A189391
The minimum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 0 to n, and each number in a higher row is the sum of the two numbers directly below it.
Original entry on oeis.org
0, 1, 3, 8, 19, 44, 98, 216, 467, 1004, 2134, 4520, 9502, 19928, 41572, 86576, 179587, 372044, 768398, 1585416, 3263210, 6711176, 13775068, 28255568, 57863214, 118430584, 242061468, 494523536, 1009105372, 2058327344, 4194213448
Offset: 0
For n = 4 consider the triangle:
....19
...8 11
..5 3 8
.4 1 2 6
3 1 0 2 4
This triangle has 19 at its apex and no other such triangle with the numbers 0 - 4 on its base has a smaller apex value, so a(4) = 19.
- Nathaniel Johnston, Table of n, a(n) for n = 0..1000
- F. Disanto and S. Rinaldi, Symmetric convex permutominoes and involutions, PU. M. A., Vol. 22 (2011), No. 1, pp. 39-60. - From _N. J. A. Sloane_, May 04 2012
- Steven R. Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018.
- Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - _N. J. A. Sloane_, Sep 20 2018
-
m:=30; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!((2*x+Sqrt(1-4*x^2)-1)/(2*(2*x-1)^2))); // G. C. Greubel, Aug 24 2018
-
a:=proc(n)return add((2*n-4*k-1)*binomial(n,k),k=0..floor((n-1)/2)): end:
seq(a(n),n=0..50);
-
CoefficientList[Series[(2*x+Sqrt[1-4*x^2]-1) / (2*(2*x-1)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 16 2014 *)
-
A189391(n)=sum(i=0,(n-1)\2,(2*n-4*i-1)*binomial(n,i)) \\ M. F. Hasler, Jan 24 2012
A189390
The maximum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 0 to n, and each number in a higher row is the sum of the two numbers directly below it.
Original entry on oeis.org
0, 1, 5, 16, 45, 116, 286, 680, 1581, 3604, 8106, 18008, 39650, 86568, 187804, 404944, 868989, 1856180, 3950194, 8376056, 17708310, 37329016, 78499620, 164682416, 344789970, 720430216, 1502768996, 3129355120, 6507087396, 13510929104
Offset: 0
For n = 4 consider the triangle:
45
21 24
8 13 11
2 6 7 4
0 2 4 3 1
This triangle has 45 at its apex and no other such triangle with the numbers 0 through 4 on its base has a larger apex value, so a(4) = 45.
-
a:= proc(n) return add((4*k+1)*binomial(n,k), k=0..floor((n-1)/2)) + `if`(n mod 2=0, n*binomial(n,n/2), 0):end:
seq(a(n), n=0..50);
-
a[n_] := Sum[(4k+1)*Binomial[n, k], {k, 0, Floor[(n-1)/2]}] + If[EvenQ[n], n*Binomial[n, n/2], 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 18 2017, translated from Maple *)
-
A189390(n)=sum(i=0, (n-1)\2, (4*i+1)*binomial(n, i), if(!bittest(n,0),n*binomial(n, n\2))) \\ - M. F. Hasler, Jan 24 2012
-
from math import comb
def A189390(n): return sum(((k<<2)|1)*comb(n,k) for k in range(n+1>>1))+(0 if n&1 else n*comb(n,n>>1)) # Chai Wah Wu, Oct 28 2024
A066411
Form a triangle with the numbers [0..n] on the base, where each number is the sum of the two below; a(n) is the number of different possible values for the apex.
Original entry on oeis.org
1, 1, 3, 5, 23, 61, 143, 215, 995, 2481, 5785, 12907, 29279, 64963, 144289, 158049, 683311, 1471123, 3166531, 6759177, 14404547, 30548713
Offset: 0
For n = 2 we have three triangles:
..4.......5.......3
.1,3.....2,3.....2,1
0,1,2...0,2,1...2,0,1
with three different values for the apex, so a(2) = 3.
-
import Data.List (permutations, nub)
a066411 0 = 1
a066411 n = length $ nub $ map
apex [perm | perm <- permutations [0..n], head perm < last perm] where
apex = head . until ((== 1) . length)
(\xs -> (zipWith (+) xs $ tail xs))
-- Reinhard Zumkeller, Jan 24 2012
-
for n=0:9
size(unique(perms(0:n)*diag(fliplr(pascal(n+1)))),1)
end % Nathaniel Johnston, Apr 20 2011
(C++)
#include
#include
#include
#include
using namespace std;
inline long long pascApx(const vector & s)
{
const int n = s.size() ;
vector scp(n) ;
for(int i=0; i s;
for(int i=0;i apx;
do
{
apx.insert( pascApx(s)) ;
} while( next_permutation(s.begin(),s.end()) ) ;
cout << n << " " << apx.size() << endl ;
}
return 0 ;
} /* R. J. Mathar, Jan 24 2012 */
-
g[s_List] := Plus @@@ Partition[s, 2, 1]; f[n_] := Block[{k = 1, lmt = 1 + (n + 1)!, lst = {}, p = Permutations[Range[0, n]]}, While[k < lmt, AppendTo[ lst, Nest[g, p[[k]], n][[1]]]; k++]; lst]; Table[ Length@ Union@ f@ n, {n, 0, 10}] (* Robert G. Wilson v, Jan 24 2012 *)
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A066411(n)={my(u=0,o=A189391(n),v,b=vector(n++,i,binomial(n-1,i-1))~);sum(k=1,n!\2,!bittest(u,numtoperm(n,k)*b-o) & u+=1<<(numtoperm(n,k)*b-o))} \\ M. F. Hasler, Jan 24 2012
-
from sympy import binomial
def partitionpairs(xlist): # generator of all partitions into pairs and at most 1 singleton, returning the sums of the pairs
if len(xlist) <= 2:
yield [sum(xlist)]
else:
m = len(xlist)
for i in range(m-1):
for j in range(i+1,m):
rem = xlist[:i]+xlist[i+1:j]+xlist[j+1:]
y = [xlist[i]+xlist[j]]
for d in partitionpairs(rem):
yield y+d
def A066411(n):
b = [binomial(n,k) for k in range(n//2+1)]
return len(set((sum(d[i]*b[i] for i in range(n//2+1)) for d in partitionpairs(list(range(n+1)))))) # Chai Wah Wu, Oct 19 2021
A189162
The maximum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 1 to n, and each number in a higher row is the sum of the two numbers directly below it.
Original entry on oeis.org
1, 3, 9, 24, 61, 148, 350, 808, 1837, 4116, 9130, 20056, 43746, 94760, 204188, 437712, 934525, 1987252, 4212338, 8900344, 18756886, 39426168, 82693924, 173071024, 361567186, 753984648, 1569877860, 3263572848, 6775522852, 14047800016, 29091783096, 60175932320
Offset: 1
For n = 5 consider the triangle:
61
29 32
12 17 15
4 8 9 6
1 3 5 4 2
This triangle has 61 at its apex and no other such triangle with the numbers 1 - 5 on its base has a larger apex value, so a(5) = 61.
-
a:=proc(n)return 2^(n-1) + add((4*k+1)*binomial(n-1,k),k=0..floor(n/2)-1) + `if`(n mod 2=1,(n-1)*binomial(n-1,(n-1)/2),0):end:
seq(a(n),n=1..50);
-
a[n_] := a[n] = Switch[n, 1, 1, 2, 3, 3, 9, 4, 24, _, (1/(n-1))*(4((4n-16)a[n-4] - (4n-16)a[n-3] - 3a[n-2] + (n-1)a[n-1]))];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, May 09 2023, after R. J. Mathar *)
A099326
Expansion of ((1-2x)*sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(5/2)).
Original entry on oeis.org
1, 4, 11, 28, 67, 156, 354, 792, 1747, 3820, 8278, 17832, 38174, 81368, 172644, 365104, 769411, 1617228, 3389838, 7090440, 14797546, 30828424, 64106716, 133113168, 275967022, 571415416, 1181585564, 2440680592, 5035637212
Offset: 0
-
CoefficientList[Series[((1-2*x)*Sqrt[1+2*x]+Sqrt[1-2*x])/(2*(1-2*x)^(5/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
A099327
Expansion of ((1-x)*sqrt(1+2x) + (1+x)*sqrt(1-2x))/(2*(1-2x)^(5/2)).
Original entry on oeis.org
1, 5, 16, 45, 117, 291, 700, 1646, 3799, 8647, 19448, 43330, 95738, 210094, 458216, 994204, 2146955, 4617439, 9893376, 21128058, 44982486, 95510090, 202278376, 427425860, 901236582, 1896594966, 3983929680, 8354539156, 17492095604
Offset: 0
-
CoefficientList[Series[((1-x)*Sqrt[1+2*x]+(1+x)*Sqrt[1-2*x])/(2*(1-2*x)^(5/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
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