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 and 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, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 16.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Andrew M. Baxter and Lara K. Pudwell, Ascent sequences avoiding pairs of patterns, Elect. J. Comb. 22(1) (2015), #P1.58.
Christian Bean, Bjarki Gudmundsson, and Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag 63(1) (1990), 3-20.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63(1) (1990), 3-20. [Annotated scanned copy]
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Vít Jelínek, Toufik Mansour, and Mark Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50(2) (2013), 292-326. See 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.
Qi Liu, Sergey Kitaev, and Philip B. Zhang, Simultaneous avoidance of length-4 patterns in ascent sequences, arXiv:2604.06735 [math.CO], 2026. See p. 4 (Table 1).
Shuzhen Lv and Sergey Kitaev, Stoimenow matchings avoiding multiple Catalan patterns simultaneously, arXiv:2509.12726 [math.CO], 2025. See pp. 3, 10.
Toufik Mansour and Mark Shattuck, On ascent sequences avoiding 021 and a pattern of length four, arXiv:2507.17947 [math.CO], 2025. See p. 11.
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.
Lara Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015 Joint Mathematics Meetings, AMS Special Session on Enumerative Combinatorics, January 11, 2015.
Lara Pudwell and Andrew 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
Row sums of triangle A134399. - Gary W. Adamson, Oct 23 2007
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
(SageMath) [2^(n-1)*n+1 for n in (0..35)] # G. C. Greubel, May 31 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
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