cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Nathan B. Shank

Nathan B. Shank's wiki page.

Nathan B. Shank has authored 6 sequences.

A364480 Table of Sprague-Grundy values for M X N Pop-It! hooks read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 1, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 1, 9, 9, 9, 9, 10, 10, 10, 10, 2, 2, 10, 10, 10, 10, 11, 11, 11, 11, 3, 3, 3, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12
Offset: 1

Views

Author

Ryan J. Blau, Michael Carrion, Djeneba Diop, Sarah Dumnich, Matt Prudente, and Nathan B. Shank, Jul 26 2023

Keywords

Comments

Pop-It! is played on an M X N lattice graph with a "pip" at each node. A move consists of popping any number of pips that are all collinear on the same horizontal or vertical line. A hook h(a,b) is defined as two lines of lengths a and b connected by a corner pip. The total number of pips of a hook h(a,b) is a+b+1.
This table seems to have similarities to A354586.

Examples

			Table begins
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, ...
   2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, ...
   3,  4,  1,  6,  7,  8,  9, 10, 11, 12, 13, ...
   4,  5,  6,  7,  8,  9, 10, 11, 12, 13, ...
   5,  6,  7,  8,  1,  2,  3, 12, 13, ...
   6,  7,  8,  9,  2,  3, 12, 13, ...
   7,  8,  9, 10,  3, 12,  1, ...
   8,  9, 10, 11, 12, 13, ...
   9, 10, 11, 12, 13, ...
  10, 11, 12, 13, ...
  11, 12, 13, ...
  12, 13, ...
  13, ...
  ...

Links

Crossrefs

Sprague-Grundy values for Toppling Dominoes L's: A354586.
Cf. A038712 (main diagonal).

A346199 a(n) is the number of permutations on [n] with at least one strong fixed point and no small descents.

Original entry on oeis.org

1, 1, 1, 5, 19, 95, 569, 3957, 31455, 281435, 2799981, 30666153, 366646995, 4751669391, 66348304849, 992975080813, 15856445382119, 269096399032035, 4836375742967861, 91766664243841393, 1833100630242606203, 38452789552631651191, 845116020421125048153
Offset: 1

Views

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

Keywords

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.
A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Examples

			For n = 4, the a(4) = 5 permutations on [4] with strong fixed points but no small descents: {(1*, 2*, 3*, 4*), (1*, 3, 4, 2), (1*, 4, 2, 3), (2, 3, 1, 4*), (3, 1, 2, 4*)} where * marks strong fixed points.

References

  • E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346198, A346204.

Programs

Formula

a(n) = b(n-1) + Sum_{i=4..n} A346189(i-1)*b(n-i) where b(n) = A000255(n).

A346198 a(n) is the number of permutations on [n] with no strong fixed points but contains at least one small descent.

Original entry on oeis.org

0, 1, 1, 8, 43, 283, 2126, 17947, 168461, 1741824, 19684171, 241506539, 3198239994, 45482655683, 691471698917, 11193266251700, 192238116358427, 3491633681792507, 66875708261486766, 1347168876070616179, 28474546456352896021, 630130731702950549248, 14570725407559756078387, 351411668456841530417027
Offset: 1

Views

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

Keywords

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.
A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Examples

			For n = 4, the a(4) = 8 permutations on [4] with no strong fixed points but has small descents: {([2, 1], [4, 3]), (2, [4, 3], 1), ([3, 2], 4, 1), (3, 4, [2, 1]), (4, 1, [3, 2]), (4, [2, 1], 3), ([4, 3], 1, 2), (<4, 3, 2, 1>)} []small descent, <>consecutive small descents.

References

  • E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346199, A346204.

Programs

Formula

For n > 2, a(n) = b(n)-c(n) where b(n) = A052186(n-1), c(n) = A346189(n).

A346189 a(n) is the number of permutations on [n] with no strong fixed points or small descents.

Original entry on oeis.org

0, 0, 2, 6, 34, 214, 1550, 12730, 116874, 1187022, 13219550, 160233258, 2100360778, 29610224590, 446789311934, 7185155686666, 122690711149290, 2217055354281582, 42269657477711198, 847998698508705834, 17857221256001240458, 393839277313540073230, 9078806210245773668990, 218340709713567352161226
Offset: 1

Views

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

Keywords

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.
A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Examples

			For n = 4, the a(4) = 6 permutations on [4] with no strong fixed points or small descents: {(2,3,4,1),(3,4,1,2),(4,1,2,3),(3,1,4,2),(2,4,1,3),(4,2,3,1)}.

References

  • E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346198, A346199, A346204.

Programs

Formula

For n > 3, a(n) = b(n) - b(n-1) - Sum{i=4..n}(a(i-1)*b(n-i)) where b(n) = A000255(n-1) and b(0) = 1.

A346204 a(n) is the number of permutations on [n] with at least one strong fixed point and at least one small descent.

Original entry on oeis.org

0, 0, 2, 5, 24, 128, 795, 5686, 46090, 418519, 4213098, 46595650, 561773033, 7333741536, 103065052300, 1551392868821, 24902155206164, 424588270621876, 7663358926666175, 145967769353476594, 2926073829112697318, 61577929208485406331, 1357369100658321844470, 31276096500003460511422
Offset: 1

Views

Author

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 10 2021

Keywords

Comments

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.
A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

Examples

			For n=4, the a(4)=5 permutations on [4] with strong fixed points and small descents: {(1*, 2*, [4, 3]), (1*, [3, 2], 4*), (1*, <4, 3, 2>), ([2, 1], 3*, 4*), (<3, 2, 1>, 4*)}. *strong fixed point, []small descent, <>consecutive small descents.

References

  • E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Links

Crossrefs

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346198, A346199.

Programs

  • Python
    import math
bn = [1,1,1]
wn = [0,0,0]
kn = [1,1,1]
def summation(n):
    final = bn[n] - bn[n-1]
    for k in range(4,n+1):
        final -= wn[k-1]*bn[n-k]
    return final
def smallsum(n):
    final = bn[n-1]
    for k in range(4,n+1):
        final += wn[k-1]*bn[n-k]
    return final
def derrangement(n):
    finalsum = 0
    for i in range(n+1):
        if i%2 == 0:
            finalsum += math.factorial(n)*1//math.factorial(i)
        else:
            finalsum -= math.factorial(n)*1//math.factorial(i)
    if finalsum != 0:
        return finalsum
    else:
        return 1
def fixedpoint(n):
    finalsum = math.factorial(n-1)
    for i in range(2,n):
        finalsum += math.factorial(i-i)*math.factorial(n-i-1)
        print(math.factorial(i-i)*math.factorial(n-i-1))
    return finalsum
def no_cycles(n):
    goal = n
    cycles = [0, 1]
    current = 2
    while current<= goal:
        new = 0
        k = 1
        while k<=current:
            new += (math.factorial(k-1)-cycles[k-1])*(math.factorial(current-k))
            k+=1
        cycles.append(new)
        current+=1
    return cycles
def total_func(n):
    for i in range(3,n+1):
        bn.append(derrangement(i+1)//(i))
        kn.append(smallsum(i))
        wn.append(summation(i))
    an = no_cycles(n)
    tl = [int(an[i]-kn[i]) for i in range(n+1)]
    factorial = [math.factorial(x) for x in range(0,n+1)]
    print("A346189 :" + str(wn[1:]))
    print("A346198 :" + str([factorial[i]-wn[i]-tl[i]-kn[i] for i in range(n+1)][1:]))
    print("A346199 :" + str(kn[1:]))
    print("A346204 :" + str(tl[1:]))
total_func(20)

Formula

a(n) = A006932(n) - A346199(n).

A076744 Related to the expected length of the shortest nonintersecting path through n points on a Sierpiński Gasket from corner to corner.

Original entry on oeis.org

4, 72, 4176, 731808, 381879360, 592267282560, 2733202405059840, 37590062966534453760, 1542797317119230338360320, 189160927199005707074274969600, 69339320764681731806436884240486400, 76034016779649931607383549266935620608000
Offset: 1

Views

Author

Nathan B. Shank, Nov 11 2002

Keywords

Comments

Original name: This sequence with the appropriate denominator (product of (2*3^k-3) k=0..n) produces the expected length of shortest nonintersecting path through n points on a Sierpiński Gasket from corner to corner.
Does b(n) / n^(1-log(2)/log(3)) -> constant?

Programs

  • Maple
    b:=n->sum(binomial(n,k)*(-1)^k/(2*3^k-3),k=0..n);
c:=n->product(2*3^k-3,k=0..n);
a:=n->b(n)*c(n);
seq(a(n),n=1..12);

Formula

a(n) = b(n) * c(n) where b(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) / (2*3^k-3) and c(n) = Product_{k=0..n} (2*3^k-3).

Extensions

Revised by Sean A. Irvine, Apr 14 2025

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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