A153130 -id:A153130 - cp's OEIS frontend

A173598 Period 6: repeat [1, 8, 7, 2, 4, 5].

1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8, 7, 2, 4, 5, 1, 8

Offset: 0

Paul Curtz, Nov 23 2010

For A141425 = 1,2,4,5,7,8 permutations, see A153130. a(n) is based on A022998. Successive differences are linked to A070366.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A022998, A070366, A141425, A153130, A166138.

&cat [[1, 8, 7, 2, 4, 5]^^20]; // Wesley Ivan Hurt, Jun 23 2016

A173598:=n->[1, 8, 7, 2, 4, 5][(n mod 6)+1]: seq(A173598(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

PadRight[{}, 100, {1, 8, 7, 2, 4, 5}] (* Wesley Ivan Hurt, Jun 23 2016 *)

a(n)=[1,8,7,2,4,5][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n) = A166138(n) mod 9.
a(2n+1) + a(2n+2) = 9.
G.f.: (1+8*x+7*x^2+2*x^3+4*x^4+5*x^5) / ((1-x)*(1+x)*(1+x+x^2)*(x^2-x+1)). - R. J. Mathar, Mar 08 2011
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (9 - cos(n*Pi) - 6*cos(2*n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3))/2. (End)

A177883 Period 6: repeat [4, 5, 7, 2, 1, 8].

Original entry on oeis.org

4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5, 7, 2, 1, 8, 4, 5

Offset: 0

Paul Curtz, Dec 14 2010

Represents also the decimal expansion of 16934/37037 and the continued fractions of 0.23839... = (sqrt(496555)-667)/158 or of 4.194699... = (667+sqrt(496555))/327. - R. J. Mathar, Dec 20 2010

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A173598, A141425, A153130 (permutations).

&cat[[4, 5, 7, 2, 1, 8]: n in [0..20]]; // Wesley Ivan Hurt, Jun 18 2016

A177883:=n->[4, 5, 7, 2, 1, 8][(n mod 6)+1]: seq(A177883(n), n=0..100); # Wesley Ivan Hurt, Jun 18 2016

PadRight[{}, 120, {4,5,7,2,1,8}] (* Harvey P. Dale, Feb 11 2016 *)

a(n)=[4, 5, 7, 2, 1, 8][n%6+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = A166304(n) mod 9 = A022998(3n+2) mod 9.
a(2n) + a(2n+1) = 9.
G.f.: (4+5*x+7*x^2+2*x^3+x^4+8*x^5) / ( (1-x)*(1+x)*(1+x+x^2)*(x^2-x+1) ). - R. J. Mathar, Dec 20 2010
From Wesley Ivan Hurt, Jun 18 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (9 -cos(n*Pi) + 3*cos(n*Pi/3) - 3*cos(2*n*Pi/3) + sqrt(3)*sin(n*Pi/3) - 3*sqrt(3)*sin(2*n*Pi/3))/2. (End)

A254076 a(n) = 2a(n-1) + a(n-2) - 2a(n-3) for n>2, a(0)=-1, a(1)=-2, a(2)=-4.

Original entry on oeis.org

-1, -2, -4, 1, 2, 13, 26, 61, 122, 253, 506, 1021, 2042, 4093, 8186, 16381, 32762, 65533, 131066, 262141, 524282, 1048573, 2097146, 4194301, 8388602, 16777213, 33554426, 67108861, 134217722, 268435453, 536870906, 1073741821, 2147483642, 4294967293

Offset: 0

Paul Curtz, Jan 29 2015

The main diagonal of the difference table is -A000079(n) = -2^n.
a(n) mod 9 is of period 6: repeat 8, 7, 5, 1, 2, 4.
a(n) + a(n+1) = -3, -6, -3, 3, 15, ...; all are multiples of 3.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000079, A010704, A141725, A153130, A156067, A157823.

a[0] = -1; a[n_] := 2^(n-1) + 3*Mod[n, 2] - 6; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Feb 04 2015 *)

Vec((9*x^3+x^2-1)/((x-1)*(x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jan 30 2015

a(2n+1) = A141725(n-1), a(2n+2) = 2*a(2n+1).
a(n+1) = 2*a(n) + (period 2: repeat 0, 9), n>0.
a(n) = -A157823(n) - (period 2: repeat 6, 3).
a(n+1) = a(n) - A156067(n).
a(n+2) = a(n) +  3*2^(n-1), n>0.
a(n+4) = a(n) + 15*2^(n-1), n>0.
a(n+6) = a(n) + 63*2^(n-1), n>0.
a(n) = (2^n - 3*(-1)^n - 9)/2 for n>0. - Colin Barker, Jan 30 2015
G.f.: (9*x^3+x^2-1) / ((x-1)*(x+1)*(2*x-1)). - Colin Barker, Jan 30 2015

A154529 A090040 mod 9.

Original entry on oeis.org

1, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8

Offset: 0

Paul Curtz, Jan 11 2009

For n>2, equal to 2^(n-2) mod 9 [From Michael B. Porter, Feb 02 2010]

Apart from leading terms the same as A146501, A153130 and A029898. [From R. J. Mathar, Apr 13 2010]

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Join[{1},LinearRecurrence[{1,0,-1,1},{5,1,2,4},101]] (* Ray Chandler, Jul 15 2015 *)

a(n)=a(n-1)-a(n-3)+a(n-4), n>4. G.f.: (6*x^4+2*x^3+4*x+1-4*x^2)/((1-x)*(1+x)*(x^2-x+1)). [From R. J. Mathar, Feb 25 2009]

Edited by N. J. A. Sloane, Jan 12 2009

Extended by Ray Chandler, Jul 15 2015

A158927 a(n) = -3a(n-1) - 3a(n-2) - 2a(n-3), n > 3.

Original entry on oeis.org

2, 2, 2, -7, 11, -16, 29, -61, 128, -259, 515, -1024, 2045, -4093, 8192, -16387, 32771, -65536, 131069, -262141, 524288, -1048579, 2097155, -4194304, 8388605, -16777213, 33554432, -67108867, 134217731, -268435456, 536870909, -1073741821, 2147483648

Offset: 0

Paul Curtz, Mar 31 2009

sign

The inverse binomial transform of A153130, after dropping A153130(0).
The inverse binomial transform of the full A153130 is A158916.
Dropping two initial terms of A153130 yields A158935, dropping three yields essentially a sign-reversed version of A158916, dropping 4 essentially the sequence here.

Muniru A Asiru, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (-3, -3, -2).

Same recurrence as A131562, A158916, A158926.

a := [2,2,2,-7];; for n in [5..10^3] do a[n] := -3*a[n-1] - 3*a[n-2] - 2*a[n-3]; od; a; # Muniru A Asiru, Jan 27 2018

a := proc(n) option remember: if n=0 then 2 elif n=1 then 2 elif n=2 then 2 elif n=3 then -7 elif n>=4 then -3*procname(n-1) - 3*procname(n-2) - 2*procname(n-3) fi; end:
seq(a(n), n=0..100); # Muniru A Asiru, Jan 27 2018

a(n) = -3a(n-1) - 3a(n-2) - 2a(n-3), with a(0)=a(1)=a(2)=2, a(3)=-7.
a(n) = (-1)^(n+1)*A157823(n) - A099838(n+3).
G.f.: (2+8*x+14*x^2+9*x^3)/((2*x+1)*(1+x+x^2)). - R. J. Mathar, Apr 09 2009
a(0)=2; a(n) = (1/2)*(-2)^n - 3*cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) for n >= 1. - Richard Choulet, Apr 23 2009

Edited and extended by R. J. Mathar, Apr 09 2009

A158935 a(n)= -3a(n-1)-3a(n-2)-2a(n-3), n>3. a(0)=4, a(1)=4, a(2)=-5, a(3)=4.

Original entry on oeis.org

4, 4, -5, 4, -5, 13, -32, 67, -131, 256, -509, 1021, -2048, 4099, -8195, 16384, -32765, 65533, -131072, 262147, -524291, 1048576, -2097149, 4194301, -8388608, 16777219, -33554435, 67108864, -134217725, 268435453, -536870912, 1073741827, -2147483651

Offset: 0

Paul Curtz, Mar 31 2009

The third column of the array of differences described in A153130. The first two columns are in A158916 and A158987. Taking differences like in A158926 keeps the recurrence.

Also the inverse binomial transform of A153130 if the first two items of A153130 are omitted.

Index entries for linear recurrences with constant coefficients, signature (-3, -3, -2).

Join[{4},LinearRecurrence[{-3,-3,-2},{4,-5,4},50]] (* Harvey P. Dale, May 25 2011 *)

a(n)= A154589(n) + A099838(n+2).

G.f.: (4+16*x+19*x^2+9*x^3)/((2*x+1)*(1+x+x^2)). - R. J. Mathar, Apr 08 2009

Partially edited and extended by R. J. Mathar, Apr 08 2009

Edited by N. J. A. Sloane, Apr 08 2009

A225539 Numbers n where 2^n and n have the same digital root.

Original entry on oeis.org

5, 16, 23, 34, 41, 52, 59, 70, 77, 88, 95, 106, 113, 124, 131, 142, 149, 160, 167, 178, 185, 196, 203, 214, 221, 232, 239, 250, 257, 268, 275, 286, 293, 304, 311, 322, 329, 340, 347, 358, 365, 376, 383, 394, 401, 412, 419, 430, 437, 448

Offset: 1

Marcus Hedbring, May 17 2013

The digital roots of n have a cycle length of 9 (A010888) and the digital roots of 2^n have a cycle length of 6 (A153130). Therefore, if n is a term so is n+18.

The only values of the digital roots of a(n) are 5 and 7 (A010718).

			For n=23, the digital root of n is 5. 2^n equals 8388608 so the digital root of 2^n is 5 as well.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A010888, A153130, A010718.

digitalRoot[n_] :=  Module[{r = n}, While[r > 9, r = Total[IntegerDigits[ r]]]; r]; Select[Range[448], digitalRoot[2^#] == digitalRoot[#] &] (* T. D. Noe, May 19 2013 *)
LinearRecurrence[{1,1,-1},{5,16,23},60] (* Harvey P. Dale, Dec 29 2018 *)

forstep(n=16,500,[7,11],print1(n", ")) \\ Charles R Greathouse IV, May 19 2013

a(n) = 9*n - 3 + (-1)^n.
a(n) = a(n-1) + 7 (odd n), a(n) = a(n-1) + 11 (even n) with a(1) = 5.
G.f. x*(5 + 11*x + 2*x^2) / ((1-x)^2 * (1+x)). - Joerg Arndt, May 17 2013

A294364 Linear recurrence with signature (1,1,-1,1,1), where the first terms are powers of 2 (1,2,4,8,16).

Original entry on oeis.org

1, 2, 4, 8, 16, 23, 37, 56, 94, 152, 250, 401, 649, 1046, 1696, 2744, 4444, 7187, 11629, 18812, 30442, 49256, 79702, 128957, 208657, 337610, 546268, 883880, 1430152, 2314031, 3744181, 6058208, 9802390, 15860600, 25662994, 41523593, 67186585, 108710174, 175896760, 284606936

Offset: 0

Jean-François Alcover and Paul Curtz, Oct 29 2017

The interest of this sequence mainly lies in the peculiarities of its array of successive differences, which begins:
1,    2,   4,   8,  16,  23,  37,  56,  94, ...
1,    2,   4,   8,   7,  14,  19,  38,  58, ...
1,    2,   4,  -1,   7,   5,  19,  20,  40, ...
1,    2,  -5,   8,  -2,  14,   1,  20,  13, ...
1,   -7,  13, -10,  16, -13,  19,  -7,  31, ...
-8,  20, -23,  26, -29,  32, -26,  38, -23, ...
28, -43,  49, -55,  61, -58,  64, -61,  67, ...
The main diagonal is A000079 (powers of 2).
The first upper subdiagonal is A254076.
The second upper subdiagonal (4, 8, 7, 14, 19, 38, ...) is not in the OEIS.
The third upper subdiagonal is A185346 (2^n-9).
Every row, once computed mod 9, is 6-periodic, repeating (1, 2, 4, 8, 7, 5) (A153130).

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,1).

Cf. A000045, A000079, A153130, A185346, A254076.

LinearRecurrence[{1, 1, -1, 1, 1}, {1, 2, 4, 8, 16}, 40]

G.f.: (1+x+x^2+3*x^3+5*x^4) / (1-x-x^2+x^3-x^4-x^5).
a(n) = (9/2)*fibonacci(n) + (-1)^n - sqrt(3)*sin(n*Pi/3).
a(n) ~ (9/2)*fibonacci(n).

cp's OEIS Frontend

A173598 Period 6: repeat [1, 8, 7, 2, 4, 5].

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A177883 Period 6: repeat [4, 5, 7, 2, 1, 8].

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A254076 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2, a(0)=-1, a(1)=-2, a(2)=-4.

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A154529 A090040 mod 9.

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A158927 a(n) = -3a(n-1) - 3a(n-2) - 2a(n-3), n > 3.

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A158935 a(n)= -3a(n-1)-3a(n-2)-2a(n-3), n>3. a(0)=4, a(1)=4, a(2)=-5, a(3)=4.

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A225539 Numbers n where 2^n and n have the same digital root.

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A254076 a(n) = 2a(n-1) + a(n-2) - 2a(n-3) for n>2, a(0)=-1, a(1)=-2, a(2)=-4.