A269915 -id:A269915 - cp's OEIS frontend

A081038 3rd binomial transform of (1,2,0,0,0,0,0,0,...).

1, 5, 21, 81, 297, 1053, 3645, 12393, 41553, 137781, 452709, 1476225, 4782969, 15411789, 49424013, 157837977, 502211745, 1592728677, 5036466357, 15884240049, 49977243081, 156905298045, 491636600541, 1537671920841

Offset: 0

Paul Barry, Mar 03 2003

a(n) is the number of distinguished parts in all compositions of n+1 in which some (possibly all or none) of the parts have been distinguished. a(1) = 2 because we have: 2', 1'+1, 1+1', 1'+1' where we see 5's marking the distinguished parts. With offset=1, a(n) = Sum_{k=1..n} A200139(n,k)*k. - Geoffrey Critzer, Jan 12 2013
For n>=1, a(n-1) the number of ternary strings of length 2n containing the block 11..12 with n ones where no runs of length larger than n are permitted. - Marko Riedel, Mar 08 2016
Binomial transform of {A001787(n + 1)}{n >= 0}. - _Wolfdieter Lang, Oct 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 13.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A001787, A001792, A081039, A081040, A086972, A200139.
First differences of A027471.
Cf. A269914, A269915, A269916, A269917.

[(2*n+3)*3^(n-1): n in [0..30]]; // Vincenzo Librandi, Jun 09 2011

A081038:=n->(2*n+3)*3^(n-1): seq(A081038(n), n=0..30); # Wesley Ivan Hurt, Mar 07 2016

LinearRecurrence[{6,-9},{1,5},40] (* Harvey P. Dale, Jun 22 2012 *)

G.f.: (1-x)/(1-3*x)^2.
a(n) = 6*a(n-1) - 9*a(n-2), with a(0)=1, a(1)=5.
a(n) = (2*n+3)*3^(n-1).
a(n) = Sum_{k=0..n} (k+1)*2^k*binomial(n, k).
a(n) = 2*A086972(n) - 1. - Lambert Herrgesell (zero815(AT)googlemail.com), Feb 10 2008
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=0} 1/a(n) = 9*(sqrt(3)*arctanh(1/sqrt(3)) - 1).
Sum_{n>=0} (-1)^n/a(n) = 9 - 3*sqrt(3)*Pi/2. (End)
E.g.f.: exp(3*x)*(1 + 2*x). - Stefano Spezia, Jan 31 2025

A269914 Number of ternary strings of length n with maximal run length two containing 112.

Original entry on oeis.org

1, 5, 20, 71, 237, 761, 2377, 7278, 21945, 65375, 192861, 564387, 1640496, 4741103, 13634501, 39042437, 111379025, 316687006, 897796581, 2538530615, 7160768785, 20156241155, 56626360256, 158804376883, 444638710925, 1243115597929, 3470779612521, 9678320566654

Offset: 3

Marko Riedel, Mar 07 2016

nonn

Math StackExchange, Words built with 0,1,2
Marko Riedel, Maple code by total enumeration and by generating function.
Index entries for linear recurrences with constant coefficients, signature (3,3,-5,-11,-8,-2).

Cf. A081038, A269915, A269916, A269917.

Drop[CoefficientList[Series[x^3 (x + 1) (x^2 + x + 1)/((2 x^2 + 2 x - 1) (x^4 + 3 x^3 + 3 x^2 + x - 1)), {x, 0, 30}], x], 3] (* Michael De Vlieger, Mar 08 2016 *)

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

A269916 Number of ternary strings of length n with maximal run length four containing 11112.

Original entry on oeis.org

1, 5, 21, 81, 296, 1043, 3585, 12095, 40221, 132225, 430633, 1391623, 4467689, 14262766, 45311977, 143343279, 451768405, 1419092951, 4444424613, 13882255419, 43256925753, 134492621659, 417322590000, 1292554593007, 3996626787973, 12338508959035, 38037021764053

Offset: 5

Marko Riedel, Mar 07 2016

nonn

Math StackExchange, Words built with 0,1,2
Marko Riedel, Maple code by total enumeration and by generating function.
Index entries for linear recurrences with constant coefficients, signature (3,3,-1,-9,-25,-39,-45,-43,-32,-18,-8,-2).

Cf. A081038, A269914, A269915, A269917.

Drop[CoefficientList[Series[x^5 (x + 1) (x^2 + 1) (x^4 + x^3 + x^2 + x + 1)/((x^8 + 3 x^7 + 5 x^6 + 7 x^5 + 7 x^4 + 5 x^3 + 3 x^2 + x - 1) (2 x^4 + 2 x^3 + 2 x^2 + 2 x - 1)), {x, 0, 31}], x], 5] (* Michael De Vlieger, Mar 08 2016 *)

G.f.: x^5*(x+1)*(x^2+1)*(x^4+x^3+x^2+x+1) / ((x^8+3*x^7+5*x^6+7*x^5 +7*x^4+5*x^3+3*x^2+x-1) * (2*x^4+2*x^3+2*x^2+2*x-1)).

A269917 Number of ternary strings of length n with maximal run length five containing 111112.

Original entry on oeis.org

1, 5, 21, 81, 297, 1052, 3635, 12333, 41255, 136449, 447147, 1454091, 4697983, 15094393, 48264551, 153678185, 487510286, 1541427097, 4859385039, 15278735029, 47923821239, 149992151725, 468512665975, 1460770946689, 4546890238683, 14131055304241, 43854326838403

Offset: 6

Marko Riedel, Mar 07 2016

nonn

Words built with 0,1,2
Marko Riedel, Maple code by total enumeration and by generating function.
Index entries for linear recurrences with constant coefficients, signature (3,3,-1,-9,-21,-41,-59,-69,-71,-65,-50,-32,-18,-8,-2).

Cf. A081038, A269914, A269915, A269916.

Drop[CoefficientList[Series[x^6 (x + 1) (x^2 + x + 1) (x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)/((2 x^5 + 2 x^4 + 2 x^3 + 2 x^2 + 2 x - 1) (x^10 + 3 x^9 + 5 x^8 + 7 x^7 + 9 x^6 + 9 x^5 + 7 x^4 + 5 x^3 + 3 x^2 + x - 1)), {x, 0, 32}], x], 6] (* Michael De Vlieger, Mar 08 2016 *)

G.f.: x^6*(x+1)*(x^2+x+1)*(x^2-x+1)*(x^4+x^3+x^2+x+1) / ((2*x^5+2*x^4 +2*x^3 +2*x^2+2*x-1) * (x^10+3*x^9+5*x^8+7*x^7+9*x^6+9*x^5 +7*x^4 +5*x^3 +3*x^2+x-1)).

A309000 Number of strings of length n from a 3-symbol alphabet (A,B,C, say) containing at least one "A" and at least two "B"s.

Original entry on oeis.org

3, 22, 105, 416, 1491, 5034, 16365, 51892, 161799, 498686, 1524705, 4635528, 14037627, 42391378, 127763925, 384536924, 1156232175, 3474201510, 10434138825, 31326533680, 94029932643, 282194655482, 846802070205, 2540859195396, 7623517110231, 22872497487694

Offset: 3

Adam Vellender, Jul 04 2019

This sequence can be thought of as the number of ways of rolling n 3-sided dice (with sides "A", "B", and "C") and obtaining at least one A and at least two B's.

The general formula is readily proved true by counting arguments.

			Suppose three-sided dice each have sides labeled A,B,C.
If there are three dice, then ABB, BAB, and BBA are the three strings resulting from rolling the dice satisfying the property of at least one A and at least two B's, hence a(3)=3 [Note a(0)=a(1)=a(2)=0].
If there are four such dice, there are 22 such permutations, hence a(4)=22: AABB, ABAB, ABBA, ABBB, ABBC, ABCB, ACBB, BAAB, BABA, BABB, BABC, BACB, BBAA, BBAB, BBAC, BBBA, BBCA, BCAB, BCBA, CABB, CBAB, CBBA.

Index entries for linear recurrences with constant coefficients, signature (9,-31,51,-40,12).

Cf. A269914, A269915, A269916, A269917, A186244, A186314.

[3^n-2^(n+1)-n*2^(n-1)+n+1: n in [3..40]]; // Vincenzo Librandi, Jul 05 2019

Array[3^# - 2^(# + 1) - # 2^(# - 1) + # + 1 &, 27, 3] (* or *)
CoefficientList[Series[(-3 + 5 x)/((-1 + 3 x) (1 - 3 x + 2 x^2)^2), {x, 0, 26}], x] (* Michael De Vlieger, Jul 04 2019 *)

[3**n-2**(n+1)-n*2**(n-1)+n+1 for n in range(3,20)]

a(n) = 3^n - 2^(n+1) - n*2^(n-1) + n + 1.
G.f.: x^3*(-3 + 5*x)/((-1 + 3*x)*(1 - 3*x + 2*x^2)^2). - Michael De Vlieger, Jul 04 2019.
a(n) = 9*a(n-1) - 31*a(n-2) + 51*a(n-3) - 40*a(n-4) + 12*a(n-5) for n > 7. - Stefano Spezia, Jul 05 2019

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A081038 3rd binomial transform of (1,2,0,0,0,0,0,0,...).

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A269914 Number of ternary strings of length n with maximal run length two containing 112.

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A269916 Number of ternary strings of length n with maximal run length four containing 11112.

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A269917 Number of ternary strings of length n with maximal run length five containing 111112.

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A309000 Number of strings of length n from a 3-symbol alphabet (A,B,C, say) containing at least one "A" and at least two "B"s.

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