A001687 -id:A001687 - cp's OEIS frontend

A369840 Number of compositions of 5*n into parts 2 and 5.

1, 1, 2, 7, 23, 68, 194, 555, 1601, 4633, 13404, 38752, 112004, 323728, 935737, 2704817, 7818464, 22599701, 65325542, 188826693, 545813094, 1577700612, 4560424135, 13182138184, 38103641048, 110140512968, 318366757185, 920255312908, 2660044812499, 7688994894381

Offset: 0

Seiichi Manyama, Feb 03 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,10,-5,1).

Cf. A369803, A369842, A369843, A369844.
Cf. A099131, A369836, A369845.
Cf. A001687.

LinearRecurrence[{5, -9, 10, -5, 1}, {1, 1, 2, 7, 23}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\2, binomial(n+3*k, n-2*k));

a(n) = A001687(5*n+1).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,n-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (1-x)^4/((1-x)^5 - x^2).
a(n) = A369843(n) - A369843(n-1). - R. J. Mathar, Feb 14 2024

A369803 Expansion of 1/(1 - x^2/(1-x)^5).

Original entry on oeis.org

1, 0, 1, 5, 16, 45, 126, 361, 1046, 3032, 8771, 25348, 73252, 211724, 612009, 1769080, 5113647, 14781237, 42725841, 123501151, 356986401, 1031887518, 2982723523, 8621714049, 24921502864, 72036871920, 208226244217, 601888555723, 1739789499591, 5028950081882

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Number of compositions of 5*n-2 into parts 2 and 5.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,10,-5,1).

Cf. A079675, A368475, A369804.
Cf. A369840, A369842, A369843, A369844.
Cf. A001687.

my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-x)^5))

a(n) = sum(k=0, n\2, binomial(n-1+3*k, n-2*k));

a(n) = A001687(5*n-1) for n > 0.
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+3*k,n-2*k).
a(n) = A369840(n)-A369840(n-1). - R. J. Mathar, Feb 14 2024

A369842 Number of compositions of 5*n-1 into parts 2 and 5.

Original entry on oeis.org

1, 3, 7, 18, 52, 154, 450, 1301, 3753, 10838, 31327, 90568, 261813, 756786, 2187496, 6323023, 18277014, 52830706, 152709940, 441415867, 1275934888, 3688154521, 10660798289, 30815580241, 89074003241, 257472939209, 744238632362, 2151259638423, 6218325456983

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,10,-5,1).

Cf. A369803, A369840, A369843, A369844.

Cf. A001687.

LinearRecurrence[{5, -9, 10, -5, 1}, {1, 3, 7, 18, 52}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\2, binomial(n+1+3*k, n-1-2*k));

a(n) = A001687(5*n).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+1+3*k,n-1-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^2/((1-x)^5 - x^2).

A369843 Number of compositions of 5*n-3 into parts 2 and 5.

Original entry on oeis.org

1, 2, 4, 11, 34, 102, 296, 851, 2452, 7085, 20489, 59241, 171245, 494973, 1430710, 4135527, 11953991, 34553692, 99879234, 288705927, 834519021, 2412219633, 6972643768, 20154781952, 58258423000, 168398935968, 486765693153, 1407021006061, 4067065818560

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,10,-5,1).

Cf. A369803, A369840, A369842, A369844.

Cf. A001687.

LinearRecurrence[{5, -9, 10, -5, 1}, {1, 2, 4, 11, 34}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\2, binomial(n+3*k, n-1-2*k));

a(n) = A001687(5*n-2).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,n-1-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x^2).

A369844 Number of compositions of 5*n-4 into parts 2 and 5.

Original entry on oeis.org

0, 1, 4, 11, 29, 81, 235, 685, 1986, 5739, 16577, 47904, 138472, 400285, 1157071, 3344567, 9667590, 27944604, 80775310, 233485250, 674901117, 1950836005, 5638990526, 16299788815, 47115369056, 136189372297, 393662311506, 1137900943868, 3289160582291, 9507486039274

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,10,-5,1).

Cf. A369803, A369840, A369842, A369843.

Cf. A001687.

LinearRecurrence[{5, -9, 10, -5, 1}, {0, 1, 4, 11, 29}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n\2, binomial(n+1+3*k, n-2-2*k));

a(n) = A001687(5*n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+1+3*k,n-2-2*k).
a(n) = 5*a(n-1) - 9*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^2*(1-x)/((1-x)^5 - x^2).

A005686 Number of Twopins positions.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 12, 14, 18, 22, 27, 34, 41, 52, 63, 79, 97, 120, 149, 183, 228, 280, 348, 429, 531, 657, 811, 1005, 1240, 1536, 1897, 2347, 2902, 3587, 4438, 5484, 6785, 8386, 10372, 12824, 15856, 19609, 24242, 29981, 37066, 45837

Offset: 0

N. J. A. Sloane

R. K. Guy, "Anyone for Twopins?" in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Natasha Blitvić, Vicente I. Fernandez, A Combinatorial Model for Heterogeneous Microbial Growth, arXiv:1901.04080 [math.CO], 2019.
S. Falcon, Generalized (k,r)-Fibonacci Numbers, Gen. Math. Notes, 25(2), 2014, 148-158.
R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
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
I. Wloch, U. Bednarz, D. Bród, A Wloch and M. Wolowiec-Musial, On a new type of distance Fibonacci numbers, Discrete Applied Math., Volume 161, Issues 16-17, November 2013, Pages 2695-2701.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 1).

Cf. A001687.

I:=[1,1,1,1,1]; [0] cat [n le 5 select I[n] else Self(n-2)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jan 19 2016

A005686 := -(z+1)*(z**3+z+1)/(-1+z**2+z**5); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 1's
a := proc(n): if n = 0 then 0 else add(binomial(floor((n+3*k-4)/5), k), k=0..floor((n-1)/2)) fi: end: seq(a(n), n=0..54); # Johannes W. Meijer, Aug 05 2013

nn=54; CoefficientList[Series[(x+x^2)/(1-x^2-x^5),{x,0,nn}],x]  (* Geoffrey Critzer, Apr 28 2013 *)
m = 5; For[n = 0, n < m, n++, a[n] = 1]; For[n = m, n < 51, n++, a[n] = a[n - m] + a[n - 2]]; Table[a[n], {n, 0, 50}] (*Sergio Falcon, Nov 12 2015 *)
Join[{0}, LinearRecurrence[{0, 1, 0, 0, 1}, {1, 1, 1, 1, 1}, 60]] (* Vincenzo Librandi, Jan 19 2016 *)

a(n)=if(n<0,polcoeff((x^3+x^4)/(1+x^3-x^5)+x^-n*O(x),-n),polcoeff((x+x^2)/(1-x^2-x^5)+x^n*O(x),n)) /* Michael Somos, Jul 15 2004 */

a(n)=sum(k=0,(n-1)\2,binomial((n+3*k-4)\5,k))

a(n) = Sum_{k=0..floor(n/2)} binomial(floor((n+3k-3)/5), k). - Paul Barry, Jul 10 2004
G.f.: (x+x^2)/(1-x^2-x^5). - Ralf Stephan, Apr 21 2004
a(n) = A001687(n)+A001687(n-1). - Ralf Stephan, Apr 21 2004
a(n) = a(n-2) + a(n-5). - Michael Somos, Jul 15 2004
a(n+1) = Sum_{k=0..floor(n/5)} A065941(n-4*k, n-5*k). - Johannes W. Meijer, Aug 05 2013

More terms from Paul Barry, Jul 10 2004

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 *)

A228571 The backwards antidiagonal sums of triangle A228570.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 4, 7, 6, 10, 10, 14, 15, 20, 24, 30, 35, 45, 53, 69, 79, 104, 120, 157, 184, 236, 281, 356, 431, 540, 656, 821, 1001, 1252, 1525, 1908, 2328, 2909, 3557, 4434, 5436, 6762

Offset: 0

Johannes W. Meijer, Aug 26 2013

The a(n) equal the backwards antidiagonal sums of triangle A228570.

Cf. A228570, A102543, A192928, A001687.

f := x -> (1/((1-x^2-x^5)) + (1+x^2+x^5)/(1-x^4-x^10))/2 : seq(coeff(series(f(x), x, n+1), x, n), n=0..50);  # End first program
a := proc(n): (A001687(n+1) + x(n) + x(n-2) + x(n-5))/2 end: A001687 := proc(n) option remember: if n=0 then 0 elif n=1 then 1 elif n=2 then 0 elif n=3 then 1 elif n=4 then 0 else procname(n-2) + procname(n-5) fi: end: x := proc(n) local x: if n <0 then return(0) fi: if type(n, even) then A001687((n+2)/2) else 0 fi: end: seq(a(n), n=0..50); # End second program

a(n) = sum(A228570(n-k, n-2*k), k=0..floor(n/2)).

G.f.: (1/2)*(1/(1-x^2-x^5) + (1+x^2+x^5)/(1-x^4-x^10)).

A376783 Expansion of 1/sqrt((1 - x^2 - x^5)^2 - 4*x^7).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 4, 1, 9, 2, 16, 10, 25, 37, 37, 101, 65, 226, 164, 443, 481, 810, 1325, 1522, 3258, 3251, 7236, 7926, 15010, 20234, 30557, 50234, 64501, 117966, 145557, 263107, 346293, 569726, 835909, 1233943, 1984730, 2740492, 4579704, 6288323, 10311571

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Cf. A246884, A376722, A376784.

Cf. A001687.

my(N=50, x='x+O('x^N)); Vec(1/sqrt((1-x^2-x^5)^2-4*x^7))

a(n) = sum(k=0, n\5, ((n-3*k)%2==0)*binomial((n-3*k)/2, k)^2);

G.f.: 1/sqrt((1 - x^2 + x^5)^2 - 4*x^5) = 1/sqrt((1 + x^2 - x^5)^2 - 4*x^2).

A007384 Number of strict 3rd-order maximal independent sets in path graph.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 0, 6, 1, 10, 4, 15, 10, 22, 20, 33, 35, 51, 57, 80, 90, 125, 141, 193, 221, 295, 346, 449, 539, 684, 834, 1045, 1283, 1600, 1967, 2451, 3012, 3752, 4612, 5738, 7063, 8770, 10815, 13403, 16553, 20488, 25323, 31326, 38726

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. Yanco and A. Bagchi, ``K-th order maximal independent sets in path and cycle graphs,'' J. Graph Theory, submitted, 1994.

R. Yanco, Letter and Email to N. J. A. Sloane, 1994

Cf. A001687.

Conjecture: a(n)= 3*a(n-2) -3*a(n-4) +a(n-5) +a(n-6) -2*a(n-7) +a(n-9) with g.f. -x^5/((x^5+x^2-1)*(x-1)^2*(1+x)^2). [From R. J. Mathar, Oct 30 2009]

a(n) = A001687(n + 6) - b(n) where b(2*n+1) = 1 and b(2*n) = n+1. - Sean A. Irvine, Jan 02 2018

More terms from Sean A. Irvine, Jan 02 2018

cp's OEIS Frontend

A369840 Number of compositions of 5*n into parts 2 and 5.

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A369803 Expansion of 1/(1 - x^2/(1-x)^5).

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A369842 Number of compositions of 5*n-1 into parts 2 and 5.

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A369843 Number of compositions of 5*n-3 into parts 2 and 5.

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A369844 Number of compositions of 5*n-4 into parts 2 and 5.

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A005686 Number of Twopins positions.

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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.

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A228571 The backwards antidiagonal sums of triangle A228570.

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