A072854 -id:A072854 - cp's OEIS frontend

A003269 a(n) = a(n-1) + a(n-4) with a(0) = 0, a(1) = a(2) = a(3) = 1.

0, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 95, 131, 181, 250, 345, 476, 657, 907, 1252, 1728, 2385, 3292, 4544, 6272, 8657, 11949, 16493, 22765, 31422, 43371, 59864, 82629, 114051, 157422, 217286, 299915, 413966, 571388, 788674, 1088589, 1502555, 2073943

Offset: 0

N. J. A. Sloane

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 0..m-1. The generating function is 1/(1-x-x^m). Also a(n) = Sum_{i=0..n/m} binomial(n-(m-1)*i, i). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that are m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.
For this family of sequences, a(n+1) is the number of compositions of n+1 into parts 1 and m. For n>=m, a(n-m+1)is the number of compositions of n in which each part is greater than m or equivalently, in which parts 1 through m are excluded. - Gregory L. Simay, Jul 14 2016
For this family of sequences, let a(m,n) = a(n-1) + a(n-m). Then the number of compositions of n having m as a least summand is a(m, n-m) - a(m+1, n-m-1). - Gregory L. Simay, Jul 14 2016
For n>=3, a(n-3) = number of compositions of n in which each part is >=4. - Milan Janjic, Jun 28 2010
For n>=1, number of compositions of n into parts == 1 (mod 4). Example: a(8)=5 because there are 5 compositions of 8 into parts 1 or 5: (1,1,1,1,1,1,1,1), (1,1,1,5), (1,1,5,1), (1,5,1,1), (5,1,1,1). - Adi Dani, Jun 16 2011
a(n+1) is the number of compositions of n into parts 1 and 4. - Joerg Arndt, Jun 25 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=4, 2*a(n-3) equals the number of 2-colored compositions of n with all parts >= 4, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
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={1,2}. - Vladimir Baltic, Mar 07 2012
a(n+4) equals the number of binary words of length n having at least 3 zeros between every two successive ones. - Milan Janjic, Feb 07 2015
From Clark Kimberling, Jun 13 2016: (Start)
Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*.
Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3, 2*x, x+1, x^2}, etc.
Let T(r) be the tree obtained by substituting r for x.
If N is a positive integer such that r = N^(1/4) is not an integer, then the number of (not necessarily distinct) integers in g(n) is A003269(n), for n > = 1. See A274142. (End)

			G.f.: x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 10*x^10 + ...
The number of compositions of 12 having 4 as a least summand is a(4, 12 -4 + 1) - a(5, 12 - 5 + 1) = A003269(9) - A003520(8) = 7-4 = 3. The compositions are (84), (48) and (444). - _Gregory L. Simay_, Jul 14 2016

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 120.
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..501
Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, and Mark Shattuck, Some New Restricted n-Color Composition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See pp. 18, 22.
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
Peter G. Anderson, More Properties of the Zeckendorf Array, Fib. Quart. 52-5 (2014), 15-21. With 2 more initial zeros.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 9.
Bruce M. Boman, Geometric Capitulum Patterns Based on Fibonacci p-Proportions, Fibonacci Quart. 58 (2020), no. 5, 91-102.
Russ Chamberlain, Sam Ginsburg and Chi Zhang, Generating Functions and Wilf-equivalence on Theta_k-embeddings, University of Wisconsin, April 2012.
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
Hùng Việt Chu, Nurettin Irmak, Steven J. Miller, László Szalay, and Sindy Xin Zhang, Schreier Multisets and the s-step Fibonacci Sequences, arXiv:2304.05409 [math.CO], 2023. See also Integers (2024) Vol. 24A, Art. No. A7, p. 3.
Adi Dani, Compositions of natural numbers over arithmetic progressions
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,3,1).
J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 377
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
S. Kitaev, Independent sets on path-schemes, JIS 9 (2006) # 06.2.2 G(x) for M={1,2,3} gives seq. shifted 4 places left
R. J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 17.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 [math.CO] (2014), eq. (23).
Flaviano Morone, Ian Leifer, and Hernán A. Makse, Fibration symmetries uncover the building blocks of biological networks, Proceedings of the National Academy of Sciences (2020) Vol. 117, No. 15, 8306-8314.
Augustine O. Munagi, Euler-type identities for integer compositions via zig-zag graphs, Integers 12 (2012), Paper No. A60, 10 pp.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.8.3.
Djamila Oudrar and Maurice Pouzet, Ordered structures with no finite monomorphic decomposition. Application to the profile of hereditary classes, arXiv:2312.05913 [math.CO], 2023. See p. 18.
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
E. Wilson, The Scales of Mt. Meru
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Cf. A000045, A000079, A000930, A003520, A005708, A005709, A005710, A005711, A017898.
Cf. A048718, A072827, A072850, A072851, A072852, A072853, A072854, A072855, A072856.
Cf. A079955 - A080014.
See A017898 for an essentially identical sequence.
Row sums of A180184.

a003269 n = a003269_list !! n
a003269_list = 0 : 1 : 1 : 1 : zipWith (+) a003269_list
                                          (drop 3 a003269_list)
-- Reinhard Zumkeller, Feb 27 2011

I:=[0,1,1,1]; [n le 4 select I[n] else Self(n-1) + Self(n-4) :n in [1..50]]; // Marius A. Burtea, Sep 13 2019

with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 3)}, unlabeled]: seq(count(SeqSetU, size=j), j=4..51);
seq(add(binomial(n-3*k,k),k=0..floor(n/3)),n=0..47); # Zerinvary Lajos, Apr 03 2007
A003269:=z/(1-z-z**4); # Simon Plouffe in his 1992 dissertation
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 3)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=3..50); # Zerinvary Lajos, Mar 26 2008
M:= Matrix(4, (i,j)-> if j=1 then [1,0,0,1][i] elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1,2]; seq(a(n), n=0..48); # Alois P. Heinz, Jul 27 2008

a[0]= 0; a[1]= a[2]= a[3]= 1; a[n_]:= a[n]= a[n-1] + a[n-4]; Table[a[n], {n,0,50}]
CoefficientList[Series[x/(1-x-x^4), {x,0,50}], x] (* Zerinvary Lajos, Mar 29 2007 *)
Table[Sum[Binomial[n-3*i-1,i], {i,0,(n-1)/3}], {n,0,50}]
LinearRecurrence[{1,0,0,1}, {0,1,1,1}, 50] (* Robert G. Wilson v, Jul 12 2014 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,a+d}; NestList[nxt,{0,1,1,1},50][[;;,1]] (* Harvey P. Dale, May 27 2024 *)

{a(n) = polcoeff( if( n<0, (1 + x^3) / (1 + x^3 - x^4), 1 / (1 - x - x^4)) + x * O(x^abs(n)), abs(n))} /* Michael Somos, Jul 12 2003 */

@CachedFunction
def a(n): return ((n+2)//3) if (n<4) else a(n-1) + a(n-4) # a = A003269
[a(n) for n in (0..50)] # G. C. Greubel, Jul 25 2022

G.f.: x/(1-x-x^4).
G.f.: -1 + 1/(1-Sum_{k>=0} x^(4*k+1)).
a(n) = a(n-3) + a(n-4) + a(n-5) + a(n-6) for n>4.
a(n) = floor(d*c^n + 1/2) where c is the positive real root of -x^4+x^3+1 and d is the positive real root of 283*x^4-18*x^2-8*x-1 (c=1.38027756909761411... and d=0.3966506381592033124...). - Benoit Cloitre, Nov 30 2002
Equivalently, a(n) = floor(c^(n+3)/(c^4+3) + 1/2) with c as defined above (see A086106). - Greg Dresden and Shuer Jiang, Aug 31 2019
a(n) = term (1,2) in the 4 X 4 matrix [1,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,0]^n. - Alois P. Heinz, Jul 27 2008
From Paul Barry, Oct 20 2009: (Start)
a(n+1) = Sum_{k=0..n} C((n+3*k)/4,k)*((1+(-1)^(n-k))/2 + cos(Pi*n/2))/2;
a(n+1) = Sum_{k=0..n} C(k,floor((n-k)/3))(2*cos(2*Pi*(n-k)/3)+1)/3. (End)
a(n) = Sum_{j=0..(n-1)/3} binomial(n-1-3*j,j) (cf. A180184). - Vladimir Kruchinin, May 23 2011
A017817(n) = a(-4 - n) * (-1)^n. - Michael Somos, Jul 12 2003
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(2*k+1 + x^3)/( x*(2*k+2 + x^3) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 29 2013
Appears a(n) = hypergeometric([1/4-n/4,1/2-n/4,3/4-n/4,1-n/4], [1/3-n/3,2/3-n/3,1-n/3], -4^4/3^3) for n>=10. - Peter Luschny, Sep 18 2014

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

Initial 0 prepended by N. J. A. Sloane, Apr 09 2008

A072827 Number of permutations satisfying i-2<=p(i)<=i+3, i=1..n.

Original entry on oeis.org

1, 2, 6, 18, 46, 115, 301, 792, 2068, 5380, 14020, 36581, 95413, 248786, 648714, 1691686, 4411530, 11503991, 29998953, 78228640, 203998184, 531969064, 1387222648, 3617479225, 9433351129, 24599481138, 64148406350, 167280683834

Offset: 1

Vladimir Baltic, Jul 21 2002

R. H. Hardin, Table of n, a(n) for n = 1..400
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135.
Index entries for linear recurrences with constant coefficients, signature (1,2,3,5,6,-1,-1,0,-1,-1).

Cf. A002524, A002525, A002526, A002527, A002528, A002529, A072850, A072851, A072852, A072853, A072854, A072855, A072856, A079955-A080014.

LinearRecurrence[{1,2,3,5,6,-1,-1,0,-1,-1},{1,2,6,18,46,115,301,792,2068,5380},30] (* Harvey P. Dale, Aug 15 2014 *)

a(n)=([0,1,0,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0,0; 0,0,0,0,1,0,0,0,0,0; 0,0,0,0,0,1,0,0,0,0; 0,0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,0,1; -1,-1,0,-1,-1,6,5,3,2,1]^(n-1)*[1;2;6;18;46;115;301;792;2068;5380])[1,1] \\ Charles R Greathouse IV, Jul 28 2015

Recurrence: a(n) = a(n-1)+2*a(n-2)+3*a(n-3)+5*a(n-4)+6*a(n-5)-a(n-6)-a(n-7)-a(n-9)-a(n-10).

G.f.: (1-x^5-x^3-x^2)/(x^10+x^9+x^7+x^6-6*x^5-5*x^4-3*x^3-2*x^2-x+1). [Corrected by Georg Fischer, May 15 2019]

A079955 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,2,3}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 2, 3, 3, 5, 6, 9, 11, 15, 19, 26, 34, 46, 60, 80, 105, 140, 185, 246, 325, 431, 570, 756, 1001, 1327, 1757, 2328, 3083, 4085, 5411, 7169, 9496, 12580, 16664, 22076, 29244, 38741, 51320, 67985, 90060, 119305, 158045, 209366, 277350, 367411

Offset: 0

Vladimir Baltic, Feb 19 2003

Number of compositions (ordered partitions) of n into elements of the set {2,5,6}.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,1,1).

Cf. A002524, A002525, A002526, A002527, A002528, A002529, A072827.
Cf. A072850, A072851, A072852, A072853, A072854, A072855, A072856.
Cf. A079955 - A080014.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x^2-x^5-x^6) )); // G. C. Greubel, Dec 11 2019

seq(coeff(series(1/(1-x^2-x^5-x^6), x, n+1), x, n), n = 0..50); # G. C. Greubel, Dec 11 2019

LinearRecurrence[{0, 1, 0, 0, 1, 1}, {1, 0, 1, 0, 1, 1}, 51] (* Jean-François Alcover, Dec 11 2019 *)

a(n) = ([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,1,0,0,1,0]^n*[1;0;1;0;1;1])[1,1] \\ Charles R Greathouse IV, Jul 28 2015

def A079955_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x^2-x^5-x^6) ).list()
A079955_list(50) # G. C. Greubel, Dec 11 2019

a(n) = a(n-2) + a(n-5) + a(n-6).

G.f.: 1/(1 - x^2 - x^5 - x^6).

A079977 Fibonacci numbers interspersed with zeros.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, 0, 13, 0, 21, 0, 34, 0, 55, 0, 89, 0, 144, 0, 233, 0, 377, 0, 610, 0, 987, 0, 1597, 0, 2584, 0, 4181, 0, 6765, 0, 10946, 0, 17711, 0, 28657, 0, 46368, 0, 75025, 0, 121393, 0, 196418, 0, 317811, 0, 514229, 0, 832040, 0, 1346269

Offset: 0

Vladimir Baltic, Feb 17 2003

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={0,2}.
Number of compositions of n into elements of the set {2,4}.
a(n-2) is the number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 1 or 3. - Ethan Patrick White, Jun 24 2020

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135.
Ethan P. White, Richard K. Guy, and Renate Scheidler, Difference Necklaces, arXiv:2006.15250 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A002524, A002525, A002526, A002527, A002528, A002529, A072827.
Cf. A072850, A072851, A072852, A072853, A072854, A072855, A072856, A099574.
Cf. A079955 - A080014.

A079977:= func< n | (1+(-1)^n)*Fibonacci(Floor((n+2)/2))/2 >;
[A079977(n): n in [0..50]]; // G. C. Greubel, Jul 25 2022

Riffle[Fibonacci[Range[50]],0] (* Harvey P. Dale, Dec 20 2015 *)

a(n)=if(n%2,0,fibonacci(n/2+1)) \\ Charles R Greathouse IV, Jun 11 2015

def A079977(n): return ((n+1)%2)*fibonacci((n+2)//2)
[A079977(n) for n in (0..50)] # G. C. Greubel, Jul 25 2022

a(n) = A000045(k+1) if n=2k, a(n)=0 otherwise.
a(n) = a(n-2) + a(n-4).
G.f.: 1/(1 - x^2 - x^4).

Editorial note: normally the alternate zeros are omitted from sequences like this. This entry is an exception. - N. J. A. Sloane

A079962 Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i=1..n, with k=1, r=5, I={1,3}.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 14, 22, 36, 58, 94, 153, 247, 399, 646, 1045, 1691, 2737, 4428, 7164, 11592, 18756, 30348, 49105, 79453, 128557, 208010, 336567, 544577, 881145, 1425722, 2306866, 3732588, 6039454, 9772042, 15811497, 25583539, 41395035

Offset: 0

Vladimir Baltic, Feb 19 2003

Number of compositions (ordered partitions) of n into elements of the set {1,3,5,6}. - Mark Dols, Aug 20 2010

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,1).

Cf. A002524, A002525, A002526, A002527, A002528, A002529, A072827, A072850, A072851, A072852, A072853, A072854, A072855, A072856, A079955 - A080014.

[Round(Fibonacci(n+3)/4): n in [0..40]]; // G. C. Greubel, Jan 21 2022

with(combinat,fibonacci): seq(round(fibonacci(n+3)/4),n=0..38) # Mircea Merca, Jan 04 2011

LinearRecurrence[{1,0,1,0,1,1}, {1,1,1,2,3,5}, 41] (* G. C. Greubel, Jan 21 2022 *)

a(n)=fibonacci(n+3)\/4 \\ Charles R Greathouse IV, Oct 07 2015

[(1/4)*(fibonacci(n+3) + chebyshev_U(n,1/2) + chebyshev_U(2*n,1/2)) for n in (0..40)] # G. C. Greubel, Jan 21 2022

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-6).
G.f.: 1/((1+x+x^2)*(1-x+x^2)*(1-x-x^2)).
a(n+1)/a(n) -> golden ratio A001622. - Roger L. Bagula, Mar 13 2006
a(n) + a(n+2) + a(n+4) = Fibonacci(n+5). - Mark Dols, Aug 20 2010
a(n) = round(Fibonacci(n+3)/4). - Mircea Merca, Jan 04 2011
a(n+6) - a(n) = A000045(n+6). - Paul Curtz, Jun 29 2013
a(n) + a(n+1) + a(n+2) = A024490(n+6). - R. J. Mathar, Jun 30 2013
a(n) - a(n-1) + a(n-2) = A094686(n). - R. J. Mathar, Jun 30 2013
4*a(n) = A057078(n) + A010892(n) + A000045(n+3). - R. J. Mathar, Nov 02 2016

cp's OEIS Frontend

A003269 a(n) = a(n-1) + a(n-4) with a(0) = 0, a(1) = a(2) = a(3) = 1.

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A072827 Number of permutations satisfying i-2<=p(i)<=i+3, i=1..n.

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A079955 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,2,3}.

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A079977 Fibonacci numbers interspersed with zeros.

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A079962 Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i=1..n, with k=1, r=5, I={1,3}.

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