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Number of labeled histories for rooted 5-furcating trees with 4n+1 leaves if simultaneous 5-furcations are allowed.
+0
0
1, 1, 126, 198198, 1552358808, 41269930621920, 2917021792126858416, 466738566750935966462976, 150642168106131265276308435840, 89930728809765858827345682838905216, 92814015425659158860323886440105229380608, 156870775305420194841270876582071899442900414976, 415352074564676036635314305973768435826840253066044416
FORMULA
a(n) = Y(4n+1), where Y(n) = Sum_{i=1..floor(n/5)} (n!/(i!*120^i*(n-5*i)!)) * Y(n-4*i), with Y(1)=1.
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add((4*n+1)!/
(i!*120^i*(4*n+1-5*i)!)*a(n-i), i=1..(4*n+1)/5))
end:
Number of labeled histories for rooted 4-furcating trees with 3n+1 leaves if simultaneous 4-furcations are not allowed.
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0
1, 1, 35, 7350, 5255250, 9564555000, 37072215180000, 271183254041700000, 3430468163627505000000, 70238835650273164875000000, 2210064963735845132791875000000, 102493972758213553878355995000000000, 6769214430816214165896021689775000000000, 618638506832293812621237422228537250000000000
FORMULA
a(n) = ((3*n+1)!/24^n) * Product_{i=1..n} (3*i-2).
MAPLE
a:= n-> (3*n+1)!/24^n*mul(3*i-2, i=1..n):
Numbers k such that 5*k+1 divides 5^k+1.
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0
0, 1, 625, 57057, 7748433, 30850281, 111494625, 393423745, 499088601, 519341361, 1051107705, 1329416385, 1616038425, 2215448001, 2433936225, 2852972265, 3399207273, 4344683849, 4961725281, 5454760185, 5485530369, 6578054145, 6678031745, 7701979761, 7807302825
COMMENTS
The numbers are called Curzon numbers by Tattersall (p. 85, exercise 43).
REFERENCES
James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005, p. 85.
EXAMPLE
5*625+1 = 3126 divides 5^625+1.
PROG
(PARI) isok(n) = my(m=5*n+1); Mod(5, m)^n==-1
Smaller of two consecutive primes p and q, both ending with 9, such that q-p = 10n, or -1 if no such primes exist.
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0
139, 3089, 5749, 20809, 60539, 110359, 173359, 618719, 1294849, 838249, 6877109, 1895359, 11188759, 7621259, 35560009, 33803689, 124956059, 92801029, 142414669, 378043979, 229316459, 390932389, 1095750599, 995151679, 2174082649, 2603726969, 3402493709, 1997191249
EXAMPLE
a(1) = 139, because 139 and 139 + 10 = 149 are two consecutive primes with the same last digit 9 and no smaller p has this property.
PROG
(PARI) a(n) = my(p=9); while (!isprime(p) || ((nextprime(p+1)-p) != 10*n), p+=10); p; \\ Michel Marcus, Feb 25 2025
Smaller of two consecutive primes p and q, both ending with 7, such that q - p = 10n, or -1 if no such primes exist.
+0
0
337, 887, 4297, 33247, 31907, 124367, 218287, 1122287, 1964987, 1313467, 1468277, 7160227, 5518687, 16525757, 13626257, 71880637, 27915737, 17051707, 394059907, 566348087, 252314747, 472865287, 1289694257, 633418787, 1588640437, 944192807, 1391048047, 7059848287
EXAMPLE
a(1) = 337, because 337 and 337 + 10 = 347 are two consecutive primes with the same last digit 9 and no smaller p has this property.
PROG
(PARI) a(n) = my(p=7); while (!isprime(p) || ((nextprime(p+1)-p) != 10*n), p+=10); p; \\ Michel Marcus, Feb 25 2025
Side lengths of the squares in the smallest simple perfect squared square whose side lengths range from 1 (smallest) to 100 (largest).
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0
1, 4, 6, 9, 13, 15, 17, 28, 29, 35, 37, 41, 43, 53, 57, 58, 60, 62, 63, 64, 66, 73, 82, 83, 91, 100
COMMENTS
Side lengths of the 26 squares inside the smallest square (k = 274) that can be dissected into unequal squares (simple perfect squared square, SPSS) such that the largest square has a side length that is a hundred times that of the smallest.
FORMULA
Sum_{n=1..26} a(n)^2 = 274^2.
Number of connected components, not counting isolated vertices, of the blet graph for n coins.
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0
3, 2, 1, 7, 2, 5, 8, 8, 6, 50, 12, 30, 61, 62, 47, 417, 102, 303, 682, 696, 532
COMMENTS
The blet graph for n coins has one vertex for each binary heads/tails-sequence of length n. Two vertices are connected by an edge if there is a legal move between them in the game of blet, i.e., if one can be obtained from the other by replacing one occurrence of a triple THT with HTH. The binary sequences are circularly connected, so such a triple is allowed to start at one of the last two elements of the sequence and continue from the beginning.
A075273(n) is the size of the component containing (HT)^n in the blet graph for 2*n coins.
EXAMPLE
For n = 4, the blet graph has A007039(4) = 6 isolated vertices TTTT, TTHH, THHT, HTTH, HHTT, HHHH, and a(4) = 2 components of size at least 2: {TTTH, THTT, THHH, HTHT, HHTH} and {TTHT, THTH, HTTT, HTHH, HHHT}.
Size of the largest component of the blet graph for n coins.
+0
0
2, 5, 20, 8, 56, 56, 74, 180, 660, 220, 2288, 2002, 2942, 7280, 24752, 8568, 93024, 77520, 120920, 298452, 1009470
COMMENTS
See A381813 for the definition of the blet graph.
FORMULA
a(2*n) >= A075273(n) (the size of the component containing the vertex (HT)^n).
EXAMPLE
For n = 4, the blet graph has 2 components of maximum size a(4) = 5: {TTTH, THTT, THHH, HTHT, HHTH} and {TTHT, THTH, HTTT, HTHH, HHHT}.
Number of moves required to reach a position with the maximum number of heads in the game of blet with 2*n coins.
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0
1, 1, 2, 5, 3, 6, 11, 7, 10, 17, 11, 16, 25, 15, 22
COMMENTS
See A075273 or Rodriguez Villegas, Sadun, and Voloch (2002) for the definition of blet. Rodriguez Villegas, Sadun, and Voloch (2002) prove that the maximum number of heads achievable is A047206(n).
LINKS
Fernando Rodriguez Villegas, Lorenzo Sadun, and José Felipe Voloch, Blet: a mathematical puzzle, The American Mathematical Monthly 109 (2002), Issue 8, 729-740.
EXAMPLE
For n = 5, a(5) = 5 moves are needed to go from HTHTHTHTHT to a sequence with the maximum number A047206(5) = 8 of heads: HTHTHTHTHT -> THHTHTHTHH -> THHTHTTHTH -> THHHTHTHTH -> THHHHTHHTH -> HHHHHTHHHT.
Number of ways to place eight distinct positive integers on a square, four on the corners and four on the sides such that the sum of the three values on each side is n.
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0
3, 9, 23, 48, 84, 132, 226, 304, 456, 629, 849, 1079, 1501, 1794, 2317, 2898, 3519, 4195, 5288, 6049, 7282, 8605, 10017, 11494, 13662, 15273, 17680, 20231, 22842, 25573, 29432, 32353, 36463, 40791, 45216, 49803, 55926, 60759, 67295, 74071, 80929, 88034, 97283, 104713, 114359, 124383, 134526, 144957, 158110
COMMENTS
Solutions differing by only rotation or reflections are not counted separately.
FORMULA
Conjecture: a(n)= -2*a(n-1) -3*a(n-2) -2*a(n-3) +3*a(n-5) +6*a(n-6) +8*a(n-7) +9*a(n-8) +7*a(n-9) +3*a(n-10) -4*a(n-11) -10*a(n-12) -15*a(n-13) -16*a(n-14) -14*a(n-15) -8*a(n-16) +8*a(n-18) +14*a(n-19) +16*a(n-20) +15*a(n-21) +10*a(n-22) +4*a(n-23) -3*a(n-24) -7*a(n-25) -9*a(n-26) -8*a(n-27) -6*a(n-28) -3*a(n-29) +2*a(n-31) +3*a(n-32) +2*a(n-33) +a(n-34). - R. J. Mathar, Mar 04 2025 Conjecture: g.f. ( -x^12 *(3045*x^12 +2826*x^11 +2520*x^10 +2079*x^9 +1625*x^8 +1173*x^7 +793*x^6 +267*x^4 +481*x^5 +98*x^22 +236*x^21 +491*x^20 +796*x^19 +1231*x^18 +1673*x^17 +2187*x^16 +2580*x^15 +2906*x^14 +3038*x^13 +127*x^3 +3 +15*x +50*x^2) ) / ( (x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+1)^2 *(1+x)^3 *(1+x+x^2)^3 *(x-1)^5 ). - R. J. Mathar, Mar 04 2025
EXAMPLE
for n = 12, one of the a(12) = 3 possible arrangements of numbers is
2 6 4
9 5
1 8 3
The 3 numbers of each side sum to 12, eg. 2+9+1 = 12.
CROSSREFS
Cf. A380853 (order 3 perimeter magic triangles), A005994 (8 elements need not be distinct), A006325 (8 elements need not be distinct, rotat+flips count separately)
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