These are Hogben's central polygonal numbers with the (two-dimensional) symbol
2
.P
1 n
The first line cuts the pancake into 2 pieces. For n > 1, the n-th line crosses every earlier line (avoids parallelism) and also avoids every previous line intersection, thus increasing the number of pieces by n. For 16 lines, for example, the number of pieces is 2 + 2 + 3 + 4 + 5 + ... + 16 = 137. These are the triangular numbers plus 1 (cf.
A000217).
m = (n-1)(n-2)/2 + 1 is also the smallest number of edges such that all graphs with n nodes and m edges are connected. -
Keith Briggs, May 14 2004
Also maximal number of grandchildren of a binary vector of length n+2. E.g., a binary vector of length 6 can produce at most 11 different vectors when 2 bits are deleted.
This is also the order dimension of the (strong) Bruhat order on the finite Coxeter group B_{n+1}. - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
Number of 132- and 321-avoiding permutations of {1,2,...,n+1}. -
Emeric Deutsch, Mar 14 2002
For n >= 1 a(n) is the number of terms in the expansion of (x+y)*(x^2+y^2)*(x^3+y^3)*...*(x^n+y^n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003
Also the number of terms in (1)(x+1)(x^2+x+1)...(x^n+...+x+1); see
A000140.
Narayana transform (analog of the binomial transform) of vector [1, 1, 0, 0, 0, ...] =
A000124; using the infinite lower Narayana triangle of
A001263 (as a matrix), N; then N * [1, 1, 0, 0, 0, ...] =
A000124. -
Gary W. Adamson, Apr 28 2005
Number of interval subsets of {1, 2, 3, ..., n} (cf.
A002662). - Jose Luis Arregui (arregui(AT)unizar.es), Jun 27 2006
Define a number of straight lines in the plane to be in general arrangement when (1) no two lines are parallel, (2) there is no point common to three lines. Then these are the maximal numbers of regions defined by n straight lines in general arrangement in the plane. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
Note that a(n) = a(n-1) +
A000027(n-1). This has the following geometrical interpretation: Suppose there are already n-1 lines in general arrangement, thus defining the maximal number of regions in the plane obtainable by n-1 lines and now one more line is added in general arrangement. Then it will cut each of the n-1 lines and acquire intersection points which are in general arrangement. (See the comments on
A000027 for general arrangement with points.) These points on the new line define the maximal number of regions in 1-space definable by n-1 points, hence this is
A000027(n-1), where for
A000027 an offset of 0 is assumed, that is,
A000027(n-1) = (n+1)-1 = n. Each of these regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n) = a(n-1) +
A000027(n-1). Cf. the comments on
A000125 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
When constructing a zonohedron, one zone at a time, out of (up to) 3-d non-intersecting parallelepipeds, the n-th element of this sequence is the number of edges in the n-th zone added with the n-th "layer" of parallelepipeds. (Verified up to 10-zone zonohedron, the enneacontahedron.) E.g., adding the 10th zone to the enneacontahedron requires 46 parallel edges (edges in the 10th zone) by looking directly at a 5-valence vertex and counting visible vertices. -
Shel Kaphan, Feb 16 2006
Binomial transform of (1, 1, 1, 0, 0, 0, ...) and inverse binomial transform of
A072863: (1, 3, 9, 26, 72, 192, ...). -
Gary W. Adamson, Oct 15 2007
If Y is a 2-subset of an n-set X then, for n >= 3, a(n-3) is the number of (n-2)-subsets of X which do not have exactly one element in common with Y. -
Milan Janjic, Dec 28 2007
It appears that a(n) is the number of distinct values among the fractions F(i+1)/F(j+1) as j ranges from 1 to n and, for each fixed j, i ranges from 1 to j, where F(i) denotes the i-th Fibonacci number. -
John W. Layman, Dec 02 2008
a(n) is the number of subsets of {1,2,...,n} that contain at most two elements. -
Geoffrey Critzer, Mar 10 2009
For n >= 2, a(n) gives the number of sets of subsets A_1, A_2, ..., A_n of n = {1, 2, ..., n} such that Meet_{i = 1..n} A_i is empty and Sum_{j in [n]} (|Meet{i = 1..n, i != j} A_i|) is a maximum. -
Srikanth K S, Oct 22 2009
The numbers along the left edge of Floyd's triangle. -
Paul Muljadi, Jan 25 2010
Let A be the Hessenberg matrix of order n, defined by: A[1,j] = A[i,i]:=1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n-1) = (-1)^(n-1)*coeff(charpoly(A,x),x). -
Milan Janjic, Jan 24 2010
Also the number of deck entries of Euler's ship. See the Meijer-Nepveu link. -
Johannes W. Meijer, Jun 21 2010
(1 + x^2 + x^3 + x^4 + x^5 + ...)*(1 + 2x + 3x^2 + 4x^3 + 5x^4 + ...) = (1 + 2x + 4x^2 + 7x^3 + 11x^4 + ...). -
Gary W. Adamson, Jul 27 2010
The number of length n binary words that have no 0-digits between any pair of consecutive 1-digits. -
Jeffrey Liese, Dec 23 2010
Let b(0) = b(1) = 1; b(n) = max(b(n-1)+n-1, b(n-2)+n-2) then a(n) = b(n+1). -
Yalcin Aktar, Jul 28 2011
Also number of distinct sums of 1 through n where each of those can be + or -. E.g., {1+2,1-2,-1+2,-1-2} = {3,-1,1,-3} and a(2) = 4. -
Toby Gottfried, Nov 17 2011
This sequence is complete because the sum of the first n terms is always greater than or equal to a(n+1)-1. Consequently, any nonnegative number can be written as a sum of distinct terms of this sequence. See
A204009,
A072638. -
Frank M Jackson, Jan 09 2012
The sequence is the number of distinct sums of subsets of the nonnegative integers, and its first differences are the positive integers. See
A208531 for similar results for the squares. -
John W. Layman, Feb 28 2012
Apparently the number of Dyck paths of semilength n+1 in which the sum of the first and second ascents add to n+1. -
David Scambler, Apr 22 2013
Without 1 and 2, a(n) equals the terminus of the n-th partial sum of sequence 1, 1, 2. Explanation: 1st partial sums of 1, 1, 2 are 1, 2, 4; 2nd partial sums are 1, 3, 7; 3rd partial sums are 1, 4, 11; 4th partial sums are 1, 5, 16, etc. -
Bob Selcoe, Jul 04 2013
Equivalently, numbers of the form 2*m^2+m+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . -
Bruno Berselli, Apr 08 2014
For n >= 2: quasi-triangular numbers; the almost-triangular numbers being
A000096(n), n >= 2. Note that 2 is simultaneously almost-triangular and quasi-triangular. -
Daniel Forgues, Apr 21 2015
n points in general position determine "n choose 2" lines, so
A055503(n) <= a(n(n-1)/2). If n > 3, the lines are not in general position and so
A055503(n) < a(n(n-1)/2). -
Jonathan Sondow, Dec 01 2015
The digital root is period 9 (1, 2, 4, 7, 2, 7, 4, 2, 1), also the digital roots of centered 10-gonal numbers (
A062786), for n > 0,
A133292. -
Peter M. Chema, Sep 15 2016
For n >= 0, a(n) is the number of weakly unimodal sequences of length n over the alphabet {1, 2}. -
Armend Shabani, Mar 10 2017
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). [Martinez and Savage, 2.4]
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) and e(i) < e(k). [Martinez and Savage, 2.4]
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). [Martinez and Savage, 2.4]
(End)
The odd prime factors != 7 occur in an interval of p successive terms either never or exactly twice, while 7 always occurs only once. If a prime factor p appears in a(n) and a(m) within such an interval, then n + m == -1 (mod p). When 7 divides a(n), then 2*n == -1 (mod 7). a(n) is never divisible by the prime numbers given in
A003625.
While all prime factors p != 7 can occur to any power, a(n) is never divisible by 7^2. The prime factors are given in
A045373. The prime terms of this sequence are given in
A055469.
(End)
a(n-1) is the greatest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
/\ The top arch has a length of 3. /\ The top arch has a length of 3.
/ \ Both bottom arches have a //\\ The middle arch has a length of 2.
//\/\\ length of 1. ///\\\ The bottom arch has a length of 1.
Example: for n = 4, a(4-1) = a(3) = 7 /\
//\\
/\ ///\\\ 1 + 3 + 2 + 1 = 7. (End)
a(n+1) is the a(n)-th smallest positive integer that has not yet appeared in the sequence. -
Matthew Malone, Aug 26 2021
For n> 0, let the n-dimensional cube {0,1}^n be, provided with the Hamming distance, d. Given an element x in {0,1}^n, a(n) is the number of elements y in {0,1}^n such that d(x, y) <= 2. Example: n = 4. (0,0,0,0), (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0) are at distance <= 2 from (0,0,0,0), so a(4) = 11. -
Yosu Yurramendi, Dec 10 2021
a(n) is the sum of the first three entries of row n of Pascal's triangle. -
Daniel T. Martin, Apr 13 2022
a(n-1) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 3 with exactly one descent. For example, sigma is one of the patterns, {132, 213, 231, 312}. -
Jessica A. Tomasko, Sep 14 2022
a(n+4) is the number of ways to tile an equilateral triangle of side length 2*n with smaller equilateral triangles of side length n and side length 1. For example, with n=2, there are 22 ways to tile an equilateral triangle of side length 4 with smaller ones of sides 2 and 1, including the one tiling with sixteen triangles of sides 1 and the one tiling with four triangles of sides 2. -
Ahmed ElKhatib and
Greg Dresden, Aug 19 2024
