A096568 -id:A096568 - cp's OEIS frontend

A096569 Number of compositions of n with first part 1 and no equal adjacent parts; this is column 1 of the array in A096568.

1, 0, 1, 2, 2, 5, 9, 14, 25, 46, 78, 136, 242, 419, 733, 1291, 2251, 3938, 6905, 12073, 21129, 37001, 64741, 113304, 198344, 347126, 607532, 1063387, 1861149, 3257410, 5701362, 9978711, 17465052, 30568232, 53501720, 93640745, 163894183

Offset: 1

Clark Kimberling, Jun 27 2004

nonn

Cf. A003242, A096568, A096570, A096571, A096572.

Define s(0)=1, T(1, 1)=1 and T(i, j)=0 for j>i. For n>=2 and 1<=k<=n, define s(n)=T(n, 1)+T(n, 2)+...+T(n, n) and T(n, k)=s(n-k)-T(n-k, k). (This is array A096568.)

A096570 Number of compositions of n with first part 2 and no equal adjacent parts; this is column 2 of the array in A096568.

Original entry on oeis.org

0, 1, 1, 0, 2, 4, 5, 10, 18, 29, 53, 95, 161, 283, 500, 869, 1524, 2673, 4665, 8170, 14313, 25032, 43817, 76710, 134228, 234938, 411242, 719720, 1259677, 2204816, 3858882, 6753956, 11821191, 20689807, 36212093, 63380145, 110930372, 194154783

Offset: 1

Clark Kimberling, Jun 27 2004

nonn

Cf. A003242, A096568, A096569, A096571, A096572.

Define s(0)=1, T(1, 1)=1 and T(i, j)=0 for j>i. For n>=2 and 1<=k<=n, define s(n)=T(n, 1)+T(n, 2)+...+T(n, n) and T(n, k)=s(n-k)-T(n-k, k). (This is array A096568.)

A096571 Number of compositions of n with first part 3 and no equal adjacent parts; this is column 3 of the array in A096568.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 6, 12, 20, 33, 59, 104, 181, 319, 557, 971, 1705, 2985, 5218, 9138, 15993, 27984, 48992, 85749, 150061, 262656, 459721, 804597, 1408263, 2464815, 4313962, 7550509, 13215258, 23129801, 40482775, 70854694, 124012664

Offset: 1

Clark Kimberling, Jun 27 2004

nonn

Cf. A003242, A096568, A096569, A096570, A096572.

Define s(0)=1, T(1, 1)=1 and T(i, j)=0 for j>i. For n>=2 and 1<=k<=n, define s(n)=T(n, 1)+T(n, 2)+...+T(n, n) and T(n, k)=s(n-k)-T(n-k, k). (This is array A096568.)

A096572 Number of compositions of n with first part 4 and no equal adjacent parts; this is column 4 of the array in A096568.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 6, 13, 20, 36, 65, 111, 194, 342, 596, 1041, 1830, 3200, 5593, 9802, 17148, 30002, 52537, 91940, 160897, 281646, 492933, 862718, 1510022, 2642890, 4625626, 8096054, 14170051, 24800873, 43407658, 75973898, 132972414

Offset: 1

Clark Kimberling, Jun 27 2004

nonn

Cf. A003242, A096568, A096569, A096570, A096571.

Define s(0)=1, T(1, 1)=1 and T(i, j)=0 for j>i. For n>=2 and 1<=k<=n, define s(n)=T(n, 1)+T(n, 2)+...+T(n, n) and T(n, k)=s(n-k)-T(n-k, k). (This is array A096568.)

A003242 Number of compositions of n such that no two adjacent parts are equal (these are sometimes called Carlitz compositions).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 14, 23, 39, 71, 124, 214, 378, 661, 1152, 2024, 3542, 6189, 10843, 18978, 33202, 58130, 101742, 178045, 311648, 545470, 954658, 1670919, 2924536, 5118559, 8958772, 15680073, 27443763, 48033284, 84069952, 147142465, 257534928, 450748483, 788918212

Offset: 0

E. Rodney Canfield

			From _Joerg Arndt_, Oct 27 2012:  (Start)
The 23 such compositions of n=7 are
[ 1]  1 2 1 2 1
[ 2]  1 2 1 3
[ 3]  1 2 3 1
[ 4]  1 2 4
[ 5]  1 3 1 2
[ 6]  1 3 2 1
[ 7]  1 4 2
[ 8]  1 5 1
[ 9]  1 6
[10]  2 1 3 1
[11]  2 1 4
[12]  2 3 2
[13]  2 4 1
[14]  2 5
[15]  3 1 2 1
[16]  3 1 3
[17]  3 4
[18]  4 1 2
[19]  4 2 1
[20]  4 3
[21]  5 2
[22]  6 1
[23]  7
(End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 191.

Alois P. Heinz, Table of n, a(n) for n = 0..4100 (first 501 terms from Christian G. Bower)
L. Carlitz, Restricted Compositions, Fibonacci Quarterly, 14 (1976) 254-264.
Sylvie Corteel, Paweł Hitczenko, Generalizations of Carlitz Compositions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.8
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 42 and 117.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 201
F. Harary & R. W. Robinson, The number of achiral trees, Jnl. Reine Angewandte Mathematik 278 (1975), 322-335. (Annotated scanned copy)
A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.
E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Chapter 8.

Cf. A106351, A114900, A114902.
Cf. A096568, A011782, A106356. - Franklin T. Adams-Watters, May 27 2010
Row sums of A232396, A241701.
Cf. A241902.
Column k=1 of A261960.
Cf. A048272.
Compositions with adjacent parts coprime are A167606.
The complement is counted by A261983.
Cf. A000740, A005251, A032020, A114901, A178470, A261041, A274174, A329738, A329863.

a003242 n = a003242_list !! n
a003242_list = 1 : f [1] where
   f xs = y : f (y : xs) where
          y = sum $ zipWith (*) xs a048272_list
-- Reinhard Zumkeller, Nov 04 2015

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(`if`(j=i, 0, b(n-j, `if`(j<=n-j, j, 0))), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 27 2014

A048272[n_] := Total[If[OddQ[#], 1, -1]& /@ Divisors[n]]; a[n_] := a[n] = Sum[A048272[k]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 38}](* Jean-François Alcover, Nov 25 2011, after Vladeta Jovovic *)
nmax = 50; CoefficientList[Series[1/(1 - Sum[x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 07 2020 *)
Table[Count[Flatten[Permutations/@IntegerPartitions[n],1],?(FreeQ[Differences[#],0]&)],{n,0,20}] (* The program generates the first 21 terms of the sequence. *) (* _Harvey P. Dale, Nov 23 2024 *)

N = 66;  x = 'x + O('x^N);  p=2;
gf = 1/(1-sum(k=1,N, x^k/(1-x^k)-p*x^(k*p)/(1-x^(k*p))));
Vec(gf)  /* Joerg Arndt, Apr 28 2013 */

a(n) = Sum_{k=1..n} A048272(k)*a(n-k), n>1, a(0)=1. - Vladeta Jovovic, Feb 05 2002
G.f.: 1/(1 - Sum_{k>0} x^k/(1+x^k)).
a(n) ~ c r^n where c is approximately 0.456387 and r is approximately 1.750243. (Formula from Knopfmacher and Prodinger reference.) - Franklin T. Adams-Watters, May 27 2010. With better precision: r = 1.7502412917183090312497386246398158787782058181381590561316586... (see A241902), c = 0.4563634740588133495321001859298593318027266156100046548066205... - Vaclav Kotesovec, Apr 30 2014
G.f. is the special case p=2 of 1/(1 - Sum_{k>0} (z^k/(1-z^k) - p*z^(k*p)/(1-z^(k*p)))), see A129922. - Joerg Arndt, Apr 28 2013
G.f.: 1/(1 - x * (d/dx) log(Product_{k>=1} (1 + x^k)^(1/k))). - Ilya Gutkovskiy, Oct 18 2018
Moebius transform of A329738. - Gus Wiseman, Nov 27 2019
For n>=2, a(n) = A128695(n) - A091616(n). - Vaclav Kotesovec, Jul 07 2020

More terms from David W. Wilson

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A096569 Number of compositions of n with first part 1 and no equal adjacent parts; this is column 1 of the array in A096568.

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A096570 Number of compositions of n with first part 2 and no equal adjacent parts; this is column 2 of the array in A096568.

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A096571 Number of compositions of n with first part 3 and no equal adjacent parts; this is column 3 of the array in A096568.

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A096572 Number of compositions of n with first part 4 and no equal adjacent parts; this is column 4 of the array in A096568.

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A003242 Number of compositions of n such that no two adjacent parts are equal (these are sometimes called Carlitz compositions).

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