A075188 -id:A075188 - cp's OEIS frontend

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090

Offset: 0

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.
In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.
On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the central square to A175655.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Johannes W. Meijer, The elephant sequences.
Wikipedia, War Elephant.
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Cf. A000244, A006130, A006138, A075188, A175655, A180140.

[n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2021

nmax:=28; m:=1; A[1]:=[0,0,0,0,1,0,0,0,1]: A[2]:=[0,0,0,1,0,1,0,0,0]: A[3]:=[0,0,0,0,1,0,1,0,0]: A[4]:=[0,1,0,0,0,0,0,1,0]: A[5]:=[0,0,1,0,0,0,1,1,1]: A[6]:=[0,1,0,0,0,0,0,1,0]: A[7]:=[0,0,1,0,1,0,0,0,0]: A[8]:=[0,0,0,1,0,1,0,0,0]: A[9]:=[1,0,0,0,1,0,0,0,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{3,1,-6}, {1,2,6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;2;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=2 and a(2)=6.
a(n) = ((6+10*A)*A^(-n-1) + (6+10*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).
a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - Vladimir Kruchinin, Aug 20 2010
a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - G. C. Greubel, Dec 08 2021
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Feb 12 2023

A075189 Number of distinct primes in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

Original entry on oeis.org

0, 1, 3, 6, 14, 20, 38, 74, 134, 232, 486, 526, 1078, 2036, 2505, 4762, 9929, 14598, 29831, 31521, 52223, 101123, 207892, 215796, 426772, 836665, 1640357, 1689653, 3401483, 3471770, 6868800, 13470379, 23182192, 45792615, 47136366

Offset: 1

T. D. Noe, Sep 08 2002

Every prime is generated eventually. For the largest generated prime, see A075226. For the smallest odd prime not generated, see A075227.

A217712(n) = number of primes occurring exactly once as numerators among the 2^n sums. - Reinhard Zumkeller, Jun 02 2013

			a(3) = 3 because 3 sums yield distinct prime numerators: 1+1/2 = 3/2, 1/2+1/3 = 5/6 and 1+1/2+1/3 = 11/6.

Cf. A001008, A075135, A075188, A075226, A075227.

Cf. A010051.

import Data.Ratio ((%), numerator)
import Data.Set (Set, empty, fromList, toList, union, size)
a075189 n = a075189_list !! (n-1)
a075189_list = f 1 empty empty where
   f x s s1 = size s1' : f (x + 1) (s `union` fromList hs) s1' where
     s1' = s1 `union` fromList
           (filter ((== 1) . a010051') $ map numerator hs)
     hs = map (+ 1 % x) $ 0 : toList s
-- Reinhard Zumkeller, May 28 2013

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[lst={}; prms={}; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], prms=Union[prms, {k}]]]; AppendTo[lst, Length[prms]]]; lst

a(21)-a(29) from Reinhard Zumkeller, May 28 2013

a(30)-a(35) from Sean A. Irvine, Feb 10 2025

A075226 Largest prime in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3,..., 1/n.

Original entry on oeis.org

3, 11, 19, 137, 137, 1019, 2143, 7129, 7129, 78167, 81401, 1085933, 1111673, 1165727, 2364487, 41325407, 41325407, 796326437, 809074601, 812400209, 822981689, 19174119571, 19652175721, 99554817251, 100483070801

Offset: 2

T. D. Noe, Sep 08 2002

For the smallest odd prime not generated, see A075227. For information about how often the numerator of these sums is prime, see A075188 and A075189. The Mathematica program also prints the subset that yields the largest prime. For n <=20, the largest prime occurs in a sum of n-2, n-1, or n reciprocals.

			a(3) =11 because 11 is largest prime numerator in the three sums that yield primes: 1+1/2 = 3/2, 1/2+1/3 = 5/6 and 1+1/2+1/3 = 11/6.

Chai Wah Wu, Table of n, a(n) for n = 2..441 (terms 2..100 from Martin Fuller)
Martin Fuller, PARI program

Cf. A001008, A075135, A075188, A075189, A075227, A010051, A217712.

import Data.Ratio (numerator)
a075226 n = a075226_list !! (n-1)
a075226_list = f 2 [recip 1] where
   f x hs = (maximum $ filter ((== 1) . a010051') (map numerator hs')) :
            f (x + 1) hs' where hs' = hs ++ map (+ recip x) hs
-- Reinhard Zumkeller, May 28 2013

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[t={}; lst={}; mx=0; i=0; n=2, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], If[k>mx, t=s]; mx=Max[mx, k]]]; Print[n, " ", t]; AppendTo[lst, mx]]; lst
Table[Max[Select[Numerator[Total/@Subsets[1/Range[n],{2,2^n}]],PrimeQ]],{n,2,30}] (* The program will take a long time to run. *) (* Harvey P. Dale, Jan 08 2019 *)

PARI
```
See Fuller link.
```

from math import gcd, lcm
from itertools import combinations
from sympy import isprime
def A075226(n):
    m = lcm(*range(1,n+1))
    c, mlist = 0, tuple(m//i for i in range(1,n+1))
    for l in range(n,-1,-1):
        if sum(mlist[:l]) < c:
            break
        for p in combinations(mlist,l):
            s = sum(p)
            s //= gcd(s,m)
            if s > c and isprime(s):
                c = s
    return c # Chai Wah Wu, Feb 14 2022

More terms from Martin Fuller, Jan 19 2008

A075227 Smallest odd prime not occurring in the numerator of any of the 2^n subset sums generated from the set 1/1, 1/2, 1/3, ..., 1/n.

Original entry on oeis.org

3, 5, 7, 17, 37, 43, 43, 151, 151, 409, 491, 491, 491, 1087, 2011, 3709, 3709, 7417, 7417, 7417, 19699, 30139, 35573, 35573, 40237, 40237, 132151, 132151, 158551, 158551, 245639, 245639, 961459, 1674769, 1674769, 1674769, 1674769, 4339207

Offset: 1

T. D. Noe, Sep 08 2002

The largest prime generated is given in A075226. For information about how often the numerator of these sums is prime, see A075188 and A075189.

			a(3) = 7 because 7 is the smallest prime not occurring in the numerator of any of the sums 1/1 + 1/2 = 3/2, 1/1 + 1/3 = 4/3, 1/2 + 1/3 = 5/6 and 1/1 + 1/2 + 1/3 = 11/6.

Cf. A001008, A075135, A075188, A075189, A075226.
Cf. A010051, A065091.
Cf. A217712.

import Data.Ratio ((%), numerator)
import Data.Set (Set, empty, fromList, toList, union)
a075227 n = a075227_list !! (n-1)
a075227_list = f 1 empty a065091_list where
   f x s ps = head qs : f (x + 1) (s `union` fromList hs) qs where
     qs = foldl (flip del)
          ps $ filter ((== 1) . a010051') $ map numerator hs
     hs = map (+ 1 % x) $ 0 : toList s
   del u vs'@(v:vs) = case compare u v
                      of LT -> vs'; EQ -> vs; GT -> v : del u vs
-- Reinhard Zumkeller, May 28 2013

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[lst={}; prms={}; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], AppendTo[prms, k]]]; prms=Union[prms]; j=2; While[MemberQ[prms, Prime[j]], j++ ]; AppendTo[lst, Prime[j]]]; lst
(* Second program; does not need Combinatorica *)
a[1] = 3; a[2] = 5; a[n_] := For[nums = (Total /@ Subsets[1/Range[n]]) // Numerator // Union // Select[#, PrimeQ]&; p = 3, p <= Last[nums], p = NextPrime[p], If[FreeQ[nums, p], Print[n, " ", p]; Return[p]]];
Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Sep 10 2017 *)

from sympy import sieve
from fractions import Fraction
fracs, newnums, primeset = {0}, {0}, set(sieve.primerange(3, 10**6+1))
for n in range(1, 24):
  newfracs = set(Fraction(1, n) + f for f in fracs)
  fracs |= newfracs
  primeset -= set(f.numerator for f in newfracs)
  print(min(primeset), end=", ") # Michael S. Branicky, May 09 2021

a(21)-a(28) from Reinhard Zumkeller, May 28 2013
a(29)-a(33) from Jon E. Schoenfield, May 09 2021
a(34)-a(36) from Michael S. Branicky, May 10 2021
a(37)-a(38) from Michael S. Branicky, May 12 2021

A217712 Number of primes occurring exactly once as numerators in sums generated from the set 1, 1/2, 1/3,..., 1/n.

Original entry on oeis.org

0, 1, 3, 3, 11, 13, 27, 54, 106, 168, 378, 142, 733, 1597, 1283, 3418, 8204, 10112, 24644, 7829, 32866, 78136, 178741, 37002, 256392, 650596, 1402914, 286854, 2053463

Offset: 1

Reinhard Zumkeller, Jun 02 2013

nonn

For information about how often the numerator of the generated sums is prime, see A075188 and A075189; for the largest generated prime, see A075226; for the smallest odd prime not generated, see A075227.

			For n=3 there are the following fractions as sums of 1, 1/2 and 1/3:
{1/3, 1/2, 5/6, 1, 4/3, 3/2, 11/6}, three numerators are prime and they occur exactly once, therefore a(3) = A075188(3) = A075189(3) = #{3, 5, 11} = 3;
n=4: adding 1/4 to the previous fractions gives together: 1/4, 1/3, 1/2, 1/3+1/4=7/12, 1/2+1/4=3/4, 5/6, 1, 5/6+1/4=13/12, 1+1/4=5/4, 4/3, 3/2, 4/3+1/4=19/12, 3/2+1/4=7/4, 11/6 and 11/6+1/4=25/12:
A075188(4) = #{7/12, 3/4, 5/6, 13/12, 5/4, 3/2, 19/12, 7/4, 11/6} = 9,
A075189(4) = #{3, 5, 7, 11, 13, 19} = 6,
a(4) = #{11, 13, 19} = 3.

Cf. A010051.

import Data.Ratio ((%), numerator)
import Data.Set (Set, empty, fromList, toList, union, size)
import Data.Set (member, delete, insert)
a217712 n = a217712_list !! (n-1)
a217712_list = f 1 empty empty where
   f x s s1 = size s1' : f (x + 1) (s `union` fromList hs) s1' where
     s1' = g s1 $ filter ((== 1) . a010051') $ map numerator hs
     g v []                    = v
     g v (w:ws) | w `member` v = g (delete w v) ws
                | otherwise    = g (insert w v) ws
     hs = map (+ 1 % x) $ 0 : toList s

A256220 Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is a Fibonacci number.

Original entry on oeis.org

1, 3, 5, 9, 11, 22, 28, 37, 45, 62, 70, 125, 133, 172, 330, 421, 450, 840, 901, 1710, 2356, 2724, 2824, 5367, 6022, 7142, 8072, 18771, 19204, 35739, 36453, 42853, 82094, 88574, 155642, 264869

Offset: 1

Michel Lagneau, Mar 19 2015

Note that for each n there are only 2^(n-1) new sums to consider. For the number of distinct Fibonacci numbers, see A256221. For the largest generated Fibonacci number, see A256222. For the smallest Fibonacci number not generated, see A256223.

			a(3) = 5 because we obtain 5 following subsets {1}, {1/2}, {1/3}, {1,1/2} and {1/2, 1/3} having 5 sums with Fibonacci numerators: 1, 1, 1, 1+1/2 = 3/2 and 1/2+1/3 = 5/6.

Cf. A000045, A075188, A010056, A256221, A256222, A256223.

<<"DiscreteMath`Combinatorica`"; maxN=22; For[cnt=0; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[IntegerQ[Sqrt[5*k^2+4]]||IntegerQ[Sqrt[5*k^2-4]], cnt++ ]]; Print[cnt]]

from math import gcd, lcm
from itertools import combinations
def A256220(n):
    m = lcm(*range(1,n+1))
    fibset, mlist = set(), tuple(m//i for i in range(1,n+1))
    a, b, c, k = 0, 1, 0, sum(mlist)
    while b <= k:
        fibset.add(b)
        a, b = b, a+b
    for l in range(1,n//2+1):
        for p in combinations(mlist,l):
            s = sum(p)
            if s//gcd(s,m) in fibset:
                c += 1
            if 2*l != n and (k-s)//gcd(k-s,m) in fibset:
                c += 1
    return c+int(k//gcd(k,m) in fibset) # Chai Wah Wu, Feb 15 2022

a(23)-a(36) from Lars Blomberg, May 06 2015

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).

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A075189 Number of distinct primes in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

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A075226 Largest prime in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3,..., 1/n.

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A075227 Smallest odd prime not occurring in the numerator of any of the 2^n subset sums generated from the set 1/1, 1/2, 1/3, ..., 1/n.

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A217712 Number of primes occurring exactly once as numerators in sums generated from the set 1, 1/2, 1/3,..., 1/n.

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Haskell

A256220 Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is a Fibonacci number.

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Python

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).