A166917 -id:A166917 - cp's OEIS frontend

A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

318, 1071, 5040, 5985, 10710, 20400, 24225, 43350, 81600, 85425, 165750, 327360, 342705, 664950, 1309440, 1324785, 2629110, 5241600, 5303025, 10524150, 20966400, 21027825, 41973750, 83880960, 84126705, 167925750, 335523840

Offset: 0

Klaus Brockhaus, Sep 05 2002

290 is conjectured (cf. A066450) to be the smallest number such that the Reverse and Add! algorithm in base 4 does not lead to a palindrome. 318 (not 255 since 255 is a base 4 palindrome) is up to now the smallest number whose base 4 trajectory provably does not contain a palindrome. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 3 in {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 3 = 0.

			318 (decimal) = 10332 -> 10332 + 23301 = 100233 = 1071 (decimal).

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
K. S. Brown, Digit Reversal Sums Leading to Palindromes
Index entries for sequences related to Reverse and Add!

Cf. A058042 (trajectory of binary number 10110 (decimal 22)), A061561 (A058042 written in base 10), A066450 (conjectured minimal k so that the trajectory of k in base n does not lead to a palindrome).
Cf. A075253 (trajectory of 77 in base 2), A075420 (trajectory of n in base 4 (presumably) does not reach a palindrome), A075421 (trajectory of n in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n), A075299 (trajectory of 290 in base 4), A075466 (trajectory of 266718 in base 4), A075467 (trajectory of 270798 in base 4), A076247 (trajectory of 1059774 in base 4), A076248 (trajectory of 1059831 in base 4), A091675 (trajectory of n in base 4 (presumably) does not join the trajectory of any m < n).
Cf. A166912 (a(6*n)/3), A166913 (a(6*n+1)/3), A166914 (a(6*n+2)/240), A166915 (a(6*n+3)/15), A166916 (a(6*n+4)/30), A166917 (a(6*n+5)/240).

trajectory:=function(init, steps, base) a:=init; S:=[a]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a,base)),base); Append(~S, a); end for; return S; end function; trajectory(318, 26, 4);

NestWhileList[# + IntegerReverse[#, 4] &, 318,  # !=
IntegerReverse[#, 4] &, 1, 26] (* Robert Price, Oct 18 2019 *)

{m=318; stop=29; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0) = 318; a(1) = 1071; for n > 1 and n = 2 (mod 6): a(n) = 5*4^(2*k+5)-5*4^(k+2) where k = (n-2)/6; n = 3 (mod 6): a(n) = 5*4^(2*k+5)+55*4^(k+2)-15 where k = (n-3)/6; n = 4 (mod 6): a(n) = 10*4^(2*k+5)+30*4^(k+2)-10 where k = (n-4)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+5)-5*4^(k+2) where k = (n-5)/6; n = 0 (mod 6): a(n) = 20*4^(2*k+5)+235*4^(k+2)-15 where k = (n-6)/6; n = 1 (mod 6): a(n) = 40*4^(2*k+5)+150*4^(k+2)-10 where k = (n-7)/6.

G.f.: 3*(106 +357*x +1680*x^2 +1465*x^3 +1785*x^4 -1600*x^5 -1900*x^6 -3400*x^7 -6800*x^8 -9780*x^9 -9860*x^10 +6720*x^11 +10064*x^12 +11088*x^13) / ((1-x)*(1+x+x^2)*(1-2*x^3)*(1+2*x^3)*(1-4*x^3)).

Two comments added, g.f. edited, MAGMA program and cross-references added by Klaus Brockhaus, Oct 26 2009

A166912 a(n) = 20a(n-1) - 64a(n-2) - 225 for n > 2; a(0) = 106, a(1) = 8075, a(2) = 114235.

Original entry on oeis.org

106, 8075, 114235, 1767675, 28042235, 447713275, 7159562235, 114537594875, 1832539914235, 29320392212475, 469125289738235, 7506000693166075, 120095995320074235, 1921535862038855675, 30744573540292362235

Offset: 0

Klaus Brockhaus, Oct 27 2009

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n) = 3*a(n).

lim_{n -> infinity} a(n)/a(n-1) = 16.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (21, -84, 64).

Cf. A075153, A166913, A166914, A166915, A166916, A166917, A167120, A167121, A167122.

Join[{106},LinearRecurrence[{21,-84,64},{8075,114235,1767675},20]] (* Harvey P. Dale, Jun 07 2012 *)

m=15; v=concat([106, 8075, 114235], vector(m-3)); for(n=4, m, v[n]=20*v[n-1]-64*v[n-2]-225); v

a(n) = (1280*16^n + 940*4^n - 15)/3 for n > 0.
G.f.: (106 + 5849*x - 46436*x^2 + 40256*x^3)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 28 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3).
E.g.f.: (1/3)*(-15*exp(x) + 940*exp(4*x) + 1280*exp(16*x)) - 629. (End)

A166913 a(n) = 20a(n-1) - 64a(n-2) - 150 for n > 2; a(0) = 357, a(1) = 14450, a(2) = 221650.

Original entry on oeis.org

357, 14450, 221650, 3508050, 55975250, 894989650, 14317376850, 229068199250, 3665051866450, 58640672576850, 938250132084050, 15011999596762450, 240191983481869650, 3843071695444596050, 61489146966052263250

Offset: 0

Klaus Brockhaus, Oct 27 2009

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+1) = 3*a(n).

lim_{n -> infinity} a(n)/a(n-1) = 16.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (21, -84, 64).

Cf. A075153, A166912, A166914, A166915, A166916, A166917, A167120, A167121, A167122.

Join[{357},LinearRecurrence[{21,-84,64},{14450,221650,3508050},20]] (* Harvey P. Dale, Jun 18 2014 *)

m=15; v=concat([357, 14450, 221650], vector(m-3)); for(n=4, m, v[n]=20*v[n-1]-64*v[n-2]-150); v

a(n) = (2560*16^n + 600*4^n - 10)/3 for n > 0.
G.f.: (357 + 6953*x - 51812*x^2 + 44352*x^3)/((1-x)*(1-4*x)*(1-16*x)).
a(0)=357, a(1)=14450, a(2)=221650, a(3)=3508050, a(n)=21*a(n-1)- 84*a(n-2)+ 64*a(n-3). - Harvey P. Dale, Jun 18 2014
E.g.f.: (1/3)*(-10*exp(x) + 600*exp(4*x) + 2560*exp(16*x)) - 693. - G. C. Greubel, May 28 2016

A166915 a(n) = 20a(n-1) - 64a(n-2) - 45 for n>1; a(0) = 399, a(1) = 5695.

Original entry on oeis.org

399, 5695, 88319, 1401855, 22384639, 357974015, 5726863359, 91626930175, 1466019348479, 23456263438335, 375300030463999, 6004799749226495, 96076793034833919, 1537228676746182655, 24595658780694282239

Offset: 0

Klaus Brockhaus, Oct 27 2009

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+3) = 15*a(n).

lim_{n -> infinity} a(n)/a(n-1) = 16.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (21, -84, 64).

Cf. A075153, A166912, A166913, A166914, A166916, A166917, A006105, A167031, A167032, A167033.

LinearRecurrence[{21, -84, 64}, {399, 5695, 88319}, 50] (* G. C. Greubel, May 28 2016 *)

m=15; v=concat([399, 5695], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]-45); v

a(n) = (1024*16^n + 176*4^n - 3)/3.
G.f.: (399 - 2684*x + 2240*x^2)/((1-x)*(1-4*x)*(1-16*x)).
From G. C. Greubel, May 28 2016: (Start)
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3).
E.g.f.: (1/3)*(1024*exp(16*x) + 176*exp(4*x) - 3*exp(x)). (End)

A166916 a(n) = 20a(n-1) - 64a(n-2) - 15 for n > 1; a(0) = 357, a(1) = 5525.

Original entry on oeis.org

357, 5525, 87637, 1399125, 22373717, 357930325, 5726688597, 91626231125, 1466016552277, 23456252253525, 375299985724757, 6004799570269525, 96076792319006037, 1537228673882871125, 24595658769241036117

Offset: 0

Klaus Brockhaus, Oct 27 2009

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+4) = 30*a(n).

lim_{n -> infinity} a(n)/a(n-1) = 16.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (21, -84, 64).

Cf. A075153, A166912, A166913, A166914, A166915, A166917, A006105, A167031, A167032, A167033.

LinearRecurrence[{21,-84,64},{357,5525,87637},20] (* Harvey P. Dale, Sep 24 2012 *)

m=15; v=concat([357, 5525], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]-15); v

a(n) = (1024*16^n + 48*4^n - 1)/3.
G.f.: (357 - 1972*x + 1600*x^2)/((1-x)*(1-4*x)*(1-16*x)).
a(0)=357, a(1)=5525, a(2)=87637, a(n)=21*a(n-1)-84*a(n-2)+64*a(n-3). - Harvey P. Dale, Sep 24 2012
E.g.f.: (1/3)*(1024*exp(16*x) + 48*exp(4*x) - exp(x)). - G. C. Greubel, May 28 2016

A166914 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 21, a(1) = 340.

Original entry on oeis.org

21, 340, 5456, 87360, 1398016, 22369280, 357912576, 5726617600, 91625947136, 1466015416320, 23456247709696, 375299967549440, 6004799497568256, 96076792028200960, 1537228672719650816, 24595658764588154880

Offset: 0

Klaus Brockhaus, Oct 27 2009

nonn

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+2) = 240*a(n).

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A075153, A166912, A166913, A166915, A166916, A166917, A166927, A166965, A166984.

[Binomial(4^(n+3), 2)/96: n in [0..30]]; // G. C. Greubel, Oct 02 2024

CoefficientList[Series[(21-80x)/((1-4x)(1-16x)),{x,0,20}],x]  (* or *) LinearRecurrence[{20,-64},{21,340},20] (* Harvey P. Dale, Feb 23 2011 & Mar 30 2012 *)

{m=15; v=concat([21, 340], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]); v}

A166914=BinaryRecurrenceSequence(20,-64,21,340)
[A166914(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

a(n) = (64*16^n - 4^n)/3.
G.f.: (21 - 80*x)/((1-4*x)*(1-16*x)).
Limit_{n -> infinity} a(n)/a(n-1) = 16.
From G. C. Greubel, May 28 2016: (Start)
a(n) = 20*a(n-1) - 64*a(n-2).
E.g.f.: (1/3)*(-exp(4*x) + 64*exp(16*x)). (End)

A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

Original entry on oeis.org

1, 20, 336, 5440, 87296, 1397760, 22368256, 357908480, 5726601216, 91625881600, 1466015154176, 23456246661120, 375299963355136, 6004799480791040, 96076791961092096, 1537228672451215360, 24595658763514413056, 393530540233410478080, 6296488643803287126016

Offset: 0

Klaus Brockhaus, Oct 26 2009

Partial sums of A166965.

First differences of A006105. - Klaus Purath, Oct 15 2020

Seiichi Manyama, Table of n, a(n) for n = 0..830 (terms 0..200 from Vincenzo Librandi)
E. Saltürk and I. Siap, Generalized Gaussian Numbers Related to Linear Codes over Galois Rings, European Journal of Pure and Applied Mathematics, Vol. 5, No. 2, 2012, 250-259; ISSN 1307-5543. - From _N. J. A. Sloane_, Oct 23 2012
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A000302, A002450, A002452, A002453, A006105, A079598, A083584.

Cf. A115490, A166914, A166917, A166927, A166965, A307695.

[n le 2 select 19*n-18 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];

LinearRecurrence[{20,-64},{1,20},30] (* Harvey P. Dale, Jul 04 2012 *)

a(n) = (4*16^n - 4^n)/3 \\ Charles R Greathouse IV, Jun 21 2022

A166984=BinaryRecurrenceSequence(20,-64,1,20)
[A166984(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

a(n) = (4*16^n - 4^n)/3.
G.f.: 1/((1-4*x)*(1-16*x)).
Limit_{n -> oo} a(n)/a(n-1) = 16.
a(n) = A115490(n+1)/3.
Sum_{n>=0} a(n) x^(2*n+4)/(2*n+4)! = ( sinh(x) )^4/4!. - Robert A. Russell, Apr 03 2013
From Klaus Purath, Oct 15 2020: (Start)
a(n) = A002450(n+1)*(A002450(n+2) - A002450(n))/5.
a(n) = (A083584(n+1)^2 - A083584(n)^2)/80. (End)
a(n) = (A079598(n) - A000302(n))/24. - César Aguilera, Jun 21 2022
a(n) = 16*a(n-1) + 4^n with a(0) = 1. - Nadia Lafreniere, Aug 08 2022
E.g.f.: (4/3)*exp(10*x)*sinh(6*x + log(2)). - G. C. Greubel, Oct 02 2024

A166927 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 18.

Original entry on oeis.org

1, 18, 296, 4768, 76416, 1223168, 19572736, 313171968, 5010784256, 80172679168, 1282763390976, 20524216352768, 328387470032896, 5254199554080768, 84067192999510016, 1345075088529031168, 21521201418611982336

Offset: 0

Klaus Brockhaus, Oct 23 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (20, -64).

Cf. A166914, A166917.

[ n le 2 select 17*n-16 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];

LinearRecurrence[{20, -64}, {1, 18}, 50] (* G. C. Greubel, May 28 2016 *)

a(n) = (7*16^n - 4^n)/6.
G.f.: (1 - 2*x)/((1-4*x)*(1-16*x)).
E.g.f.: (1/6)*(7*exp(16*x) - exp(4*x)). - G. C. Greubel, May 28 2016

cp's OEIS Frontend

A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

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A166912 a(n) = 20*a(n-1) - 64*a(n-2) - 225 for n > 2; a(0) = 106, a(1) = 8075, a(2) = 114235.

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A166913 a(n) = 20*a(n-1) - 64*a(n-2) - 150 for n > 2; a(0) = 357, a(1) = 14450, a(2) = 221650.

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A166915 a(n) = 20*a(n-1) - 64*a(n-2) - 45 for n>1; a(0) = 399, a(1) = 5695.

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A166916 a(n) = 20*a(n-1) - 64*a(n-2) - 15 for n > 1; a(0) = 357, a(1) = 5525.

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A166914 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 21, a(1) = 340.

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A166984 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 1, a(1) = 20.

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A166912 a(n) = 20a(n-1) - 64a(n-2) - 225 for n > 2; a(0) = 106, a(1) = 8075, a(2) = 114235.

A166913 a(n) = 20a(n-1) - 64a(n-2) - 150 for n > 2; a(0) = 357, a(1) = 14450, a(2) = 221650.

A166915 a(n) = 20a(n-1) - 64a(n-2) - 45 for n>1; a(0) = 399, a(1) = 5695.

A166916 a(n) = 20a(n-1) - 64a(n-2) - 15 for n > 1; a(0) = 357, a(1) = 5525.

A166914 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 21, a(1) = 340.

A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

A166927 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 18.