|
| |
|
|
A005183
|
|
a(n) = n*2^(n-1) + 1.
(Formerly M1434)
|
|
28
|
|
|
|
1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689, 245761, 524289, 1114113, 2359297, 4980737, 10485761, 22020097, 46137345, 96468993, 201326593, 419430401, 872415233, 1811939329, 3758096385, 7784628225, 16106127361, 33285996545
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
a(n-1) is the number of permutations of length n which avoid the patterns 132, 4312. - Lara Pudwell, Jan 21 2006
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) <= e(j) >= e(k) and e(i) != e(k). [Martinez and Savage, 2.11] - Eric M. Schmidt, Jul 17 2017
Indices of records in A066099. Also, indices of "cusps" in the graph of A030303 giving positions of 1's in the binary Champernowne word A030190. - M. F. Hasler, Oct 12 2020
|
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Stephan Baier, Pallab Kanti Dey, Prime powers dividing products of consecutive integer values of x^2^n + 1, arXiv:1905.13003 [math.NT], 2019. See p. 7.
Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, The Electronic Journal of Combinatorics, Volume 22, Issue 1 (2015) Paper #P1.58.
Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
V. Jelinek, T. Mansour, M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, Example 4.16, H_{1223} and Example 4.17 L_{1232} and propositions 4.20 and 4.22, all shifted with an additional leading a(0)=1.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
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
Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005.
L. Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015 Joint Mathematics Meetings, AMS Special Session on Enumerative Combinatorics, January 11, 2015.
L. Pudwell, A. Baxter, Ascent sequences avoiding pairs of patterns, Permutation Patterns 2014, East Tennessee State University, July 7, 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).
|
|
|
FORMULA
|
Main diagonal of the array defined by T(0, j)=j+1 j>=0, T(i, 0)=i+1 i>=0, T(i, j)=T(i-1, j-1)+T(i-1, j)-1. - Benoit Cloitre, Jun 17 2003
G.f.: (1 -3*x +3*x^2)/((1-x)*(1-2*x)^2). - Lara Pudwell, Jan 21 2006
E.g.f.: exp(x) +x*exp(2*x). - Joerg Arndt, May 22 2013
Binomial transform of A028310. a(n) = 1 + Sum{k=0..n} C(n, k)*k = 1 + A001787(n). - Paul Barry, Jul 21 2003
a(n) = Sum_{k=0..2^n} A000120(k) = A000788(2^n). - Benoit Cloitre, Sep 25 2003
Row sums of triangle A134399. - Gary W. Adamson, Oct 23 2007
a(n) = A000788(A000079(n)). - Reinhard Zumkeller, Mar 04 2010
a(n) = 2*a(n-1) +2^(n-1) -1 (with a(0)=1). - Vincenzo Librandi, Dec 31 2010
|
|
|
MAPLE
|
A005183 := (1-3*z+3*z**2)/(1-z)/(1-2*z)**2; # Generating function conjectured by Simon Plouffe in his 1992 dissertation.
|
|
|
MATHEMATICA
|
Table[(n+1)*2^n+1, {n, 1, 30}] (* Alexander Adamchuk, Sep 09 2006 *)
LinearRecurrence[{5, -8, 4}, {1, 2, 5}, 30] (* Harvey P. Dale, Jul 29 2015 *)
|
|
|
PROG
|
(PARI) a(n)=n*2^(n-1)+1 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [n*2^(n-1)+1: n in [0..35]]; // Vincenzo Librandi, May 14 2017
(Sage) [2^(n-1)*n+1 for n in (0..35)] # G. C. Greubel, May 31 2019
|
|
|
CROSSREFS
|
Cf. A134399.
Sequence in context: A027929 A001659 A088921 * A005348 A210496 A067676
Adjacent sequences: A005180 A005181 A005182 * A005184 A005185 A005186
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane, R. K. Guy
|
|
|
EXTENSIONS
|
More terms from Lara Pudwell, Jan 21 2006
Edited by N. J. A. Sloane at the suggestion of Jim Propp, Jul 14 2007
|
|
|
STATUS
|
approved
|
| |
|
|
