A063196 -id:A063196 - cp's OEIS frontend

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

Reinhard Zumkeller, Aug 01 2005

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)
First differences: A010673; partial sums: A000982;
A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);
A167875(n) = Sum_{k = 0..n} a(k)*A005408(n-k);
A171218(n) = Sum_{k = 0..n} a(k)*A005843(n-k);
A008794(n+2) = Sum_{k = 0..n} a(k)*A059841(n-k). (End)
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013
For n > 4: a(n) = A230584(n) - A230584(n-2). - Reinhard Zumkeller, Feb 10 2015
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

			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

Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, 3rd printing 2012, pp. (260-281).

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).

Cf. A000217, A063196, A110660, A186421, A186422, A211955, A230584, A290988.
Complement of A052928 with respect to the universe A004526. - Guenther Schrack, Aug 21 2018
First differences of A000982, A061925, A074148, A105343, A116940, and A179207. - Guenther Schrack, Aug 21 2018

a109613 = (+ 1) . (* 2) . (`div` 2)
a109613_list = 1 : 1 : map (+ 2) a109613_list
-- Reinhard Zumkeller, Oct 27 2012, Feb 21 2011

A109613:=n->2*floor(n/2)+1; seq(A109613(k), k=0..100); # Wesley Ivan Hurt, Oct 22 2013

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 *)

A109613(n)=n>>1<<1+1 \\ Charles R Greathouse IV, Feb 24 2011

def a(n) : return( len( CuspForms( Gamma0( 5), 2*n + 4, prec=1). basis())); # Michael Somos, May 29 2013

((1 to 49) by 2) flatMap { List.fill(2)() } // _Alonso del Arte, Sep 11 2019

a(n) = 2*floor(n/2) + 1.
a(n) = A052928(n) + 1 = 2*A004526(n) + 1.
a(n) = A028242(n) + A110654(n).
a(n) = A052938(n-2) + A084964(n-2) for n > 1. - Reinhard Zumkeller, Aug 27 2005
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) = A001477(n) + A059841(n). - Philippe Deléham, Mar 31 2009
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).
a(n) = A005408(A004526(n)). (End)
a(n) = A000217(n) / A004526(n+1), n > 0. - Torlach Rush, Nov 10 2023

A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 3, 6, 7, 0, 0, 1, 0, 9, 0, 15, 0, 1, 0, 3, 12, 31, 0, 33, 8, 0, 1, 0, 17, 0, 107, 0, 77, 0, 1, 0, 5, 22, 81, 0, 395, 410, 181, 0, 0, 1, 0, 27, 0, 397, 0, 1525, 0, 443, 0, 1, 0, 5, 34, 171, 0, 2073, 4508, 6095, 0, 1113, 58, 0, 1, 0, 41, 0, 1081, 0

Offset: 1

R. H. Hardin, May 05 2013

Table starts
.0...1...0.....1....0......1.....0.......1.....0........1......0........1
.0...1...0.....1....0......1.....0.......1.....0........1......0........1
.0...1...0.....3....0......3.....0.......5.....0........5......0........7
.2...3...6.....9...12.....17....22......27....34.......41.....48.......57
.0...7...0....31....0.....81.....0.....171.....0......309......0......509
.0..15...0...107....0....397.....0....1081.....0.....2399......0.....4675
.0..33...0...395....0...2073.....0....7261.....0....19709......0....45385
.8..77.410..1525.4508..11291.25056...50659.95130...168289.283338...457627
.0.181...0..6095....0..63121.....0..364051.....0..1478059......0..4749875
.0.443...0.24893....0.360909.....0.2676331.....0.13280209......0.50435657

			Some solutions for n=4, k=4
.-1.-1.-1..1...-1.-1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1
.-1..1..1..1...-1..1..1..1...-1.-1.-1..1....1..1..1..1....1..1..1..1
.-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1....1..1..1..1...-1.-1.-1..1
.-1.-1.-1..1...-1..1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..3132

Column 1 is A063074(n/4).
Row 3 is A063196(n/2+1).
Row 4 is A008810(n+1).
Row 5 is A202254(n/2).

Empirical for row n:
n=1: a(n) = a(n-2);
n=2: a(n) = a(n-2);
n=3: a(n) = a(n-2) +a(n-4) -a(n-6);
n=4: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5);
n=5: a(n) = 3*a(n-2) -2*a(n-4) -2*a(n-6) +3*a(n-8) -a(n-10);
n=6: [order 26, even n];
n=7: [order 42, even n];
n=8: [order 28];
n=9: [order 58, even n];
n=10: [order 90, even n];
n=11: [order 102, even n];
n=12: [order 66].

A200181 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum and no two consecutive declines, no adjacent equal elements, and no element more than one greater than the previous (random base sawtooth pattern).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 3, 3, 1, 4, 5, 6, 2, 1, 5, 5, 11, 12, 6, 1, 6, 7, 14, 15, 15, 10, 1, 7, 7, 19, 24, 29, 29, 7, 1, 8, 9, 26, 31, 48, 78, 72, 12, 1, 9, 9, 31, 48, 72, 100, 160, 133, 28, 1, 10, 11, 38, 53, 103, 186, 280, 283, 214, 29, 1, 11, 11, 47, 74, 141, 246, 460, 608, 574, 394

Offset: 1

R. H. Hardin Nov 13 2011

Table starts
..1...1...1....1....1....1....1....1.....1.....1.....1.....1.....1.....1.....1
..1...2...3....4....5....6....7....8.....9....10....11....12....13....14....15
..3...3...5....5....7....7....9....9....11....11....13....13....15....15....17
..3...6..11...14...19...26...31...38....47....54....63....74....83....94...107
..2..12..15...24...31...48...53...74....83...108...119...148...159...196...209
..6..15..29...48...72..103..141..186...244...309...385...472...572...685...813
.10..29..78..100..186..246..380..464...686...798..1096..1276..1658..1878..2408
..7..72.160..280..460..700.1010.1430..1954..2592..3392..4348..5470..6826..8392
.12.133.283..608..891.1573.2152.3430..4429..6531..8124.11410.13787.18525.21952
.28.214.574.1094.1934.3247.5014.7552.11060.15511.21380.29006.38248.49885.64294

			Some solutions for n=7 k=6
..2....6....3...-1....1....3....4....2....1....4....6....5....1....1....5....6
..3....1....4....0....2....1....5....3....0....0....0....4....2....2....0...-1
..1....2....2....1...-1....2....6....1....1....1....1....5....3....3....1....0
..2....3....3....2....0....1...-4....2...-1...-1...-2...-4...-1...-2....2...-1
..3...-4...-4...-1....1....2...-3...-3....0....0...-1...-3....0...-1...-2....0
.-6...-3...-3....0...-2...-5...-2...-2...-1....1....0...-2...-3...-2...-1....1
.-5...-5...-5...-1...-1...-4...-6...-3....0...-5...-4...-5...-2...-1...-5...-5

R. H. Hardin, Table of n, a(n) for n = 1..958

Row 3 is A063196(n+2)

A086500 Group the natural numbers such that the n-th group sum is divisible by the n-th triangular number: (1), (2, 3, 4), (5, 6, 7), (8, 9, 10, 11, 12), (13, 14, 15, 16, 17), (18, 19, 20, 21, 22, 23, 24), ... Sequence contains the group sum.

Original entry on oeis.org

1, 9, 18, 50, 75, 147, 196, 324, 405, 605, 726, 1014, 1183, 1575, 1800, 2312, 2601, 3249, 3610, 4410, 4851, 5819, 6348, 7500, 8125, 9477, 10206, 11774, 12615, 14415, 15376, 17424, 18513, 20825, 22050, 24642, 26011, 28899, 30420, 33620, 35301

Offset: 1

Amarnath Murthy, Jul 28 2003

The number of terms in the groups is given by A063196. i.e., 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, ...

Also the arithmetic mean of the n-th group is T(n), the n-th triangular number.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A001082, A022998, A063196, A181900.

a086500 n = a086500_list !! (n-1)
a086500_list = scanl1 (+) $ tail a181900_list
-- Reinhard Zumkeller, Mar 31 2012

Table[n*(n + 1)*(2*n + 1 + (-1)^n)/4, {n, 1, 40}] (* Amiram Eldar, Feb 22 2022 *)

Vec(x*(x^4+8*x^3+6*x^2+8*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 20 2014

a(n) = n*(n+1)*(2*n+1+(-1)^n)/4. - Wesley Ivan Hurt, Sep 19 2014
a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>7. - Colin Barker, Sep 19 2014
G.f.: x*(x^4+8*x^3+6*x^2+8*x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Sep 19 2014
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(1-log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 4. (End)

More terms from Ray Chandler, Sep 17 2003

cp's OEIS Frontend

A109613 Odd numbers repeated.

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A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A200181 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum and no two consecutive declines, no adjacent equal elements, and no element more than one greater than the previous (random base sawtooth pattern).

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A086500 Group the natural numbers such that the n-th group sum is divisible by the n-th triangular number: (1), (2, 3, 4), (5, 6, 7), (8, 9, 10, 11, 12), (13, 14, 15, 16, 17), (18, 19, 20, 21, 22, 23, 24), ... Sequence contains the group sum.

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