A166912 -id:A166912 - 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

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)

A166917 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 85, a(1) = 1364.

Original entry on oeis.org

85, 1364, 21840, 349504, 5592320, 89478144, 1431654400, 22906486784, 366503854080, 5864061927424, 93824991887360, 1501199874392064, 24019198007050240, 384307168179912704, 6148914691147038720, 98382635059426361344, 1574122160955116748800, 25185954575299047849984

Offset: 0

Klaus Brockhaus, Oct 27 2009

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+5) = 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, A166914, A166915, A166916, A166927, A166965, A166984.

[Binomial(4^(n+4), 2)/384: n in [0..30]]; // G. C. Greubel, Oct 02 2024

LinearRecurrence[{20,-64}, {85, 1364}, 50] (* G. C. Greubel, May 28 2016 *)

{m=15; v=concat([85, 1364], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]); v}

A166917=BinaryRecurrenceSequence(20,-64,85,1364)
[A166917(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

a(n) = (256*16^n - 4^n)/3.
G.f.: (85 - 336*x)/((1-4*x)*(1-16*x)).
Limit_{n -> infinity} a(n)/a(n-1) = 16.
E.g.f.: (1/3)*(256*exp(16*x) - exp(4*x)). - G. C. Greubel, May 28 2016

A167120 a(n) = 20a(n-1) - 64a(n-2) + 1 for n > 2; a(0) = 1, a(1) = 22, a(2) = 376.

Original entry on oeis.org

1, 22, 376, 6113, 98197, 1572709, 25169573, 402738085, 6443909029, 103102943141, 1649648684965, 26394385338277, 422310190927781, 6756963156905893, 108111410918739877, 1729782576332820389, 27676521227857055653

Offset: 0

Klaus Brockhaus, Oct 27 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 (21,-84,64).

Cf. A166912, A166913, A167121, A167122.

[ n le 2 select 21*n-20 else n eq 3 select 376 else 20*Self(n-1)-64*Self(n-2)+1: n in [1..17] ];

CoefficientList[Series[(1 + x - 2*x^2 + x^3)/((1 - x)*(1 - 4*x)*(1 - 16*x)), {x, 0, 10}], x] (* G. C. Greubel, Jun 04 2016 *)
LinearRecurrence[{21,-84,64},{1,22,376,6113},20] (* Harvey P. Dale, Jul 13 2018 *)

a(n) = (4321*16^n - 1460*4^n + 64)/2880, for n > 0.
G.f.: (1 + x - 2*x^2 + x^3)/((1-x)*(1-4*x)*(1-16*x)).
E.g.f.: (1/2880)*(-45 + 64*exp(x) - 1460*exp(4*x) + 4321*exp(16*x)). - G. C. Greubel, Jun 04 2016
a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3). - Wesley Ivan Hurt, Aug 04 2025

A167121 a(n) = 20a(n-1) - 64a(n-2) + 2 for n > 2; a(0) = 1, a(1) = 22, a(2) = 377.

Original entry on oeis.org

1, 22, 377, 6134, 98554, 1578506, 25262666, 404228938, 6467768138, 103484710730, 1655757053770, 26492119588682, 423873940332362, 6781983152971594, 108511730878160714, 1736187695773032266, 27779003139258359626

Offset: 0

Klaus Brockhaus, Oct 27 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 (21,-84,64).

Cf. A166912, A166913, A167120, A167122.

[ n le 2 select 21*n-20 else n eq 3 select 377 else 20*Self(n-1)-64*Self(n-2)+2: n in [1..17] ];

CoefficientList[Series[(1 + x - x^2 + x^3)/((1-x)*(1-4*x)*(1-16*x)), {x, 0, 10}], x] (* G. C. Greubel, Jun 04 2016 *)
Join[{1},RecurrenceTable[{a[1]==22,a[2]==377,a[n]==20a[n-1]-64a[n-2]+2}, a,{n,20}]] (* Harvey P. Dale, Apr 01 2019 *)

a(n) = (4337*16^n - 1540*4^n + 128)/2880, for n > 0.
G.f.: (1 + x - x^2 + x^3)/((1-x)*(1-4*x)*(1-16*x)).
E.g.f.: (1/2880)*(-45 + 128*exp(x) - 1540*exp(4*x) + 4337*exp(16*x)). - G. C. Greubel, Jun 04 2016

A167122 a(n) = 20a(n-1) - 64a(n-2) + 3 for n > 2; a(0) = 1, a(1) = 22, a(2) = 378.

Original entry on oeis.org

1, 22, 378, 6155, 98911, 1584303, 25355759, 405719791, 6491627247, 103866478319, 1661865422575, 26589853839087, 425437689736943, 6807003149037295, 108912050837581551, 1742592815213244143, 27881485050659663599

Offset: 0

Klaus Brockhaus, Oct 27 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 (21, -84, 64).

Cf. A166912, A166913, A167120, A167121.

[ n le 2 select 21*n-20 else n eq 3 select 378 else 20*Self(n-1)-64*Self(n-2)+3: n in [1..17] ];

CoefficientList[Series[(1 + x + x^3)/((1-x)*(1-4*x)*(1-16*x)), {x, 0, 10}], x] (* G. C. Greubel, Jun 04 2016 *)
LinearRecurrence[{21,-84,64},{1,22,378,6155},20] (* Harvey P. Dale, Sep 26 2023 *)

a(n) = (1451*16^n - 540*4^n + 64)/960, for n > 0.
G.f.: (1 + x + x^3)/((1-x)*(1-4*x)*(1-16*x)).
E.g.f.: (1/960)*(-15 + 64*exp(x) - 540*exp(4*x) + 1451*exp(16*x)). - G. C. Greubel, Jun 04 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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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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A166917 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 85, a(1) = 1364.

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A167120 a(n) = 20*a(n-1) - 64*a(n-2) + 1 for n > 2; a(0) = 1, a(1) = 22, a(2) = 376.

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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.

A166917 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 85, a(1) = 1364.

A167120 a(n) = 20a(n-1) - 64a(n-2) + 1 for n > 2; a(0) = 1, a(1) = 22, a(2) = 376.

A167121 a(n) = 20a(n-1) - 64a(n-2) + 2 for n > 2; a(0) = 1, a(1) = 22, a(2) = 377.

A167122 a(n) = 20a(n-1) - 64a(n-2) + 3 for n > 2; a(0) = 1, a(1) = 22, a(2) = 378.