Displaying 91-100 of 209 results found.
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1, 34, 1156, 39304, 1336336, 45435424, 1544804416, 52523350144, 1785793904896, 60716992766464, 2064377754059776, 70188843638032384, 2386420683693101056, 81138303245565435904, 2758702310349224820736
COMMENTS
Same as Pisot sequences E(1, 34), L(1, 34), P(1, 34), T(1, 34). Essentially same as Pisot sequences E(34, 1156), L(34, 1156), P(34, 1156), T(34, 1156). 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 34-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
1, 35, 1225, 42875, 1500625, 52521875, 1838265625, 64339296875, 2251875390625, 78815638671875, 2758547353515625, 96549157373046875, 3379220508056640625, 118272717781982421875, 4139545122369384765625
COMMENTS
Same as Pisot sequences E(1, 35), L(1, 35), P(1, 35), T(1, 35). Essentially same as Pisot sequences E(35, 1225), L(35, 1225), P(35, 1225), T(35, 1225). 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} such that for fixed y_1,y_2,...,y_n in {1,2,3,4,5,6} 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 35-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
1, 38, 1444, 54872, 2085136, 79235168, 3010936384, 114415582592, 4347792138496, 165216101262848, 6278211847988224, 238572050223552512, 9065737908494995456, 344498040522809827328, 13090925539866773438464, 497455170514937390661632
COMMENTS
Same as Pisot sequences E(1, 38), L(1, 38), P(1, 38), T(1, 38). Essentially same as Pisot sequences E(38, 1444), L(38, 1444), P(38, 1444), T(38, 1444). 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 38-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011. [See A000244 for a proof.]
1, 40, 1600, 64000, 2560000, 102400000, 4096000000, 163840000000, 6553600000000, 262144000000000, 10485760000000000, 419430400000000000, 16777216000000000000, 671088640000000000000, 26843545600000000000000
COMMENTS
Same as Pisot sequences E(1, 40), L(1, 40), P(1, 40), T(1, 40). Essentially same as Pisot sequences E(40, 1600), L(40, 1600), P(40, 1600), T(40, 1600). 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 40-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
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, 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 *)
Pisot sequence E(4,13): a(n) = floor( a(n-1)^2/a(n-2) + 1/2 ).
+10
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(3,7), P(3,7).
+10
3
3, 7, 16, 37, 86, 200, 465, 1081, 2513, 5842, 13581, 31572, 73396, 170625, 396655, 922111, 2143648, 4983377, 11584946, 26931732, 62608681, 145547525, 338356945, 786584466, 1828587033, 4250949112, 9882257736, 22973462017, 53406819691, 124155792775
FORMULA
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (3-2*x+x^2)/(1-3*x+2*x^2-x^3). - Colin Barker, Feb 19 2012 Since Pisot (1938) showed that E(3,k) always satisfies a linear recurrence, presumably it would not be difficult to prove that the above conjectures are correct. - N. J. A. Sloane, Jul 30 2016 Theorem: a(n) = 3 a(n - 1) - 2 a(n - 2) + a(n - 3) for n>=3. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016
PROG
(Magma) XY:=[3, 7]; [n le 2 select XY[n] else Ceiling(Self(n-1)^2/Self(n-2)-1/2): n in [1..32]]; // Klaus Brockhaus, Nov 17 2010 (Magma) a:=1; b:=1; c:=1; S:=[]; for n in [1..32] do a+:=b; b+:=c; c+:=a; Append(~S, c); end for; S; // Klaus Brockhaus, Nov 17 2010 (PARI) Vec((3-2*x+x^2)/(1-3*x+2*x^2-x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020
CROSSREFS
See A008776 for definitions of Pisot sequences.
Pisot sequence T(2,5), a(n) = floor(a(n-1)^2/a(n-2)).
+10
3
2, 5, 12, 28, 65, 150, 346, 798, 1840, 4242, 9779, 22543, 51967, 119796, 276157, 636604, 1467515, 3382951, 7798460, 17977197, 41441465, 95531857, 220222323, 507661769, 1170274058, 2697743762, 6218903474, 14335965099, 33047609788, 76182140871, 175616894078
FORMULA
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) - a(n-6) (holds at least up to n = 1000 but is not known to hold in general).
MATHEMATICA
RecurrenceTable[{a[0] == 2, a[1] == 5, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 04 2016 *)
PROG
(Magma) Iv:=[2, 5]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..40]]; // Bruno Berselli, Feb 04 2016 (PARI) pisotT(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]));
a
}
CROSSREFS
See A008776 for definitions of Pisot sequences.
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
COMMENTS
Pisot sequences E(4,5), P(4,5), T(4,5).
FORMULA
a(n) = 2*a(n-1) - a(n-2).
G.f.: (4 - 3*x)/(1 - x)^2.
E.g.f.: (4 + x)*exp(x).
CROSSREFS
See A008776 for definitions of Pisot sequences.
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