A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.
1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761
Offset: 0
Keywords
A109613 Odd numbers repeated.
1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73
Offset: 0
Comments
The number of rounds in a round-robin tournament with n competitors. - A. Timothy Royappa, Aug 13 2011
Diagonal sums of number triangle A113126. - Paul Barry, Oct 14 2005
When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - Tanya Khovanova, Apr 06 2008
The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - Johannes W. Meijer, Nov 12 2009
From Reinhard Zumkeller, Dec 05 2009: (Start)
A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013
The arithmetic function v+-(n,2) as defined in A290988. - Robert Price, Aug 22 2017
For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - Eric W. Weisstein, Nov 17 2017
a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - Wolfdieter Lang, Feb 19 2020
Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - Peter Bala, Aug 30 2022
The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - R. J. Mathar, Feb 25 2023
Examples
G.f. = 1 + x + 3*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 9*x^9 + ...
Complete coach system for (a composite) b = 2*n + 1 = 33: Sigma(33) ={[1; 5], [5, 7, 13; 2, 1, 2]} (the first two rows are here 1 and 5, 7, 13), a_{up}(33) = a(15) = 15. But 15 is not in the reduced residue system modulo 33, so the maximal (odd) a number is 13. For the prime b = 31, a_{up}(31) = a(14) = 15 appears as maximum of the first rows. - _Wolfdieter Lang_, Feb 19 2020
References
- Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, 3rd printing 2012, pp. (260-281).
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
- Eric Weisstein's World of Mathematics, Chromatic Number
- Eric Weisstein's World of Mathematics, Johnson Graph
- Eric Weisstein's World of Mathematics, Triangular Graph
- Wikipedia, Round-robin tournament
- Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Crossrefs
Programs
-
Haskell
a109613 = (+ 1) . (* 2) . (`div` 2) a109613_list = 1 : 1 : map (+ 2) a109613_list -- Reinhard Zumkeller, Oct 27 2012, Feb 21 2011
-
Maple
A109613:=n->2*floor(n/2)+1; seq(A109613(k), k=0..100); # Wesley Ivan Hurt, Oct 22 2013
-
Mathematica
Flatten@ Array[{2# - 1, 2# - 1} &, 37] (* Robert G. Wilson v, Jul 07 2012 *) (# - Boole[EvenQ[#]] &) /@ Range[80] (* Alonso del Arte, Sep 11 2019 *) With[{c=2*Range[0,40]+1},Riffle[c,c]] (* Harvey P. Dale, Jan 02 2020 *) -
PARI
A109613(n)=n>>1<<1+1 \\ Charles R Greathouse IV, Feb 24 2011
-
Sage
def a(n) : return( len( CuspForms( Gamma0( 5), 2*n + 4, prec=1). basis())); # Michael Somos, May 29 2013
-
Scala
((1 to 49) by 2) flatMap { List.fill(2)() } // _Alonso del Arte, Sep 11 2019
Formula
a(n) = 2*floor(n/2) + 1.
G.f.: (1 + x + x^2 + x^3)/(1 - x^2)^2. - Paul Barry, Oct 14 2005
a(n) = 2*a(n-2) - a(n-4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 3. - Philippe Deléham, Nov 03 2008
a(n) = 2*n - a(n-1), with a(0) = 1. - Vincenzo Librandi, Nov 13 2010
a(n) = R(n, -2), where R(n, x) is the n-th row polynomial of A211955. a(n) = (-1)^n + 2*Sum_{k = 1..n} (-1)^(n - k - 2)*4^(k-1)*binomial(n+k, 2*k). Cf. A084159. - Peter Bala, May 01 2012
a(n) = A182579(n+1, n). - Reinhard Zumkeller, May 06 2012
G.f.: ( 1 + x^2 ) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Jul 12 2016
E.g.f.: x*exp(x) + cosh(x). - Ilya Gutkovskiy, Jul 12 2016
From Guenther Schrack, Sep 10 2018: (Start)
a(-n) = -a(n-1).
a(n) = A047270(n+1) - (2*n + 2).
A047838 a(n) = floor(n^2/2) - 1.
1, 3, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647, 683, 721, 759, 799, 839, 881, 923, 967, 1011, 1057, 1103, 1151, 1199, 1249, 1299, 1351, 1403
Offset: 2
Comments
Define the organization number of a permutation pi_1, pi_2, ..., pi_n to be the following. Start at 1, count the steps to reach 2, then the steps to reach 3, etc. Add them up. Then the maximal value of the organization number of any permutation of [1..n] for n = 0, 1, 2, 3, ... is given by 0, 1, 3, 7, 11, 17, 23, ... (this sequence). This was established by Graham Cormode (graham(AT)research.att.com), Aug 17 2006, see link below, answering a question raised by Tom Young (mcgreg265(AT)msn.com) and Barry Cipra, Aug 15 2006
From Dmitry Kamenetsky, Nov 29 2006: (Start)
This is the length of the longest non-self-intersecting spiral drawn on an n X n grid. E.g., for n=5 the spiral has length 17:
1 0 1 1 1
1 0 1 0 1
1 0 1 0 1
1 0 0 0 1
1 1 1 1 1 (End)
It appears that a(n+1) is the maximum number of consecutive integers (beginning with 1) that can be placed, one after another, on an n-peg Towers of Hanoi, such that the sum of any two consecutive integers on any peg is a square. See the problem: http://online-judge.uva.es/p/v102/10276.html. - Ashutosh Mehra, Dec 06 2008
a(n) = number of (w,x,y) with all terms in {0,...,n} and w = |x+y-w|. - Clark Kimberling, Jun 11 2012
The same sequence also represents the solution to the "pigeons problem": maximal value of the sum of the lengths of n-1 line segments (connected at their end-points) required to pass through n trail dots, with unit distance between adjacent points, visiting all of them without overlaping two or more segments. In this case, a(0)=0, a(1)=1, a(2)=3, and so on. - Marco Ripà, Jan 28 2014
Also the longest path length in the n X n white bishop graph. - Eric W. Weisstein, Mar 27 2018
a(n) is the number of right triangles with sides n*(h-floor(h)), floor(h) and h, where h is the hypotenuse. - Andrzej Kukla, Apr 14 2021
Examples
x^2 + 3*x^3 + 7*x^4 + 11*x^5 + 17*x^6 + 23*x^7 + 31*x^8 + 39*x^9 + 49*x^10 + ...
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
- Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
- Graham Cormode, Notes on the organization number of a permutation.
- Eric Weisstein's World of Mathematics, Longest Path Problem.
- Eric Weisstein's World of Mathematics, White Bishop Graph.
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Crossrefs
Programs
-
Magma
[Floor(n^2/2)-1 : n in [2..100]]; // Wesley Ivan Hurt, Aug 06 2015
-
Maple
seq(floor((n^2+4*n+2)/2), n=0..20) # Gary Detlefs, Feb 10 2010
-
Mathematica
Table[Floor[n^2/2] - 1, {n, 2, 60}] (* Robert G. Wilson v, Aug 31 2006 *) LinearRecurrence[{2, 0, -2, 1}, {1, 3, 7, 11}, 60] (* Harvey P. Dale, Jan 16 2015 *) Floor[Range[2, 20]^2/2] - 1 (* Eric W. Weisstein, Mar 27 2018 *) Table[((-1)^n + 2 n^2 - 5)/4, {n, 2, 20}] (* Eric W. Weisstein, Mar 27 2018 *) CoefficientList[Series[(-1 - x - x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *) -
PARI
a(n) = n^2\2 - 1
Formula
a(2)=1; for n > 2, a(n) = a(n-1) + n - 1 + (n-1 mod 2). - Benoit Cloitre, Jan 12 2003
a(n) = T(n-1) + floor(n/2) - 1 = T(n) - floor((n+3)/2), where T(n) is the n-th triangular number (A000217). - Robert G. Wilson v, Aug 31 2006
Equals (n-1)-th row sums of triangles A134151 and A135152. Also, = binomial transform of [1, 2, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Nov 21 2007
G.f.: x^2*(1+x+x^2-x^3)/((1-x)^3*(1+x)). - R. J. Mathar, Sep 09 2008
a(n) = floor((n^2 + 4*n + 2)/2). - Gary Detlefs, Feb 10 2010
a(n) = abs(A188653(n)). - Reinhard Zumkeller, Apr 13 2011
a(n) = (2*n^2 + (-1)^n - 5)/4. - Bruno Berselli, Sep 14 2011
a(n) = a(-n) = A007590(n) - 1.
a(n) = A080827(n) - 2. - Kevin Ryde, Aug 24 2013
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4. - Wesley Ivan Hurt, Aug 06 2015
From Guenther Schrack, May 12 2018: (Start)
Set a(0) = a(1) = -1, a(n) = a(n-2) + 2*n - 2 for n > 1.
a(n) = A000982(n-1) + n - 2 for n > 1.
a(n) = 2*A033683(n) - 3 for n > 1.
a(n) = A061925(n-1) + n - 3 for n > 1.
a(n) = A074148(n) - n - 1 for n > 1.
a(n) = A105343(n-1) + n - 4 for n > 1.
a(n) = A116940(n-1) - n for n > 1.
a(n) = A179207(n) - n + 1 for n > 1.
a(n) = A183575(n-2) + 1 for n > 2.
a(n) = A265284(n-1) - 2*n + 1 for n > 1.
a(n) = 2*A290743(n) - 5 for n > 1. (End)
E.g.f.: 1 + x + ((x^2 + x - 2)*cosh(x) + (x^2 + x - 3)*sinh(x))/2. - Stefano Spezia, May 06 2021
Sum_{n>=2} 1/a(n) = 3/2 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)) - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 15 2022
Extensions
Edited by Charles R Greathouse IV, Apr 23 2010
A061925 a(n) = ceiling(n^2/2) + 1.
1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406
Offset: 0
Comments
a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004
First term in each group in A074148. - Amarnath Murthy, Aug 28 2002
From Christian Barrientos, Jan 01 2021: (Start)
For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.
For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
(End)
Except for a(2), a(n) agrees with the lower matching number of the (n+1) X (n+1) bishop graph up to at least n = 13. - Eric W. Weisstein, Dec 23 2024
Links
- Harry J. Smith, Table of n, a(n) for n = 0..1000
- Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
- Eric Weisstein's World of Mathematics, Bishop Graph.
- Eric Weisstein's World of Mathematics, Lower Matching Number.
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Programs
-
Maple
seq(floor((n^2+3)/2),n=0..25); # Gary Detlefs, Feb 13 2010
-
Mathematica
Table[Ceiling[n^2/2]+1,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *) LinearRecurrence[{2,0,-2,1},{1,2,3,6},60] (* Harvey P. Dale, Jan 03 2024 *) -
PARI
a(n) = { ceil(n^2/2) + 1 } \\ Harry J. Smith, Jul 29 2009
Formula
a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010
From R. J. Mathar, Feb 19 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011
a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021
Extensions
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007
A099392 a(n) = floor((n^2 - 2*n + 3)/2).
1, 1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405
Offset: 1
Links
- Guenther Schrack, Table of n, a(n) for n = 1..10010
- Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Crossrefs
From Guenther Schrack, Apr 17 2018: (Start)
First differences: A052928.
Partial sums: A212964(n) + n for n > 0.
Programs
-
Mathematica
Array[Floor[(#^2 - 2 # + 3)/2] &, 54] (* or *) Rest@ CoefficientList[Series[x (-1 + x - x^2 - x^3)/((1 + x) (x - 1)^3), {x, 0, 54}], x] (* Michael De Vlieger, Apr 21 2018 *) -
PARI
a(n)=(n^2+3)\2-n \\ Charles R Greathouse IV, Aug 01 2013
Formula
a(n) = ceiling(n^2/2)-n+1. - Paul Barry, Jul 16 2006; index shifted by R. J. Mathar, Jul 29 2007
a(n) = ceiling(A002522(n-1)/2). - Branko Curgus, Sep 02 2007
From R. J. Mathar, Feb 20 2011: (Start)
G.f.: x *( -1+x-x^2-x^3 ) / ( (1+x)*(x-1)^3 ).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n+1) = (3 + 2*n^2 + (-1)^n)/4. (End)
a(n) = A007590(n-1) + 1 for n >= 2. - Richard R. Forberg, Aug 01 2013
From Guenther Schrack, Apr 17 2018: (Start)
a(n) = (2*n^2 - 4*n + 5 -(-1)^n)/4.
a(n+2) = a(n) + 2*n for n > 0.
a(n) = 2*A033683(n-1) - 1 for n > 0.
a(n) = A047838(n-1) + 2 for n > 2.
a(n) = A074148(n-1) - n + 2 for n > 1.
a(n) = A183575(n-3) + 3 for n > 3.
a(n) = 2*A290743(n-1) - 3 for n > 0.
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) + 1/2. - Amiram Eldar, Sep 16 2022
E.g.f.: ((2 - x + x^2)*cosh(x) + (3 - x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 28 2024
A074147 (2n-1) odd numbers followed by 2n even numbers.
1, 2, 4, 3, 5, 7, 6, 8, 10, 12, 9, 11, 13, 15, 17, 14, 16, 18, 20, 22, 24, 19, 21, 23, 25, 27, 29, 31, 26, 28, 30, 32, 34, 36, 38, 40, 33, 35, 37, 39, 41, 43, 45, 47, 49, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 62, 64, 66, 68, 70, 72
Offset: 1
Examples
As a triangular array: 1, 2, 4, 3, 5, 7, 6, 8, 10, 12, 9, 11, 13, 15, 17, 14, 16, 18, 20, 22, 24, ...
Links
- Ivan Neretin, Table of n, a(n) for n = 1..5050
Programs
-
Maple
A074147 := proc(n,k) ceil((n-1)^2/2)+1+2*k ; end proc: seq(seq( A074147(n,k),k=0..n-1) ,n=1..12) ; # R. J. Mathar, Nov 02 2023
-
Mathematica
Flatten@Table[Ceiling[(n - 1)^2/2] + 2 k - 1, {n, 12}, {k, n}] (* Ivan Neretin, Dec 15 2016 *)
Formula
T(n,k) = A061925(n-1)+2k, 0<=kR. J. Mathar, Jul 17 2007, corrected Nov 02 2023
a(n) = A116941(n-1)+1. - Robert G. Wilson v, Mar 09 2017
A188236 T(n,k)=Number of nondecreasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero and not more than two numbers equal.
1, 1, 2, 1, 3, 4, 1, 4, 7, 15, 1, 5, 12, 30, 58, 1, 6, 17, 52, 119, 245, 1, 7, 24, 81, 221, 527, 1082, 1, 8, 31, 121, 374, 1019, 2395, 5020, 1, 9, 40, 172, 598, 1818, 4818, 11376, 24040, 1, 10, 49, 234, 903, 3047, 8964, 23522, 55368, 118154, 1, 11, 60, 311, 1317, 4859
Offset: 1
Comments
Table starts
......1......1......1.......1.......1.......1.......1........1........1
......2......3......4.......5.......6.......7.......8........9.......10
......4......7.....12......17......24......31......40.......49.......60
.....15.....30.....52......81.....121.....172.....234......311......403
.....58....119....221.....374.....598.....903....1317.....1852.....2540
....245....527...1019....1818....3047....4859....7435....10994....15791
...1082...2395...4818....8964...15696...26123...41748....64370....96346
...5020..11376..23522...45225...81981..141519..234413...374820...581280
..24040..55368.117209..231596..432491..769915.1316060..2171675..3475284
.118154.275735.594789.1202495.2302608.4209720.7395049.12546170.20642874
Examples
Some solutions for n=6 k=4 .-7...-8...-5...-8...-5...-7...-7...-3...-8...-8...-6...-6...-5...-7...-6...-5 .-4...-8...-3...-5...-5...-7...-4...-3...-4...-8...-6...-4...-4...-4...-2...-5 ..0...-4...-2...-4...-1...-1....1....1....0....0....2...-4....1....0...-1....1 ..2....6....0....3....1....0....2....1....3....2....2....2....1....1...-1....1 ..3....7....4....7....3....7....2....2....4....7....3....4....3....3....3....4 ..6....7....6....7....7....8....6....2....5....7....5....8....4....7....7....4
Links
- R. H. Hardin, Table of n, a(n) for n = 1..521
Crossrefs
Row 3 is A074148(n+1)
A229093 The clubs patterns appearing in n X n coins.
0, 0, 1, 2, 4, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867, 902, 937, 972, 1009, 1046
Offset: 0
Comments
On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Kival Ngaokrajang, Illustration for initial terms
- Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Programs
-
Mathematica
CoefficientList[Series[(x^7 - 2 x^6 + x^5 - x^4 + x^3 - x^2 - 1)/((x - 1)^3 (x^2 + x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 08 2013 *) LinearRecurrence[{2,-1,1,-2,1},{0,0,1,2,4,6,9,12,17,22},70] (* Harvey P. Dale, Feb 05 2020 *) -
PARI
Vec(x^2*(x^7-2*x^6+x^5-x^4+x^3-x^2-1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 08 2013
-
PARI
a(n) = ceil((n-1)^2/3) \\ Charles R Greathouse IV, Jan 06 2016
Formula
a(n) = ceiling((n-1)^2/3), a(0) = 0, a(4) = 4.
G.f.: x^2*(x^7-2*x^6+x^5-x^4+x^3-x^2-1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Oct 07 2013
Extensions
More terms from Colin Barker, Oct 08 2013
A105636 Transform of n^3 by the Riordan array (1/(1-x^2), x).
0, 1, 8, 28, 72, 153, 288, 496, 800, 1225, 1800, 2556, 3528, 4753, 6272, 8128, 10368, 13041, 16200, 19900, 24200, 29161, 34848, 41328, 48672, 56953, 66248, 76636, 88200, 101025, 115200, 130816, 147968, 166753, 187272, 209628, 233928, 260281, 288800, 319600
Offset: 0
Comments
Recurrence a(n) = a(n-2) + n^3, starting with a(0)=0, a(1)=1. Also, in physics, a(n)/4 is the trace of the spin operator |S_z|^3 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and therefore the sum of the absolute values of their 3rd powers is 2*28/8 = a(3)/4. - Stanislav Sykora, Nov 07 2013
Also the number of 3-cycles in the (n+1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
With zero prepended and offset 1, the sequence starts 0,0,1,8,28,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q). Then b(n) is the total volume of the family of cubes with side length |q - p|. - Wesley Ivan Hurt, Apr 14 2018
Links
- Stanislav Sykora, Table of n, a(n) for n = 0..9999
- Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
- Eric Weisstein's World of Mathematics, Graph Cycle.
- Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).
Crossrefs
Programs
-
GAP
List([0..30], n -> (2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16); # G. C. Greubel, Dec 16 2018
-
Magma
[(2*n^4+8*n^3+8*n^2-1)/16+(-1)^n/16: n in [0..50]]; // Vincenzo Librandi, Oct 27 2014
-
Mathematica
LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 1, 8, 28, 72, 153}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *) CoefficientList[Series[x (1 + 4 x + x^2)/((1 + x) (1 - x)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 26 2012 *) Table[((-1)^n + 2 n^2 (n + 2)^2 - 1)/16, {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *) -
PARI
my(x='x+O('x^99)); concat(0, Vec(x*(1+4*x+x^2)/((1+x)*(1-x)^5))) \\ Altug Alkan, Apr 16 2018 -
Sage
[(2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16 for n in range(30)] # G. C. Greubel, Dec 16 2018
Formula
G.f.: x*(1+4*x+x^2)/((1+x)*(1-x)^5).
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
a(n) = (2*n^4 + 8*n^3 + 8*n^2 - 1 + (-1)^n)/16.
a(n) = Sum_{k=0..floor((n-1)/2)} (n-2*k)^3.
a(n+1) = Sum_{k=0..n} k^3*(1 - (-1)^(n+k-1))/2.
a(n) = ((((x^2 - (x mod 2) - 4)/4)^2 - (((x^2 - (x mod 2) - 4)/4) mod 2))/8) = floor(((floor(x^2/4) - 1)^2)/8) where x = 2*n + 2. Replace x with 2*n - 1 to obtain A050534(n) = 3*A000332(n+1). Note that a(2*n) = A060300(n)/2 and a(2*n + 1) = A002593(n+1). - Raphie Frank, Jan 30 2014
a(n) = floor(1/(exp(2/n^2) - 1)^2)/2. Also a(n) = A007590(n+1)*A074148(n-1)/2. - Richard R. Forberg, Oct 26 2014
Sum_{n>=1} 1/a(n) = -cot(Pi/sqrt(2))*Pi/sqrt(2) - 1/2. - Amiram Eldar, Aug 25 2022
A209344 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero with no three beads in a row equal.
1, 1, 2, 1, 3, 1, 1, 4, 4, 4, 1, 5, 7, 15, 5, 1, 6, 12, 35, 40, 14, 1, 7, 17, 72, 145, 146, 21, 1, 8, 24, 128, 400, 770, 514, 51, 1, 9, 31, 205, 883, 2698, 4029, 2032, 102, 1, 10, 40, 311, 1724, 7358, 18646, 22739, 8076, 249, 1, 11, 49, 448, 3045, 16968, 62853, 136000
Offset: 1
Examples
Table starts: ..1....1.....1......1......1.......1.......1........1........1........1 ..2....3.....4......5......6.......7.......8........9.......10.......11 ..1....4.....7.....12.....17......24......31.......40.......49.......60 ..4...15....35.....72....128.....205.....311......448......618......829 ..5...40...145....400....883....1724....3045.....5026.....7827....11684 .14..146...770...2698...7358...16968...34720....64942...113288...186906 .21..514..4029..18646..62853..172610..409199...870122..1699831..3104474 .51.2032.22739.136000.563109.1830872.5016681.12099880.26438711.53392286 Some solutions for n=6, k=8: .-4...-4...-4...-8...-7...-6...-6...-8...-7...-8...-7...-7...-8...-8...-8...-4 .-3...-3...-3...-3....0....1....1....0...-2....0....1...-2....3...-8...-4...-4 ..5...-1...-4....4...-4...-1....1....1....8....3....0....8...-4...-4....0...-2 .-2....3...-3....1....2....8....6....4...-5....5...-6....1....0....6....7....5 .-1...-1....6....3....3...-5...-6....0....5...-4....8...-7....6....7....3....7 ..5....6....8....3....6....3....4....3....1....4....4....7....3....7....2...-2
Links
- R. H. Hardin, Table of n, a(n) for n = 1..165
Crossrefs
Row 3 is A074148.
Formula
Empirical for row n:
n=2: a(k) = 2*a(k-1) - a(k-2).
n=3: a(k) = 2*a(k-1) - 2*a(k-3) + a(k-4).
n=4: a(k) = 3*a(k-1) - 3*a(k-2) + 2*a(k-3) - 3*a(k-4) + 3*a(k-5) - a(k-6).
n=5: a(k) = 2*a(k-1) + a(k-2) - 3*a(k-3) - a(k-4) + a(k-5) + 3*a(k-6) - a(k-7) - 2*a(k-8) + a(k-9).
n=6: a(k) = 4*a(k-1) - 5*a(k-2) + a(k-3) + a(k-4) + a(k-5) + a(k-6) - 5*a(k-7) + 4*a(k-8) - a(k-9).
Comments
Examples
Links
Crossrefs
Programs
Mathematica
t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]] c[n_] := Count[t[n], 0] t = Table[c[n], {n, 0, 70}] (* A211422 *) (t - 1)/8 (* A120486 *)