Displaying 51-60 of 209 results found.
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1, 42, 1764, 74088, 3111696, 130691232, 5489031744, 230539333248, 9682651996416, 406671383849472, 17080198121677824, 717368321110468608, 30129469486639681536, 1265437718438866624512, 53148384174432398229504
COMMENTS
Same as Pisot sequences E(1, 42), L(1, 42), P(1, 42), T(1, 42). Essentially same as Pisot sequences E(42, 1764), L(42, 1764), P(42, 1764), T(42, 1764). See A008776 for definitions of Pisot sequences. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 42-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
1, 43, 1849, 79507, 3418801, 147008443, 6321363049, 271818611107, 11688200277601, 502592611936843, 21611482313284249, 929293739471222707, 39959630797262576401, 1718264124282290785243, 73885357344138503765449
COMMENTS
Same as Pisot sequences E(1, 43), L(1, 43), P(1, 43), T(1, 43). Essentially same as Pisot sequences E(43, 1849), L(43, 1849), P(43, 1849), T(43, 1849). See A008776 for definitions of Pisot sequences. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 43-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011 Numbers n such that sigma(43*n) = 43*n + sigma(n). - Jahangeer Kholdi, Nov 24 2013
1, 44, 1936, 85184, 3748096, 164916224, 7256313856, 319277809664, 14048223625216, 618121839509504, 27197360938418176, 1196683881290399744, 52654090776777588736, 2316779994178213904384, 101938319743841411792896
COMMENTS
Same as Pisot sequences E(1, 44), L(1, 44), P(1, 44), T(1, 44). Essentially same as Pisot sequences E(44, 1936), L(44, 1936), P(44, 1936), T(44, 1936). See A008776 for definitions of Pisot sequences. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 44-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
1, 45, 2025, 91125, 4100625, 184528125, 8303765625, 373669453125, 16815125390625, 756680642578125, 34050628916015625, 1532278301220703125, 68952523554931640625, 3102863559971923828125, 139628860198736572265625
COMMENTS
Same as Pisot sequences E(1, 45), L(1, 45), P(1, 45), T(1, 45). Essentially same as Pisot sequences E(45, 2025), L(45, 2025), P(45, 2025), T(45, 2025). See A008776 for definitions of Pisot sequences. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 45-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
1, 46, 2116, 97336, 4477456, 205962976, 9474296896, 435817657216, 20047612231936, 922190162669056, 42420747482776576, 1951354384207722496, 89762301673555234816, 4129065876983540801536, 189937030341242876870656
COMMENTS
Same as Pisot sequences E(1, 46), L(1, 46), P(1, 46), T(1, 46). Essentially same as Pisot sequences E(46, 2116), L(46, 2116), P(46, 2116), T(46, 2116). See A008776 for definitions of Pisot sequences. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 46-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
MATHEMATICA
46^Range[0, 20] (* or *) NestList[46#&, 1, 20] (* Harvey P. Dale, Jan 15 2017 *)
1, 47, 2209, 103823, 4879681, 229345007, 10779215329, 506623120463, 23811286661761, 1119130473102767, 52599132235830049, 2472159215084012303, 116191483108948578241, 5460999706120583177327
COMMENTS
Same as Pisot sequences E(1, 47), L(1, 47), P(1, 47), T(1, 47). Essentially same as Pisot sequences E(47, 2209), L(47, 2209), P(47, 2209), T(47, 2209). See A008776 for definitions of Pisot sequences. The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 47-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011 Numbers n such that sigma(47*n) = 47*n + sigma(n). - Jahangeer Kholdi, Nov 24 2013
Powers of 48: a(n) = 48^n.
+0
11
1, 48, 2304, 110592, 5308416, 254803968, 12230590464, 587068342272, 28179280429056, 1352605460594688, 64925062108545024, 3116402981210161152, 149587343098087735296, 7180192468708211294208, 344649238497994142121984, 16543163447903718821855232
COMMENTS
Same as Pisot sequences E(1, 48), L(1, 48), P(1, 48), T(1, 48). Essentially same as Pisot sequences E(48, 2304), L(48, 2304), P(48, 2304), T(48, 2304). See A008776 for definitions of Pisot sequences. If X_1, X_2, ..., X_n is a partition of the set {1,2,...,2*n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1,2,..., 2*n}->{1,2,3,4,5,6,7} such that for fixed y_1,y_2,...,y_n in {1,2,3,4,5,6,7} we have f(X_i)<>{y_i}, (i=1,2,...,n). - Milan Janjic, May 24 2007 The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 48-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
PROG
(Python) for n in range(0, 20): print(48**n, end=', ') # Stefano Spezia, Nov 21 2018
Pisot sequence E(4,13): a(n) = floor( a(n-1)^2/a(n-2) + 1/2 ).
+0
3
4, 13, 42, 136, 440, 1424, 4609, 14918, 48285, 156284, 505844, 1637264, 5299328, 17152321, 55516872, 179691313, 581606398, 1882483892, 6093030640, 19721296176, 63831867233, 206604436042, 668716032329, 2164431415224, 7005609443657, 22675037578854
COMMENTS
According to David Boyd his last use (as of April, 2006) of his Pisot number finding program was to prove that in fact this sequence does not satisfy a linear recurrence. He remarks "This took a couple of years in background on various Sun workstations." - Gene Ward Smith, Apr 11 2006
REFERENCES
Cantor, D. G. "Investigation of T-numbers and E-sequences." In Computers in Number Theory, ed. AOL Atkin and BJ Birch, Acad. Press, NY (1971); pp. 137-140.
Cantor, D. G. (1976). On families of Pisot E-sequences. In Annales scientifiques de l'École Normale Supérieure (Vol. 9, No. 2, pp. 283-308).
FORMULA
It is known that this does not satisfy any linear recurrence [Boyd].
MATHEMATICA
nxt[{a_, b_}]:={b, Floor[b^2/a+1/2]}; NestList[nxt, {4, 13}, 30][[All, 1]] (* Harvey P. Dale, Jun 24 2018 *)
PROG
(PARI) pisotE(nmax, a1, a2) = {
a=vector(nmax); a[1]=a1; a[2]=a2;
for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));
a
}
CROSSREFS
See A008776 for definitions of Pisot sequences.
Pisot sequences E(4,7), P(4,7).
+0
2
4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572, 55405, 97229, 170625, 299426, 525456, 922111, 1618192, 2839729, 4983377, 8745217, 15346786, 26931732, 47261895, 82938844, 145547525, 255418101, 448227521, 786584466
COMMENTS
See A008776 for definitions of Pisot sequences.
FORMULA
a(n) = 2a(n-1) - a(n-2) + a(n-3) for n>=3. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016
PROG
(PARI) pisotE(nmax, a1, a2) = {
a=vector(nmax); a[1]=a1; a[2]=a2;
for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));
a
}
Pisot sequence E(14,23), a(n) = floor( a(n-1)^2/a(n-2)+1/2 ).
+0
2
14, 23, 38, 63, 104, 172, 284, 469, 775, 1281, 2117, 3499, 5783, 9558, 15797, 26109, 43152, 71320, 117875, 194819, 321989, 532170, 879548, 1453680, 2402581, 3970885, 6562912, 10846905, 17927308, 29629500, 48970390, 80936199, 133767942, 221086022, 365401668
FORMULA
It is known (Boyd, 1977) that this sequence does not satisfy a linear recurrence. - N. J. A. Sloane, Aug 07 2016
MATHEMATICA
RecurrenceTable[{a[1] == 14, a[2] == 23, a[n] == Floor[a[n-1]^2/a[n-2]+1/2]}, a, {n, 40}] (* Vincenzo Librandi, Aug 09 2016 *)
PROG
(PARI) pisotE(nmax, a1, a2) = {
a=vector(nmax); a[1]=a1; a[2]=a2;
for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));
a
}
(Python)
a, b = 14, 23
for i in range(1000):
c, d = divmod(b**2, a)
a, b = b, c + (0 if 2*d < a else 1)
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