John Lien - cp's OEIS frontend

A165221 The Padovan sequence analog of the Fibonacci "rabbit" constant binary expansion. Starting with 0 and using the transitions 0->1,1->10,10->01 the subsequences 0,1,10,01,110,1001,01110,1101001,100101110,011101101001... are formed where each subsequence has P sub n ones and length P sub (n-1) binary digits, where P sub n is the n-th Padovan number. This sequence is the concatenation of all the subsequences. Also note that the n-th subsequence is the concatenation of the n-th-3 and n-th-2 subsequences.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1

Offset: 1

John Lien, Sep 08 2009

nonn

Ian Stewart, Tales of a Neglected Number
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
E. Wilson, The Scales of Mt. Meru (1999)

A165263 A sequence similar to the Fibonacci rabbit sequence for the Padovan sequence.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0

Offset: 1

John Lien, Sep 12 2009

nonn

Starting with 0,1 and using the maps 0->1 1->.10 and 10->01 Gives the subsequences 0,1,10,01,1.10,10.01,01.1.10,1.10.10.01 etc. The n-th subsequence has a 1 count equal to P(n) where P is the n-th Padovan sequence number (A000931) and a digit length P(n+2). This sequence represents the binary number formed by concatenating these sebsequences. Similar to how the Fibonacci rabbit constant is formed by the maps 0->1 1->10.

A126772 Padovan factorials: a(n) is the product of the first n terms of the Padovan sequence. Similar to the Fibonacci factorial.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 48, 240, 1680, 15120, 181440, 2903040, 60963840, 1706987520, 63158538240, 3094768373760, 201159944294400, 17299755209318400, 1972172093862297600, 297797986173206937600, 59559597234641387520000

Offset: 1

John Lien, Feb 17 2007

Ian Stewart, Tales of a Neglected Number
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Eric Weisstein's World of Mathematics, Padovan Sequence
E. Wilson, The Scales of Mt. Meru

Cf. A000931, A003266, A060006, A253924.

From R. J. Mathar, Sep 14 2010: (Start)
A000931 := proc(n) option remember; if n = 0 then 1; elif n <=2 then 0; else procname(n-2)+procname(n-3) ; end if; end proc:
A126772 := proc(n) mul( A000931(i),i=5..n+4) ; end proc: seq(A126772(n),n=1..40) ; (End)

Rest[FoldList[Times,1,LinearRecurrence[{0,1,1},{1,1,1},30]]] (* Harvey P. Dale, Apr 29 2013 *)

a(n) ~ c * d^(n/2) * r^(n^2/2), where r = 1.324717957244746... (see A060006) is the root of the equation r^3 = r + 1, d = 0.393641282401116385386658448446561... is the root of the equation 1 + 7*d + 184*d^2 - 529*d^3 = 0, c = 1.25373683131537208838997864311903035079685338006712312402418098138010834953... (see A253924). - Vaclav Kotesovec, Jan 26 2015

More terms from R. J. Mathar, Sep 14 2010

A100283 a(n) = floor(p(n+1)) - floor(p(n)) - 1 where p = Padovan plastic number = 1.324718... (cf. A060006).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 0

John Lien, Dec 28 2004

nonn

A rabbit-like sequence generated by the Padovan plastic number.

The well-known rabbit sequence is generated by taking the difference between the nearest integer less than phi*(n+1) minus the nearest integer less than phi*(n). If this value is 2, then the n-th rabbit sequence value is one. If this value is 1, the n-th rabbit sequence is 0. The sequence given is calculated in a similar manner, but using the plastic constant = 1.324717957244... instead of phi = 1.618033... = (1+sqrt(5))/2. It is 0001 followed by 11 copies of 001 followed by 0001 followed by 12 copies of 001 followed by 11 copies of 001 followed by similar patterns of 0001 followed by n copies of 001 where n is 11 or 12.

Midhat J. Gazale, Gnomon: From Pharaohs to Fractals, Princeton University Press, 1999

Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.

Cf. A000931, A005614, A060006.

p=(sqrt(23/108)+.5)^(1/3) + (abs( sqrt(23/108) -.5))^(1/3); for(n = 0, n = 200, r = floor(p*(n+1)) - floor(p*n) -1; print (r ))

Partially edited by N. J. A. Sloane, Jun 13 2007

A100538 Volume of the 3-dimensional box of sides of length equal to consecutive Padovan numbers (A000931). These boxes form a spiral in three dimensions similar to the spiral of Fibonacci boxes in two dimensions.

Original entry on oeis.org

1, 2, 4, 12, 24, 60, 140, 315, 756, 1728, 4032, 9408, 21756, 50764, 117845, 273910, 637260, 1480404, 3442800, 8003000, 18603000, 43251975, 100540440, 233735040, 543371136, 1263161472, 2936540824, 6826574552, 15869878969, 36893076570

Offset: 1

John Lien, Nov 27 2004

nonn

a(n)^(1/3) rounded to the nearest integer equals A000931(n+5). - Peter M. Chema, Apr 24 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
I. Stewart, Tales of a Neglected Number
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Index entries for linear recurrences with constant coefficients, signature (1,2,3,-2,4,-4,-1,-1,0,-1).

Cf. A000931.

LinearRecurrence[{1, 2, 3, -2, 4, -4, -1, -1, 0, -1}, {1, 2, 4, 12, 24, 60, 140, 315, 756, 1728}, 50] (* Vincenzo Librandi, Apr 24 2017 *)

For large n a(n+1) -> a(n) * p^3 where p is the plastic number = 1.324718... a(n+1) = a(n)+ (a(n)/P(n))*P(n+1 ) where P are the Padovan numbers (A000931) starting 1, 1, 1, 2, 2, 3, 4, 5, 7, etc.

a(n) = +a(n-1) +2*a(n-2) +3*a(n-3) -2*a(n-4) +4*a(n-5) -4*a(n-6) -a(n-7) -a(n-8) -a(n-10) = A000931(n+4)*A000931(n+5)*A000931(n+6). G.f.: x*(1+x+x^3) / ( (x-1)*(x^3-2*x^2+3*x-1)*(x^6+3*x^5+5*x^4+5*x^3+5*x^2+3*x+1) ). - R. J. Mathar, Sep 14 2010

More terms from R. J. Mathar, Sep 14 2010

A100891 Prime Padovan numbers.

Original entry on oeis.org

2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473, 1558877695141608507751098941899265975115403618621811951868598809164180630185566719

Offset: 1

John Lien, Jan 10 2005

nonn

Next term corresponds to Padovan(1262) and has 154 decimal digits.

Midhat J. Gazale, "Gnomon: From Pharaohs to Fractals", Princeton University Press, 1999.

Hugo Pfoertner, Table of n, a(n) for n = 1..20
Ian Stewart, Tales of a Neglected Number.
Eric Weisstein's World of Mathematics, Padovan Sequence

Cf. A000931, A122498, A291216, A291673.

Indices of prime Padovan numbers are A112882.

Rest[Select[LinearRecurrence[{0,1,1},{1,1,2},1000],PrimeQ]] (* Harvey P. Dale, Mar 31 2012 *)

More terms from Robert G. Wilson v, Jan 14 2005

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A165263 A sequence similar to the Fibonacci rabbit sequence for the Padovan sequence.

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A126772 Padovan factorials: a(n) is the product of the first n terms of the Padovan sequence. Similar to the Fibonacci factorial.

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A100283 a(n) = floor(p*(n+1)) - floor(p*(n)) - 1 where p = Padovan plastic number = 1.324718... (cf. A060006).

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A100538 Volume of the 3-dimensional box of sides of length equal to consecutive Padovan numbers (A000931). These boxes form a spiral in three dimensions similar to the spiral of Fibonacci boxes in two dimensions.

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A100891 Prime Padovan numbers.

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Links

Crossrefs

Programs

Mathematica

Extensions

A100283 a(n) = floor(p(n+1)) - floor(p(n)) - 1 where p = Padovan plastic number = 1.324718... (cf. A060006).