Displaying 81-90 of 208 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, 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
Pisot sequence E(8,10), a(n) = floor( a(n-1)^2/a(n-2) + 1/2 ).
+10
4
8, 10, 13, 17, 22, 28, 36, 46, 59, 76, 98, 126, 162, 208, 267, 343, 441, 567, 729, 937, 1204, 1547, 1988, 2555, 3284, 4221, 5425, 6972, 8960, 11515, 14799, 19020, 24445, 31417, 40377, 51892, 66691, 85711, 110155, 141570, 181944, 233832, 300518, 386222, 496368, 637926
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.
FORMULA
It is not true that a(n) = a(n-1) + a(n-6), which holds just for n <= 37 (see A275627). E.g. a(38) = 110155 = 85711 + 24445 - 1 = a(37) + a(32) - 1. Sequence is believed to be non-recurring.
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.
5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131
COMMENTS
Values of n such that a regular polygon with n sides can be formed by tying knots in a strip of paper. - Robert A. J. Matthews (rajm(AT)compuserve.com)
These polygons fill in many of the gaps left by the Greeks, who were restricted to compass and ruler. Specifically, they make possible construction of the regular 7-sided heptagon, 9-sided nonagon, 11-gon and 13-gon. The 14-gon becomes the first to be impossible by either ruler, compass or knotting.
Continued fraction expansion of 2/(exp(2)-7). - Thomas Baruchel, Nov 04 2003 Sun conjectures that any member of this sequence is of the form m^2 + m + p, where p is prime. Blanco-Chacon, McGuire, & Robinson prove that the primes of this form have density 1. - Charles R Greathouse IV, Jun 20 2019
REFERENCES
F. V. Morley, Proc. Lond. Math. Soc., Jun 1923
F. V. Morley, "Inversive Geometry" (George Bell, 1933; reprinted Chelsea Publishing Co. 1954)
FORMULA
a(n) = 2*n + 3.
G.f.: x*(5-3*x)/(1-2*x+x^2). a(n) = 2*a(n-1)-a(n-2). - Colin Barker, Jan 31 2012
CROSSREFS
Subsequence of A005408. See A008776 for definitions of Pisot sequences.
7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139
FORMULA
a(n) = 2n+7. a(n) = 2a(n-1) - a(n-2).
MATHEMATICA
T[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2]]; Table[a[n], {n, 0, z}]]; T[7, 9, 66] (* Michael De Vlieger, Aug 08 2016 *)
PROG
(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
}
3, 5, 8, 12, 18, 27, 40, 59, 87, 128, 188, 276, 405, 594, 871, 1277, 1872, 2744, 4022, 5895, 8640, 12663, 18559, 27200, 39864, 58424, 85625, 125490, 183915, 269541, 395032, 578948, 848490, 1243523, 1822472, 2670963, 3914487, 5736960, 8407924, 12322412, 18059373
FORMULA
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (3-x+x^2-2*x^3)/(1-x)/(1-x-x^3). [ Colin Barker, Feb 19 2012]
MATHEMATICA
RecurrenceTable[{a[0] == 3, a[1] == 5, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 50}] (* Bruno Berselli, Feb 04 2016 *)
PROG
(Magma) Iv:=[3, 5]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..50]]; // Bruno Berselli, Feb 04 2016
CROSSREFS
See A008776 for definitions of Pisot sequences.
2, 9, 40, 178, 792, 3524, 15680, 69768, 310432, 1381264, 6145920, 27346208, 121676672, 541399104, 2408949760, 10718597248, 47692288512, 212206348544, 944209971200, 4201252581888, 18693430269952, 83176226243584, 370091765514240, 1646719514544128
FORMULA
Pisot sequence P(x, y): a(0) = x, a(1) = y, a(n) = roundDown(a(n-1)^2/a(n-2)) = ceiling(a(n-1)^2/a(n-2) - 1/2).
Appears to satisfy a(n) = 4*a(n-1) + 2*a(n-2).
MATHEMATICA
RecurrenceTable[{a[0] == 2, a[1] == 9, a[n] == Ceiling[a[n - 1]^2/a[n - 2]-1/2]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 04 2016 *)
PROG
(PARI) lista(nn) = {print1(x = 2, ", ", y = 9, ", "); for (n=1, nn, z = ceil(y^2/x -1/2); print1(z, ", "); x = y; y = z; ); } \\ Michel Marcus, Feb 04 2016 (Magma) Iv:=[2, 9]; [n le 2 select Iv[n] else Ceiling(Self(n-1)^2/Self(n-2)-1/2): n in [1..30]]; // Bruno Berselli, Feb 04 2016
CROSSREFS
See A008776 for definitions of Pisot sequences.
3, 10, 34, 116, 396, 1352, 4616, 15760, 53808, 183712, 627232, 2141504, 7311552, 24963200, 85229696, 290992384, 993510144, 3392055808, 11581202944, 39540700160, 135000394752, 460920178688, 1573679925248, 5372879343616, 18344157523968, 62630871408640
FORMULA
a(n) = 4*a(n-1) - 2*a(n-2) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (3-2*x)/(1-4*x+2*x^2). [ Colin Barker, Feb 21 2012]
MATHEMATICA
RecurrenceTable[{a[0] == 3, a[1] == 10, a[n] == Ceiling[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 05 2016 *)
PROG
(Magma) Lxy:=[3, 10]; [n le 2 select Lxy[n] else Ceiling(Self(n-1)^2/Self(n-2)): n in [1..30]]; // Bruno Berselli, Feb 05 2016 (PARI) pisotL(nmax, a1, a2) = {
a=vector(nmax); a[1]=a1; a[2]=a2;
for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]));
a
}
CROSSREFS
It appears that this is a subsequence of A007052. See A008776 for definitions of Pisot sequences.
-1, -2, -2, 10, 94, 538, 2638, 12010, 52414, 222778, 930478, 3840010, 15714334, 63920218, 258869518, 1045044010, 4208873854, 16921588858, 67944635758, 272553384010, 1092538058974, 4377125804698, 17529423925198, 70180457820010, 280910117637694, 1124205329623738, 4498515895713838
FORMULA
G.f.: -(1-5*x)/((1-3*x)*(1-4*x)).
a(n) = 7*a(n-1) - 12*a(n-2) for n > 1. (End)
E.g.f.: exp(3*x)*(exp(x) - 2).
MATHEMATICA
Table[4^n - 2 3^n, {n, 0, 30}] (* or *) CoefficientList[Series[-(1 - 5 x) / ((1 - 3 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 20 2013 *)
Pisot sequence E(10,219): a(n) = nearest integer to a(n-1)^2 / a(n-2), starting 10, 219, ... Deviates from A007698 at 1403rd term.
(Formerly M4747)
+10
3
10, 219, 4796, 105030, 2300104, 50371117, 1103102046, 24157378203, 529034393290, 11585586272312, 253718493496142, 5556306986017175, 121680319386464850, 2664737596978110299, 58356408797678883616, 1277975907130111287030, 27987027523701766535844
COMMENTS
a(n+1)/a(n) -> 21.8994954189323... which is very near to a root of 11*x^4 - 18*x^3 + 3*x^2 - 22*x + 1. This is only an approximation since the sequence does not satisfy any known recurrence. The difference between the root of the equation and the real value is 1.1357748460267988*10^(-1877). - Simon Plouffe, Feb 26 2012
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Wroblewski, personal communication.
EXAMPLE
a(1403) is
1943708471314943308059445452657010940487450311864066842732596790939279068191\
168021439671095304800683519756645143142801766345115405789059172602192426\
024357604507643919310528104572431148473422703387902120314696316682603735\
267692111685622339243356242260056059336217912799059786079481997806631913\
955493134941095358770263918313025848373581726054928149011342047774528154\
248287433782463237576416857026309254788755903742777139477594456385042020\
381315538604379941789590322666368814892780385046811477655985825537894431\
894143994712043942268394043823543450207513886190799409707531632679517052\
869104335940723488960240770470438470434329535343866330429132657179201894\
810776495469936998716229270764904917198741365340242782600909003168195629\
553831589770365472687705483796661474238920271726070390505179067208859490\
817765494636249793643314197295308500154814706778732034270622318621910522\
030142040283435992446877395852252468365235219657327211742475429216859612\
898009146799397834207588995393930733511691021384920256724554594857336855\
550714963221355049079118765001875374835520434138927516201876958496564958\
805765202364476313555615826884516631224599151532590504446541236893625713\
832620042439077419006777861484860386048975978762433100742439296700782881\
889486380714070148887484098410694218233687263042755465493793927981497199\
521026920386200848153568287674310343346371498689283968784694184354766679\
111870702565268681491357079215569781219694309328629243757829281537544222\
305623084962270299300645420182502879046175714261919397771509700298570157\
891004711917373029290386303109701959096841328964650889891682871446978568\
692922345060182670103628056600403977432916893829069098732545636174794446\
362475483205590674696119315488543667867514676786440758126850754300452964\
368265133082563202580908171650074203739290735941387946242005524276316413\
356912394816492851593842390985938520048268384592849898513622096090183587\
01821
A007698(1403) = 22*a(1402) - 3*a(1401) + 18*a(1400) - 11*a(1399) = a(1403) + 1. - M. F. Hasler, Feb 09 2014. This is one more than the number displayed above.
MAPLE
a := proc(n) options remember; if n = 1 then RETURN(10); elif n = 2 then RETURN(219); else RETURN(round(a(n-1)^2/a(n-2))); fi; end:
MATHEMATICA
a = {10, 219}; Do[AppendTo[a, Round[a[[k - 1]]^2/a[[k - 2]]]], {k, 3, 17}]; a (* Michael De Vlieger, Feb 08 2016 *) nxt[{a_, b_}]:={b, Round[b^2/a]}; NestList[nxt, {10, 219}, 20][[All, 1]] (* Harvey P. Dale, Jan 01 2022 *)
PROG
(PARI) A007699(n, a=10, b=100/219)=for(k=2, n, a=(a^2+b\2)\(b+0*b=a)); a \\ M. F. Hasler, Feb 09 2014 (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.
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