A117373 -id:A117373 - cp's OEIS frontend

A003520 a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 106, 140, 185, 245, 325, 431, 571, 756, 1001, 1326, 1757, 2328, 3084, 4085, 5411, 7168, 9496, 12580, 16665, 22076, 29244, 38740, 51320, 67985, 90061, 119305, 158045, 209365, 277350, 367411, 486716, 644761

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.
Also counts ordered partitions such that no part is less than 5. For example, a(12) = a(11) + a(7) where a(7) counts 11,6+5 and 5+6 and a(11) counts 15,10+5, 9+6,8+7,7+8,6+9,5+10 and 5+5+5. Thus a(12) = 3 + 8 = 11. a(12) counts 16,11+5,10+6,9+7,8+8,7+9,6+10 and 6+5+5 but also 5+11,5+6+5 and 5+5+6. Similar results hold for the other sequences formed by a(n) = a(n-1) + a(n-k). - Alford Arnold, Aug 06 2003
Number of compositions of n into parts 1 and 5. - 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>=5, 2*a(n-5) equals the number of 2-colored compositions of n with all parts >= 5, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
a(n+4) equals the number of binary words of length n having at least 4 zeros between every two successive ones. - Milan Janjic, Feb 07 2015
Number of tilings of a 5 X n rectangle with 5 X 1 pentominoes. - M. Poyraz Torcuk, Mar 26 2022

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 119.
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
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.
Roland Bacher, On the number of perfect lattices, arXiv:1704.02234 [math.NT], 2017. See Section 6.
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.
Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, Why do Fibonacci numbers appear in patterns of growth in nature?, in Fibonacci Quarterly, 55(5): pp 30-41, (2017).
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
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,4,1).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 378
S. Kitaev, Independent sets on path-schemes, JIS 9 (2006) # 06.2.2 G(x) for M={1,2,3,4} gives seq. shifted 4 places left
T. G. Lewis, B. J. Smith and M. Z. Smith, Fibonacci sequences and money management, Fib. Quart., 14 (1976), 37-41.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
R. J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 33.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO] (2016), Section 4.4.
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), Article 16.8.3.
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
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1)

Apart from initial terms, same as A017899.

Cf. A000045, A000079, A000930, A003269, A005708, A005709, A005710, A005711.

a[0]:=1:a[1]:=1:a[2]:=1:a[3]:=1:a[4]:=1:for n from 5 to 60 do a[n]:=a[n-1]+a[n-5] od:seq(a[n],n=0..60);
with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 4)}, unlabeled]: seq(count(SeqSetU, size=j), j=5..55); # Zerinvary Lajos, Oct 10 2006
A003520:=-1/(z**3+z**2-1)/(z**2-z+1); # Simon Plouffe in his 1992 dissertation
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 4)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=4..54); # Zerinvary Lajos, Mar 26 2008
M := Matrix(5, (i,j)-> if j=1 then [1, 0, 0, 0, 1][i] elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1,1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 27 2008

a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = a[n - 1] + a[n - 5]; Table[ a[n], {n, 0, 49}] (* Robert G. Wilson v, Dec 09 2004 *)
CoefficientList[Series[1/(1 - x - x^5), {x, 0, 51}], x] (* Zerinvary Lajos, Mar 29 2007 *)
LinearRecurrence[{1, 0, 0, 0, 1}, {1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,e+a}; NestList[nxt,{1,1,1,1,1},50][[;;,1]] (* Harvey P. Dale, Sep 27 2023 *)

a(n):=sum(binomial(n-1+(-4)*j,j),j,0,(n-1)/4); /* Vladimir Kruchinin, May 23 2011 */

my(x='x+O('x^66)); Vec(x/(1-(x+x^5))) /* Joerg Arndt, Jun 25 2011 */

G.f.: 1/(1-x-x^5) = 1/((1-x+x^2)(1-x^2-x^3)).
a(n) = Sum_{j=0..(n-1)/4} binomial(n-1+(-4)*j,j).
For n>5, a(n) = floor( d*c^n + 1/2) where c is the positive real root of x^5-x^4-1 and d is the positive real root of 161*x^3-23*x^2-12*x-1 ( c=1.32471795724474602... and d=0.3811571478326847...) - Benoit Cloitre, Nov 30 2002
a(n) = term (1,1) in the 5 X 5 matrix [1,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,0,0,0]^n. - Alois P. Heinz, Jul 27 2008
For positive integers n and k such that k <= n <= 5*k, and 4 divides n-k, define c(n,k) = binomial(k,(n-k)/4), and c(n,k)=0, otherwise. Then, for n >= 1,  a(n) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([-1/5*n, 1/5-1/5*n, 2/5-1/5*n, 3/5-1/5*n, 4/5-1/5*n], [-1/4*n, 1/4-1/4*n, 1/2-1/4*n, 3/4-1/4*n], -5^5/4^4) for n>=16. - Peter Luschny, Sep 18 2014
7*a(n) = A117373(n+4) +5*b(n) +4*b(n-1) +b(n-2) where b(n) = A182097(n). - R. J. Mathar, Aug 07 2017

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

A136674 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial p(n,x) with p(0,x) = 1, p(1,x) = 2 - x, p(2,x) = 1 - 4x + x^2 and p(n,x) = (2-x)p(n-1,x) - p(n-2,x) if n>2.

Original entry on oeis.org

1, 2, -1, 1, -4, 1, 0, -8, 6, -1, -1, -12, 19, -8, 1, -2, -15, 44, -34, 10, -1, -3, -16, 84, -104, 53, -12, 1, -4, -14, 140, -258, 200, -76, 14, -1, -5, -8, 210, -552, 605, -340, 103, -16, 1, -6, 3, 288, -1056, 1562, -1209, 532, -134, 18, -1, -7, 20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1

Offset: 0

Roger L. Bagula, Apr 05 2008

Row sums: A117373(n-1).

			Triangle begins as:
   1;
   2,  -1;
   1,  -4,   1;
   0,  -8,   6,    -1;
  -1, -12,  19,    -8,    1;
  -2, -15,  44,   -34,   10,    -1;
  -3, -16,  84,  -104,   53,   -12,    1;
  -4, -14, 140,  -258,  200,   -76,   14,   -1;
  -5,  -8, 210,  -552,  605,  -340,  103,  -16,   1;
  -6,   3, 288, -1056, 1562, -1209,  532, -134,  18,  -1;
  -7,  20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

A136674aux := proc(n) option remember; if n = 0 then 1; elif n= 1 then 2-x ; elif n= 2 then 1-4*x+x^2 ; else (2-x)*procname(n-1)-procname(n-2) ; end if; end proc:
A136674 := proc(n,k) coeftayl(A136674aux(n),x=0,k) ; end proc: # R. J. Mathar, Jan 12 2011

(* tridiagonal matrix code*)
T[n_, m_, d_]:= If[n==m, 2, If[n==d && m==d-1, -3, If[(n==m-1 || n==m+1), -1, 0]]];
M[d_]:= Table[T[n, m, d], {n,d}, {m,d}];
Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]//Flatten
(* polynomial recursion: three initial terms necessary*)
p[x, 0]:= 1; p[x, 1]:= (2-x); p[x, 2]:= 1 -4*x +x^2;
p[x_, n_]:= p[x, n]= (2-x)*p[x, n-1] - p[x, n-2];
Table[ExpandAll[p[x, n]], {n, 0, Length[g] -1}]
(* Third program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, (-1)^n, If[k==0, 3-n, 2*T[n-1, k] -T[n-2, k] -T[n-1, k-1] ]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return (-1)^n
    elif (k==0): return 3-n
    else: return 2*T(n-1,k) - T(n-2,k) - T(n-1,k-1)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

T(n,k) = 2*T(n-1,k) - T(n-2,k) - T(n-1,k-1). - R. J. Mathar, Jan 12 2011

A070365 a(n) = 5^n mod 7.

Original entry on oeis.org

1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2, 3, 1, 5, 4, 6, 2

Offset: 0

N. J. A. Sloane, May 12 2002

From Klaus Brockhaus, May 23 2010: (Start)
Period 6: repeat [1, 5, 4, 6, 2, 3].
Continued fraction expansion of (221+11*sqrt(1086))/490.
Decimal expansion of 199/1287.
First bisection is A153727. (End)

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

Cf. A178229 (decimal expansion of (221+11*sqrt(1086))/490), A178141 (repeat 4, -1, 2, -4, 1, -2), A117373 (repeat 1, -2, -3, -1, 2, 3), A153727 (trajectory of 3x+1 sequence starting at 1).

Cf. A000351, A010876.

[Modexp(5, n, 7): n in [0..100]]; // Vincenzo Librandi, Mar 24 2016 - after Bruno Berselli

A070365:=n->[1, 5, 4, 6, 2, 3][(n mod 6)+1]: seq(A070365(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

PowerMod[5, Range[0, 110], 7] (* or *) LinearRecurrence[{1, 0, -1, 1}, {1, 5, 4, 6}, 110] (* Harvey P. Dale, Apr 26 2011 *)
Table[Mod[5^n, 7], {n, 0, 100}] (* G. C. Greubel, Mar 05 2016 *)
PadRight[{}, 100, {1, 5, 4, 6, 2, 3}] (* or *) CoefficientList[Series[(1 + 5 x + 4 x^2 + 6 x^3 + 2 x^4 + 3 x^5) / (1 - x^6), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 24 2016 *)

a(n)=lift(Mod(5,7)^n) \\ Charles R Greathouse IV, Mar 22 2016

From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
G.f.: (1+4*x-x^2+3*x^3)/ ((1-x)*(1+x)*(x^2-x+1)). (End)
From Klaus Brockhaus, May 23 2010: (Start)
a(n+1)-a(n) = A178141(n).
a(n+2)-a(n) = A117373(n+5). (End)
From G. C. Greubel, Mar 05 2016: (Start)
a(n) = a(n-6) for n>5.
E.g.f.: (1/3)*(7*cosh(x) + 14*sinh(x) + 2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) - 4*exp(x/2)*cos(sqrt(3)*x/2)). (End)
a(n) = (21 - 7*cos(n*Pi) - 8*cos(n*Pi/3) + 4*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 23 2016
a(n) = A010876(A000351(n)). - Michel Marcus, Jun 27 2016

A028289 Expansion of (1+x^2+x^3+x^5)/((1-x)(1-x^3)(1-x^4)(1-x^6)).

Original entry on oeis.org

1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658, 707, 763, 812, 868, 932, 988, 1052, 1124, 1188

Offset: 0

N. J. A. Sloane

C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.
B. N. Cyvin et al., Enumeration of conjugated hydrocarbons: Hollow hexagons revisited, Structural Chem., 6 (1995), 85-88, equations (6) and (22).
W. C. Huffman, The biweight enumerator of self-orthogonal binary codes, Discr. Math. Vol. 26 1979, pp. 129-143.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-2,-2,1,1,1,-1).

A117373 := proc(n) op(1+(n mod 6),[1,-2,-3,-1,2,3]) ; end proc:
A076118 := proc(n) coeftayl( x*(1-x)/(1-x+x^2)^2,x=0,n) ; end proc:
A028289 := proc(n) 1/108*n^3 +1/8*n^2 +55/108*n +29/48 +1/16*(-1)^n -2*(-1)^n*A117373(n+2)/27 +(-1)^n*A076118(n+1)/9; end proc:
seq(A028289(n),n=0..20) ; # R. J. Mathar, Mar 22 2011

CoefficientList[Series[(1+x^2+x^3+x^5)/((1-x)(1-x^3)(1-x^4) (1-x^6)),{x,0,50}],x]  (* Harvey P. Dale, Apr 20 2011 *)

Vec((1+x^2+x^3+x^5)/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

G.f.: 1 / ( (1+x)*(1+x+x^2)^2*(x-1)^4 ). - R. J. Mathar, Mar 22 2011

A099559 a(n) = Sum_{k=0..floor(n/5)} C(n-4k,k+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 10, 14, 19, 25, 33, 44, 59, 79, 105, 139, 184, 244, 324, 430, 570, 755, 1000, 1325, 1756, 2327, 3083, 4084, 5410, 7167, 9495, 12579, 16664, 22075, 29243, 38739, 51319, 67984, 90060, 119304, 158044, 209364, 277349, 367410, 486715

Offset: 0

Paul Barry, Oct 22 2004

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Minerva Catral, P. L. Ford, P. E. Harris, S. J. Miller, et al. Legal Decompositions Arising from Non-positive Linear Recurrences, arXiv preprint arXiv:1606.09312 [math.CO], 2016. See Table 2.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-1).

Cf. A098578.

LinearRecurrence[{2,-1,0,0,1,-1},{0,1,2,3,4,5},50] (* Harvey P. Dale, Feb 20 2017 *)

a(n) = sum(k=0,n\5, binomial(n-4*k, k+1)); \\ Michel Marcus, Jul 11 2018

Partial sums of A003520 (with leading zero).
G.f.: x / ( (x-1)*(x^2-x+1)*(x^3+x^2-1) ).
a(n) = 2a(n-1)-a(n-2)+a(n-5)-a(n-6).
7*a(n) = A117373(n+2) -7 +10*b(n) +15*b(n-1) +9*b(n-2), where b(n) = A182097(n). - R. J. Mathar, Aug 07 2017
a(n) = A003520(n+4) -1. - R. J. Mathar, Aug 07 2017

Values from a(5) on corrected by R. J. Mathar, Jul 29 2008

A117372 Riordan array (1-3x,x(1-x)).

Original entry on oeis.org

1, -3, 1, 0, -4, 1, 0, 3, -5, 1, 0, 0, 7, -6, 1, 0, 0, -3, 12, -7, 1, 0, 0, 0, -10, 18, -8, 1, 0, 0, 0, 3, -22, 25, -9, 1, 0, 0, 0, 0, 13, -40, 33, -10, 1, 0, 0, 0, 0, -3, 35, -65, 42, -11, 1, 0, 0, 0, 0, 0, -16, 75, -98, 52, -12, 1

Offset: 0

Paul Barry, Mar 10 2006

Row sums are A117373. Diagonal sums are A117374. Inverse is A117375.

			Triangle begins
1,
-3, 1,
0, -4, 1,
0, 3, -5, 1,
0, 0, 7, -6, 1,
0, 0, -3, 12, -7, 1,
0, 0, 0, -10, 18, -8, 1,
0, 0, 0, 3, -22, 25, -9, 1

Number triangle T(n,k)=(-1)^(n-k)(C(k,n-k)+3*C(k, n-k-1))

A135350 a(n) = 2a(n-1) - a(n-3) + 2a(n-4).

Original entry on oeis.org

0, 1, 3, 8, 15, 29, 56, 113, 227, 456, 911, 1821, 3640, 7281, 14563, 29128, 58255, 116509, 233016, 466033, 932067, 1864136, 3728271, 7456541, 14913080, 29826161, 59652323, 119304648, 238609295, 477218589, 954437176, 1908874353, 3817748707

Offset: 0

Paul Curtz, Feb 16 2008

nonn

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

Cf. A117373.

A117373 := proc(n) coeftayl( (1-3*x)/(1-x+x^2),x=0,n) ; end: A135350 := proc(n) 2*(-1)^(n+1)/9+2^(n+3)/9+A117373(n+1)/3 ; end: seq(A135350(n),n=0..10) ; # R. J. Mathar, Feb 19 2008

LinearRecurrence[{2, 0, -1, 2}, {0, 1, 3, 8}, 25] (* G. C. Greubel, Oct 11 2016 *)

From R. J. Mathar, Feb 19 2008: (Start)
O.g.f.: (1/9)*(-3*(x+2)/(x^2-x+1) - 8/(2*x-1) - 2/(x+1)).
a(n) = (1/9)*(2*(-1)^(n+1) + 2^(n+3) + 3*A117373(n+1)). (End)

More terms from R. J. Mathar, Feb 19 2008

A132798 Period 6: repeat [0, 2, 1, 0, -2, -1].

Original entry on oeis.org

0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1

Offset: 0

Paul Curtz, Nov 21 2007

Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Cf. A002264, A080425, A117373, A129339, A130151.

[(-n mod 3)*(-1)^Floor(n/3) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

A132798:=n->(-n mod 3)*(-1)^floor(n/3); seq(A132798(n), n=0..50); # Wesley Ivan Hurt, Jun 20 2014

Table[Mod[-n, 3]*(-1)^Floor[n/3], {n, 0, 50}] (* Wesley Ivan Hurt, Jun 20 2014 *)

G.f.: x*(2+x)/((x+1)*(x^2-x+1)) = (1/3)*(4*x+1)/(x^2-x+1)-(1/3)/(x+1). - R. J. Mathar, Nov 28 2007
a(n) + a(n+1) = A117373(n+4). - R. J. Mathar, Jul 22 2009
a(n) = (-n mod 3) * (-1)^floor(n/3) = A080425(n) * (-1)^A002264(n) = A080425(n) * A130151(n). - Wesley Ivan Hurt, Jun 20 2014
From Wesley Ivan Hurt, Jun 21 2016: (Start)
a(n) + a(n-3) = 0 for n>2.
a(n) = sin(n*Pi/3) * (3*sqrt(3) + 2*sin(2*n*Pi/3))/3. (End)

A178873 Partial sums of round(5^n/7).

Original entry on oeis.org

0, 1, 5, 23, 112, 558, 2790, 13951, 69755, 348773, 1743862, 8719308, 43596540, 217982701, 1089913505, 5449567523, 27247837612, 136239188058, 681195940290, 3405979701451, 17029898507255, 85149492536273

Offset: 0

Mircea Merca, Dec 28 2010

			a(6) = 0 + 1 + 4 + 18 + 89 + 446 + 2232 = 2790.

Vincenzo Librandi, Table of n, a(n) for n = 0..160
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (7,-12,11,-5).

[Floor((5*5^n+19)/28): n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

A178873 := proc(n) add( round(5^i/7),i=0..n) ; end proc:

Accumulate[Round[5^Range[0,25]/7]] (* Harvey P. Dale, Feb 01 2011 *)

a(n) = round((5*5^n + 7)/28).
a(n) = floor((5*5^n + 19)/28).
a(n) = ceiling((5*5^n - 5)/28).
a(n) = a(n-6) + 558*5^(n-5), n>5.
a(n) = 5*a(n-1) + a(n-6) - 5*a(n-7), n>6.
a(n) = 7*a(n-1) - 12*a(n-2) + 11*a(n-3) - 5*a(n-4), n>3.
G.f.: -(2*x^2-x)/((x-1)*(5*x-1)*(x^2-x+1)).
a(n) = 5^(n+1)/28 + 1/4 + A117373(n+2)/7 = (5*5^n+7)/28 - ((9-i*sqrt(3))*(1-i*sqrt(3))^n + (9+i*sqrt(3))*(1+i*sqrt(3))^n) / (42*2^n) where i is the imaginary unit. - Bruno Berselli, Jan 12 2011

cp's OEIS Frontend

A144476 Four interlaced copies of A117373.

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A003520 a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.

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Maxima

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A136674 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial p(n,x) with p(0,x) = 1, p(1,x) = 2 - x, p(2,x) = 1 - 4*x + x^2 and p(n,x) = (2-x)*p(n-1,x) - p(n-2,x) if n>2.

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A070365 a(n) = 5^n mod 7.

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A028289 Expansion of (1+x^2+x^3+x^5)/((1-x)(1-x^3)(1-x^4)(1-x^6)).

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A099559 a(n) = Sum_{k=0..floor(n/5)} C(n-4k,k+1).

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A117372 Riordan array (1-3x,x(1-x)).

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A135350 a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4).

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A136674 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial p(n,x) with p(0,x) = 1, p(1,x) = 2 - x, p(2,x) = 1 - 4x + x^2 and p(n,x) = (2-x)p(n-1,x) - p(n-2,x) if n>2.

A135350 a(n) = 2a(n-1) - a(n-3) + 2a(n-4).