The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A027306 a(n) = 2^(n-1) + ((1 + (-1)^n)/4)*binomial(n, n/2). 71
 1, 1, 3, 4, 11, 16, 42, 64, 163, 256, 638, 1024, 2510, 4096, 9908, 16384, 39203, 65536, 155382, 262144, 616666, 1048576, 2449868, 4194304, 9740686, 16777216, 38754732, 67108864, 154276028, 268435456, 614429672, 1073741824, 2448023843 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Inverse binomial transform of A027914. Hankel transform (see A001906 for definition) is {1, 2, 3, 4, ..., n, ...}. - Philippe Deléham, Jul 21 2005 Number of walks of length n on a line that starts at the origin and ends at or above 0. - Benjamin Phillabaum, Mar 05 2011 Number of binary integers (i.e., with a leading 1 bit) of length n+1 which have a majority of 1-bits. E.g., for n+1=4: (1011, 1101, 1110, 1111) a(3)=4. - Toby Gottfried, Dec 11 2011 Number of distinct symmetric staircase walks connecting opposite corners of a square grid of side n > 1. - Christian Barrientos, Nov 25 2018 From Gus Wiseman, Aug 20 2021: (Start) Also the number of integer compositions of n + 1 with alternating sum > 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A345917. For example, the a(0) = 1 through a(4) = 11 compositions are: (1) (2) (3) (4) (5) (21) (31) (32) (111) (112) (41) (211) (113) (122) (212) (221) (311) (1121) (2111) (11111) The following relate to these compositions: - The unordered version is A027193. - The complement is counted by A058622. - The reverse unordered version is A086543. - The version for alternating sum >= 0 is A116406. - The version for alternating sum < 0 is A294175. - Ranked by A345917. (End) The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022 REFERENCES A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.1.6) LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 F. Disanto, A. Frosini, and S. Rinaldi, Square involutions, J. Int. Seq. 14 (2011) # 11.3.5. Zachary Hamaker and Eric Marberg, Atoms for signed permutations, arXiv:1802.09805 [math.CO], 2018. Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3. Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. FORMULA a(n) = Sum_{k=0..floor(n/2)} binomial(n,k). Odd terms are 2^(n-1). Also a(2n) - 2^(2n-1) is given by A001700. a(n) = 2^n + (n mod 2)*binomial(n, (n-1)/2). E.g.f.: (exp(2x) + I_0(2x))/2. O.g.f.: 2*x/(1-2*x)/(1+2*x-((1+2*x)*(1-2*x))^(1/2)). - Vladeta Jovovic, Apr 27 2003 a(n) = A008949(n, floor(n/2)); a(n) + a(n-1) = A248574(n), n > 0. - Reinhard Zumkeller, Nov 14 2014 From Peter Bala, Jul 21 2015: (Start) a(n) = [x^n]( 2*x - 1/(1 - x) )^n. O.g.f.: (1/2)*( 1/sqrt(1 - 4*x^2) + 1/(1 - 2*x) ). Inverse binomial transform is (-1)^n*A246437(n). exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + ... is the o.g.f. for A001405. (End) a(n) = Sum_{k=1..floor((n+1)/2)} binomial(n-1,(2n+1-(-1)^n)/4 -k). - Anthony Browne, Jun 18 2016 D-finite with recurrence: n*a(n) + 2*(-n+1)*a(n-1) + 4*(-n+1)*a(n-2) + 8*(n-2)*a(n-3) = 0. - R. J. Mathar, Aug 09 2017 EXAMPLE From Gus Wiseman, Aug 20 2021: (Start) The a(0) = 1 through a(4) = 11 binary numbers with a majority of 1-bits (Gottfried's comment) are: 1 11 101 1011 10011 110 1101 10101 111 1110 10110 1111 10111 11001 11010 11011 11100 11101 11110 11111 The version allowing an initial zero is A058622. (End) MAPLE a:= proc(n) add(binomial(n, j), j=0..n/2) end: seq(a(n), n=0..32); # Zerinvary Lajos, Mar 29 2009 MATHEMATICA Table[Sum[Binomial[n, k], {k, 0, Floor[n/2]}], {n, 1, 35}] (* Second program: *) a = a = 1; a = 3; a[n_] := a[n] = (2(n-1)(2a[n-2] + a[n-1]) - 8(n-2) a[n-3])/n; Array[a, 33, 0] (* Jean-François Alcover, Sep 04 2016 *) PROG (PARI) a(n)=if(n<0, 0, (2^n+if(n%2, 0, binomial(n, n/2)))/2) (Haskell) a027306 n = a008949 n (n `div` 2) -- Reinhard Zumkeller, Nov 14 2014 (Magma) [2^(n-1)+(1+(-1)^n)/4*Binomial(n, n div 2): n in [0..40]]; // Vincenzo Librandi, Jun 19 2016 (GAP) List([0..35], n->Sum([0..Int(n/2)], k->Binomial(n, k))); # Muniru A Asiru, Nov 27 2018 CROSSREFS a(n) = Sum{(k+1)T(n, m-k)}, 0<=k<=[ (n+1)/2 ], T given by A008315. Column k=2 of A226873. - Alois P. Heinz, Jun 21 2013 Cf. A008949, A248574, A001405, A246437. The even bisection is A000302. The odd bisection appears to be A032443. Cf. A000984, A001700, A001791, A008549, A011782, A088218, A097805, A163493, A182616, A345197. Sequence in context: A001641 A007382 A127804 * A239024 A026676 A142870 Adjacent sequences: A027303 A027304 A027305 * A027307 A027308 A027309 KEYWORD nonn,easy,walk AUTHOR Clark Kimberling EXTENSIONS Better description from Robert G. Wilson v, Aug 30 2000 and from Yong Kong (ykong(AT)curagen.com), Dec 28 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 27 20:41 EDT 2023. Contains 365714 sequences. (Running on oeis4.)