(n+3) - a(n+1)*a(n+2) for all n in Z. - Greg Dresden, May 07 2025

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

Entry revised by N. J. A. Sloane, May 11 2025, making use of comments from Michael Somos, May 03 2011 and Greg Dresden, May 11 2025

A071675 Array read by antidiagonals of trinomial coefficients.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 3, 3, 1, 0, 0, 2, 6, 4, 1, 0, 0, 1, 7, 10, 5, 1, 0, 0, 0, 6, 16, 15, 6, 1, 0, 0, 0, 3, 19, 30, 21, 7, 1, 0, 0, 0, 1, 16, 45, 50, 28, 8, 1, 0, 0, 0, 0, 10, 51, 90, 77, 36, 9, 1, 0, 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1, 0, 0, 0, 0, 1, 30, 141, 266, 266

Offset: 0

Henry Bottomley, May 30 2002

Read as a number triangle, this is the Riordan array (1, x(1+x+x^2)) with T(n,k) = Sum_{i=0..floor((n+k)/2)} C(k,2i+2k-n)*C(2i+2k-n,i). Rows start {1}, {0,1}, {0,1,1}, {0,1,2,1}, {0,0,3,3,1},... Row sums are then the trinomial numbers A000073(n+2). Diagonal sums are A013979. - Paul Barry, Feb 15 2005

Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213742. For example, s_1(n)=binomial(n,1)=n is the first column of A213742 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213742 for n>1, etc. In particular (see comment in A213742) in cases k=4,5,6,7,8, s_k(n) is A005718(n+2), A005719(n), A005720(n), A001919(n), A064055(n+3), respectively. - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

			Rows start
1, 0,  0,  0,  0,  0, ...;
1, 1,  1,  0,  0,  0,  0, ...;
1, 2,  3,  2,  1,  0,  0, ...;
1, 3,  6,  7,  6,  3,  1, 0, ...;
1, 4, 10, 16, 19, 16, 10, 4, 1, ...; etc.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Visible version of A027907. Row sums are 3^n, i.e. A000244. Central diagonal is A002426. Cf. A071676 for a slight variation.

T[n_, k_] := Sum[Binomial[n - k - j, j]*Binomial[k, n - k - j], {j, 0,
Floor[(n - k)/2]}]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) starting with T(0, 0)=1. See A027907 for more.

As a number triangle, T(n, k) = Sum_{i=0..floor((n-k)/2)} C(n-k-i, i) * C(k, n-k-i). - Paul Barry, Apr 26 2005

A111006 Another version of Fibonacci-Pascal triangle A037027.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144

Offset: 0

Philippe Deléham, Oct 02 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.
Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, ...).
Triangle read by rows: T(n,n-k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)). - Philippe Deléham, Feb 17 2014
Diagonal sums are A013979(n). - Philippe Deléham, Feb 17 2014
T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and 1 X 2 tiles. - Emeric Deutsch, Aug 14 2014

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 1, 5,  5;
  0, 0, 0, 3, 10,  8;
  0, 0, 0, 1,  9, 20, 13;
  0, 0, 0, 0,  4, 22, 38,  21;
  0, 0, 0, 0,  1, 14, 51,  71,  34;
  0, 0, 0, 0,  0,  5, 40, 111, 130,  55;
  0, 0, 0, 0,  0,  1, 20, 105, 233, 235,  89;
  0, 0, 0, 0,  0,  0,  6,  65, 256, 474, 420, 144;

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000045, A000129, A001045, A037027, A038112, A038149, A084938, A128100 (reversed version).

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A114197, A162741, A228074.

a111006 n k = a111006_tabl !! n !! k
a111006_row n = a111006_tabl !! n
a111006_tabl =  map fst $ iterate (\(us, vs) ->
   (vs, zipWith (+) (zipWith (+) ([0] ++ us ++ [0]) ([0,0] ++ us))
                    ([0] ++ vs))) ([1], [0,1])
-- Reinhard Zumkeller, Aug 15 2013

T(0, 0) = 1, T(n, k) = 0 for k < 0 or for n < k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).
T(n, k) = A037027(k, n-k). T(n, n) = A000045(n+1). T(3n, 2n) = (n+1)*A001002(n+1) = A038112(n).
G.f.: 1/(1-yx(1-x)-x^2*y^2). - Paul Barry, Oct 04 2005
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 02 2006
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 respectively. - Philippe Deléham, Feb 17 2014

A060945 Number of compositions (ordered partitions) of n into 1's, 2's and 4's.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 18, 31, 55, 96, 169, 296, 520, 912, 1601, 2809, 4930, 8651, 15182, 26642, 46754, 82047, 143983, 252672, 443409, 778128, 1365520, 2396320, 4205249, 7379697, 12950466, 22726483, 39882198, 69988378, 122821042, 215535903, 378239143, 663763424, 1164823609

Offset: 0

Len Smiley, May 07 2001

Diagonal sums of A038137. - Paul Barry, Oct 24 2005
From Gary W. Adamson, Oct 28 2010: (Start)
INVERT transform of the aerated Fibonacci sequence (1, 0, 1, 0, 2, 0, 3, 0, 5, ...).
a(n) = term (4,4) in the n-th power of the matrix [0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,1,1]. (End)
Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=3, I={2}. - Vladimir Baltic, Mar 07 2012
Number of compositions of n if the summand 2 is frozen in place or equivalently, if the ordering of the summand 2 does not count. - Gregory L. Simay, Jul 18 2016
a(n) - a(n-2) = number of compositions of n with no 2's = A005251(n+1). - Gregory L. Simay, Jul 18 2016
In general, the number of compositions of n with summand k frozen in place is equal to the number of compositions of n with only summands 1,...,k,2k. - Gregory L. Simay, May 10 2017
In the same way that the sum of any two alternating terms of A006498 produces a term from A000045 (the Fibonacci sequence), so it could be thought of as a "meta-Fibonacci," and the sum of any two alternating terms of A013979 produces a term from A000930 (Narayana’s cows), so it could analogously be called "meta-Narayana’s cows," this sequence embeds (can generate) A000931 (the Padovan sequence), as the odd terms of A000931 are generated by the sum of successive elements (e.g. 1+2=3, 2+3=5, 3+6=9, 6+10=16) and its even terms are generated by the difference of "supersuccessive" (second-order successive or "alternating," separated by a single other term) terms (e.g. 10-3=7, 18-6=12, 31-10=21, 55-18=37) — or, equivalently, adding "supersupersuccessive" terms (separated by 2 other terms, e.g. 1+6=7, 2+10=12, 3+18=21, 6+31=37) — so it could be dubbed the "metaPadovan." - Michael Cohen and Yasuyuki Kachi, Jun 13 2024

			There are 18=a(6) compositions of 6 with the summand 2 frozen in place: (6), (51), (15), (4[2]), (33), (411), (141), (114), (3[2]1), (1[2]3), ([222]), (3111), (1311), (1131), (1113), ([22]11), ([2]1111), (111111). Equivalently, the position of the summand 2 does not affect the composition count. For example, (321)=(231)=(312) and (123)=(213)=(132).

Harry J. Smith, Table of n, a(n) for n = 0..500
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135
Michael Cohen and Yasuyuki Kachi (2024). Recurrence Relations Rhythm. In: Noll, T., Montiel, M., Gómez, F., Hamido, O.C., Besada, J.L., Martins, J.O. (eds) Mathematics and Computation in Music. MCM 2024. Lecture Notes in Computer Science, vol. 14639. Springer, Cham.
K. Edwards, M. A. Allen, Strongly restricted permutations and tiling with fences, Disc. Appl. Math. 187 (2015) 82-90, Sect. 4.3
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1).

Cf. A000045 (1's and 2's only), A023359 (all powers of 2)
Same as unsigned version of A077930.
All of A060945, A077930, A181532 are variations of the same sequence. - N. J. A. Sloane, Mar 04 2012

a060945 n = a060945_list !! (n-1)
a060945_list = 1 : 1 : 2 : 3 : 6 : zipWith (+) a060945_list
   (zipWith (+) (drop 2 a060945_list) (drop 3 a060945_list))
-- Reinhard Zumkeller, Mar 23 2012

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( 1/(1-x-x^2-x^4) )); // G. C. Greubel, Apr 09 2021

m:= 40; S:= series( 1/(1-x-x^2-x^4), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Apr 09 2021

LinearRecurrence[{1,1,0,1}, {1,1,2,3}, 39] (* or *)
CoefficientList[Series[1/(1-x-x^2-x^4), {x, 0, 38}], x] (* Michael De Vlieger, May 10 2017 *)

N=66; my(x='x+O('x^N));
Vec(1/(1-x-x^2-x^4))
/* Joerg Arndt, Oct 21 2012 */

def A060945_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x-x^2-x^4) ).list()
A060945_list(40) # G. C. Greubel, Apr 09 2021

a(n) = a(n-1) + a(n-2) + a(n-4).
G.f.: 1 / (1 - x - x^2 - x^4).
a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..n-k} C(i, n-k-i)*C(2*i-n+k, 3*k-2*n+2*i). - Paul Barry, Oct 24 2005
a(2n) = A238236(n), a(2n+1) = A097472(n). - Philippe Deléham, Feb 20 2014
a(n) + a(n+1) = A005314(n+2). - R. J. Mathar, Jun 17 2020

a(0) = 1 prepended by Joerg Arndt, Oct 21 2012

A135851 a(n) = n-1, if n <= 2, otherwise A107458(n-1) + A107458(n-2).

Original entry on oeis.org

-1, 0, 1, 0, 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925

Offset: 0

N. J. A. Sloane, Mar 08 2008

			G.f. = -1 + x^2 + x^5 + x^6 + x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 6*x^11 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f_n.]
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000930, A013979, A077961, A078012, A107458.

a135851 n = a135851_list !! n
a135851_list = -1 : 0 : 1 : zipWith (+) a135851_list (drop 2 a135851_list)
-- Reinhard Zumkeller, Mar 23 2012

[n le 3 select n-2 else Self(n-1) + Self(n-3): n in [1..61]]; // G. C. Greubel, Aug 01 2022

LinearRecurrence[{1,0,1},{-1,0,1},50] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)
a[ n_] := If[ n < 3, SeriesCoefficient[ 1 / (1 + x^2 - x^3), {x, 0, 2 - n}], SeriesCoefficient[ x^5 / (1 - x - x^3), {x, 0, n}]]; (* Michael Somos, Jan 08 2014 *)

{a(n) = if( n<3, polcoeff( 1 / (1 + x^2 - x^3) + x * O(x^(2-n)), 2-n), polcoeff( x^5 / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, Jan 08 2014 */

def A000930(n): return sum(binomial(n-2*j,j) for j in (0..(n//3)))
def A135851(n): return A000930(n+2) -2*A000930(n)
[A135851(n) for n in (0..60)] # G. C. Greubel, Aug 01 2022

From R. J. Mathar, Jul 26 2010: (Start)
a(n) = +a(n-1) +a(n-3).
a(n) = A078012(n-2), for n>=2.
G.f.: (-1 + x + x^2) / (1 - x - x^3). (End)
From Michael Somos, Jan 08 2014: (Start)
a(n) = A077961(2-n) for all n in Z.
a(n)^2 - a(n-1)*a(n+1) = A077961(n-5). (End)
a(n) = A000930(n+2) - 2*A000930(n). - G. C. Greubel, Aug 01 2022

A001634 a(n) = a(n-2) + a(n-3) + a(n-4), with initial values a(0) = 0, a(1) = 2, a(2) = 3, a(3) = 6.

Original entry on oeis.org

0, 2, 3, 6, 5, 11, 14, 22, 30, 47, 66, 99, 143, 212, 308, 454, 663, 974, 1425, 2091, 3062, 4490, 6578, 9643, 14130, 20711, 30351, 44484, 65192, 95546, 140027, 205222, 300765, 440795, 646014, 946782, 1387574, 2033591, 2980370, 4367947, 6401535, 9381908

Offset: 0

N. J. A. Sloane

E.-B. Escott, Reply to Query 1484, L'Intermédiaire des Mathématiciens, 8 (1901), 63-64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..500
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Gregory T. Minton, Linear recurrence sequences satisfying congruence conditions, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2337--2352. MR3195758.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1).

Cf. A013979, A107458.

a001634 n = a001634_list !! n
a001634_list = 0 : 2 : 3 : 6 : zipWith (+) a001634_list
   (zipWith (+) (tail a001634_list) (drop 2 a001634_list))
-- Reinhard Zumkeller, Mar 23 2012

A001634:=-z*(2+3*z+4*z**2)/(1+z)/(z**3+z-1); # Simon Plouffe in his 1992 dissertation
a:= n-> (Matrix([[0,4,-1,-1]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [0,1,1,1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..40); # Alois P. Heinz, Aug 01 2008

LinearRecurrence[{0, 1, 1, 1}, {0, 2, 3, 6}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
CoefficientList[Series[x (2+3x+4x^2)/(1-x^2-x^3-x^4),{x,0,50}],x] (* Harvey P. Dale, Mar 26 2023 *)

a(n):=(sum(sum(binomial(j,n-2*k-j-1)*binomial(k+1,j),j,0,k+1)/(k+1),k,0,(n-1)/2))*(n+1); /* Vladimir Kruchinin, Mar 22 2016 */

a(n)=if(n<0,0,polcoeff(x*(2+3*x+4*x^2)/(1-x^2-x^3-x^4)+x*O(x^n),n))

G.f.: x(2 + 3x + 4x^2)/(1 - x^2 - x^3 - x^4).

a(n) = Sum_{k=0..(n-1)/2}(Sum_{j=0..k+1}(binomial(j,n-2*k-j-1)*binomial(k+1,j))/(k+1))*(n+1). - Vladimir Kruchinin, Mar 22 2016

A023435 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-5).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 7, 11, 16, 24, 35, 52, 76, 112, 164, 241, 353, 518, 759, 1113, 1631, 2391, 3504, 5136, 7527, 11032, 16168, 23696, 34728, 50897, 74593, 109322, 160219, 234813, 344135, 504355, 739168, 1083304, 1587659, 2326828, 3410132, 4997792, 7324620, 10734753

Offset: 0

N. J. A. Sloane

nonn

Diagonal sums of Riordan array (1/(1-x), x(1+x+x^2)) yield a(n+1). - Paul Barry, Feb 16 2005
The Ca2 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 lead to this sequence. - Johannes W. Meijer, Jul 20 2011
Number of ordered partitions of (n-1) into parts less than or equal to 3, where the order of the 2's is unimportant. (see example). - David Neil McGrath, Apr 26 2015
Number of ordered partitions of (n-1) into parts less than or equal to 4, where the order of the 1's is unimportant.(see example). - David Neil McGrath, May 05 2015
List the partitions of n in nonincreasing order. Freeze the 1's and 2's in place and allow the other summands to vary their order without disturbing the 1's and 2's. The result is a(n+1). - Gregory L. Simay (based on correspondence with George E. Andrews), Jul 11 2016
Number of ordered partitions of n-1 where the order of the 1's and the 2's are unimportant. - Gregory L. Simay, Jul 18 2016

			There are 11 partitions of 6 into parts less than or equal to 3, where the order of 2's is unimportant, a(7)=11. These are (33),(321=231=312),(132=123=213),(3111),(1311),(1131),(1113),(222),(2211=1122=1221=2112=2121=1212),(21111=12111=11211=11121=11112),(111111). - _David Neil McGrath_, Apr 26 2015
There are 11 partitions of 6 into parts less than equal to 4, where the order of 1's is unimportant. These are (42),(24),(411=141=114),(33),(321=312=132),(231=213=123),(3111=1311=1131=1113),(222),(2211=1122=2112=1221=1212=2121),(21111=12111=11211=11121=11112),(111111). - _David Neil McGrath_, May 05 2015
There are a(9)=24 partitions of 8 where the 1's and 2's are frozen []: (8), (7[1]), (6[2]), (53), (35) (44), (6[1][1]), (5,[2][1]), (43[1]), (34[1]), (4[2][2]), (33[2][2]) (5[1][1][1]), (4[2][1][1]), (33[1][1]), (3[2][2][1]), ([2][2][2][2]), (4[1][1][1][1]), (3[2][1][1][1]), ([2][2][2][1][1]), (3[1][1][1][1][1]), ([2][2][1][1][1][1]), ([2][1][1][1][1][1][1]),([1][1][1][1][1][1][1][1]). - _Gregory L. Simay_, Jul 11 2016

Michael De Vlieger, Table of n, a(n) for n = 0..6024
John H. E. Cohn, Letter to the editor, Fib. Quart. 2 (1964), 108.
Verner E. Hoggatt, Jr. and Douglas A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.
Zoltán Kása, On scattered subword complexity, arXiv:1104.4425 [cs.DM], 2011.
William J. Keith, Robert Schneider, and Andrew V. Sills, Composition-theoretic series and false theta functions, Integers (2024) Vol. 24A, Art. No. A11. See p. 11.
Anthony Shannon, François Dubeau, Mine Uysal, and Engin Özkan, A Difference Equation Model of Infectious Disease, Int. J. Bioautomation (2022) Vol. 26, No. 4, 339-352.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1).

First differences are in A013979.

Cf. A077864 (bisection).

I:=[0,1,1,2,3]; [n le 5 select I[n] else Self(n-1)+Self(n-2)-Self(n-5): n in [1..45]]; // Vincenzo Librandi, Apr 27 2015

LinearRecurrence[{1, 1, 0, 0, -1}, {0, 1, 1, 2, 3}, 50] (* Vincenzo Librandi, Apr 27 2015 *)

x='x+O('x^99); concat(0, Vec(x/((x-1)*(1+x)*(x^3+x-1)))) \\ Altug Alkan, Apr 09 2018

G.f.: x / ( (x-1)*(1+x)*(x^3+x-1) ). - R. J. Mathar, Nov 28 2011

More terms from Vincenzo Librandi, Apr 27 2015

A107458 Expansion of g.f.: (1-x^2-x^3)/( (1+x)*(1-x-x^3) ).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 17, 24, 36, 52, 77, 112, 165, 241, 354, 518, 760, 1113, 1632, 2391, 3505, 5136, 7528, 11032, 16169, 23696, 34729, 50897, 74594, 109322, 160220, 234813, 344136, 504355, 739169, 1083304, 1587660, 2326828, 3410133, 4997792, 7324621

Offset: 0

N. J. A. Sloane, Mar 08 2008

The sequence can be interpreted as the top-left entry of the n-th power of a 4 X 4 (0,1) matrix. There are 12 different choices (out of 2^16) for that (0,1) matrix. - R. J. Mathar, Mar 19 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
C. Kenneth Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f^2_n.]
Renata Passos Machado Vieira, and Francisco Regis Vieira Alves, Sequences of Tridovan and their identities, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 185-197. Sequence (T_n) is a subsequence of this sequence.
Renata Passos Machado Vieira, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino, A note on the Tetrarrin sequence, Braz. Elect. J. Math. (2024).
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1).

Cf. A001634, A013979, A078012, A135851.

a:=[1,0,0,0];; for n in [5..50] do a[n]:=a[n-2]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 03 2020

a107458 n = a107458_list !! n
a107458_list = 1 : 0 : 0 : 0 : zipWith (+) a107458_list
   (zipWith (+) (tail a107458_list) (drop 2 a107458_list))
-- Reinhard Zumkeller, Mar 23 2012

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(1-x^2-x^3)/( (1+x)*(1-x-x^3))); // Marius A. Burtea, Jan 02 2020

seq(coeff(series( (1-x^2-x^3)/( (1+x)*(1-x-x^3) ), x, n+1), x, n), n = 0..50); # G. C. Greubel, Jan 03 2020

CoefficientList[Series[(1-x^2-x^3)/(1-x^2-x^3-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,1,1},{1,0,0,0},50] (* Harvey P. Dale, Jun 20 2011 *)

my(x='x+O('x^50)); Vec((1-x^2-x^3)/((1+x)*(1-x-x^3))) \\ G. C. Greubel, Apr 27 2017

def A107458_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2-x^3)/((1+x)*(1-x-x^3)) ).list()
A107458_list(50) # G. C. Greubel, Jan 03 2020

a(n) = a(n-2) + a(n-3) + a(n-4); a(0)=1, a(1)=0, a(2)=0, a(3)=0. - Harvey P. Dale, Jun 20 2011

a(n) + a(n-1) = A000930(n-4). - R. J. Mathar, Mar 19 2014

A123149 Triangle T(n,k), 0<=k<=n, read by rows given by [1, 0, -1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, -1, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 3, 5, 5, 3, 1, 0, 1, 3, 6, 7, 6, 3, 1, 0, 1, 4, 9, 13, 13, 9, 4, 1, 0, 1, 4, 10, 16, 19, 16, 10, 4, 1, 0, 1, 5, 14, 26, 35, 35, 26, 14, 5, 1, 0, 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 0, 1, 6, 20, 45, 75, 96, 96, 75, 45, 20, 6, 1, 0

Offset: 0

Philippe Deléham, Nov 05 2006

A169623 is a very similar triangle except it does not have the outer diagonal of 0's. - N. J. A. Sloane, Nov 23 2017

			Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  1,  1,  0;
  1,  2,  2,  1,  0;
  1,  2,  3,  2,  1,  0;
  1,  3,  5,  5,  3,  1,  0;
  1,  3,  6,  7,  6,  3,  1,  0;
  1,  4,  9, 13, 13,  9,  4,  1,  0;

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

Cf. A002426, A005773, A013979, A027907, A038754, A084938, A135528, A169623, A182522 (row sums).

function T(n,k) // T = A123149
  if k lt 0 or k gt n then return 0;
  elif k eq 0 or k eq n-1 then return 1;
  elif k eq n then return 0;
  else return T(n-2,k) +T(n-2,k-1) +T(n-2,k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 17 2023

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n-1, 1, If[k==n, 0, T[n-2,k] +T[n-2,k-1] +T[n-2,k-2] ]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 17 2023 *)

def T(n,k): # T = A123149
    if (k<0 or k>n): return 0
    elif (k==0 or k==n-1): return 1
    elif (k==n): return 0
    else: return T(n-2,k) +T(n-2,k-1) +T(n-2,k-2)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 17 2023

T(n,k) = T(n-1,k-1) + T(n-1,k) if n even, T(n,k) = T(n-1,k-1) + T(n-2,k) if n odd, T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k < 0 or if k > n.
T(n,k) = T(n,n-k-1).
Sum_{k=0..n} T(n,k) = A038754(n-1), for n>=1.
T(2*n,n) = A005773(n).
T(2*n+1,n) = A002426(n).
From Philippe Deléham, May 04 2012: (Start)
G.f.: (1+x-y^2*x^2)/(1-x^2-y*x^2-y^2*x^2).
T(n,k) = T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.
Sum_{k=0..n} T(n,k) = A182522(n). (End)
From G. C. Greubel, Jul 17 2023: (Start)
Sum_{k=0..n} (-1)^k*T(n,k) = A135528(n).
Sum_{k=0..floor(n/2)} T(n-k,k) = [n==0] + A013979(n+1). (End)

A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 01 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.
Main diagonal gives A246691.
Cf. A246688, A246720 (the same for partitions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A078012 a(n) = a(n-1) + a(n-3) for n >= 3, with a(0) = 1, a(1) = a(2) = 0. This recurrence can also be used to define a(n) for n < 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A071675 Array read by antidiagonals of trinomial coefficients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A111006 Another version of Fibonacci-Pascal triangle A037027.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A060945 Number of compositions (ordered partitions) of n into 1's, 2's and 4's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A135851 a(n) = n-1, if n <= 2, otherwise A107458(n-1) + A107458(n-2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

SageMath

Formula

A001634 a(n) = a(n-2) + a(n-3) + a(n-4), with initial values a(0) = 0, a(1) = 2, a(2) = 3, a(3) = 6.

Views

Author

Keywords

References

Links

Crossrefs