A153130 -id:A153130 - cp's OEIS frontend

A070403 a(n) = 7^n mod 9.

1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4, 1, 7, 4

Offset: 0

N. J. A. Sloane, May 12 2002

Also the digital root of 7^n. If we convert this to a repeating decimal 0.174174..., we get the rational number 58/333. - Cino Hilliard, Dec 31 2004
A141722 (1, 25, 121, 505, 2041, 8185) mod 9. Note A141722 = 10*A000975(2n) + A000975(2n+1). - Paul Curtz, Sep 15 2008
Digital root of the powers of any number congruent to 7 mod 9. - Alonso del Arte, Jan 26 2014

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 1).

Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 4, A100402; c = 5, A070366; c = 8, A010689.

Cf. A000975, A010888, A049347, A099837, A141722.

[Modexp(7, n, 9): n in [0..110]]; // Bruno Berselli, Mar 22 2016

A070403:=n->4-2*sqrt(3)*sin(2*(n+1)*Pi/3): seq(A070403(n), n=0..100); # Wesley Ivan Hurt, Jun 09 2016

Table[PowerMod[7, n, 9], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

a(n)=7^n%9 \\ Charles R Greathouse IV, Oct 07 2015

[power_mod(7,n,9)for n in range(0,105)] # Zerinvary Lajos, Nov 03 2009

From R. J. Mathar, Feb 23 2009: (Start)
G.f.: (1+7*x+4*x^2)/((1-x)*(1+x+x^2)).
a(n+1) - a(n) = 3*A099837(n+3).
a(n) = 4 - 3*A049347(n). (End)
a(n) = a(n-3) for n>3. - G. C. Greubel, Mar 19 2016
a(n) = 4-2*sqrt(3)*sin((2*n+2)*Pi/3). - Wesley Ivan Hurt, Jun 09 2016
a(n) = A010888(7*a(n-1)). - Stefano Spezia, Mar 20 2025

A166304 Third trisection of A022998.

Original entry on oeis.org

4, 5, 16, 11, 28, 17, 40, 23, 52, 29, 64, 35, 76, 41, 88, 47, 100, 53, 112, 59, 124, 65, 136, 71, 148, 77, 160, 83, 172, 89, 184, 95, 196, 101, 208, 107, 220, 113, 232, 119, 244, 125, 256, 131, 268, 137, 280, 143, 292, 149, 304, 155, 316, 161, 328, 167, 340, 173, 352, 179

Offset: 0

Paul Curtz, Oct 11 2009

The sequence read modulo 9 is the periodic sequence 4, 5, 7, 2, 1, 8 (repeat..)
The same set of numbers in a period of length 6 is in A153130,
A165355 read modulo 9, A165367 read modulo 9, and A166138 read modulo 9.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, -1).

Cf. A165988 (first trisection), A166138 (2nd trisection).

LinearRecurrence[{0, 2, 0, -1}, {4, 5, 16, 11}, 100] (* G. C. Greubel, May 09 2016 *)

a(n) = A022998(3*n+2).
a(n) = 2*a(n-2)-a(n-4).
G.f.: (4+5*x+8*x^2+x^3)/((x-1)^2 *(1+x)^2 ).
a(2*n) = A017569(n). a(2n+1) = A016969(n) .

Edited and extended by R. J. Mathar, Oct 14 2009

A154127 Period 6: repeat [1, 2, 5, 8, 7, 4].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Jan 05 2009

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

Cf. A020806, A029898, A070366, A141425, A146501, A153130.

&cat[[1, 2, 5, 8, 7, 4]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016

A154127:=n->(27-cos(n*Pi)-20*cos(n*Pi/3)-4*sqrt(3)*sin(n*Pi/3))/6: seq(A154127(n), n=0..100); # Wesley Ivan Hurt, Jun 17 2016

Flatten[Table[{1, 2, 5, 8, 7, 4}, {20}]] (* Wesley Ivan Hurt, Jun 17 2016 *)

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

From R. J. Mathar, Feb 25 2009, Mar 09 2009: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
G.f.: (1+x+3*x^2+4*x^3)/((1-x)*(1+x)*(x^2-x+1)). (End)
a(n) = (27-cos(n*Pi)-20*cos(n*Pi/3)-4*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 17 2016

Corrected numerator in g.f R. J. Mathar, Mar 09 2009

A153990 Period 6: repeat [1, 2, 5, 4, 7, 8].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Jan 04 2009

Shares digits with other 6-periodic sequences, see the list in A153130.

Also the decimal expansion of the constant 13942/111111. [R. J. Mathar, Jan 23 2009]

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

Cf. A131533, A141425, A153130, A154811.

&cat[[1, 2, 5, 4, 7, 8]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016

A153990:=n->(27-cos(n*Pi)-8*sqrt(3)*cos((1-4*n)*Pi/6)-16*sin((1+2*n)*Pi/6))/6: seq(A153990(n), n=0..100); # Wesley Ivan Hurt, Jun 17 2016

Flatten[Table[{1, 2, 5, 4, 7, 8}, {20}]] (* Wesley Ivan Hurt, Jun 17 2016 *)
PadRight[{},120,{1,2,5,4,7,8}] (* Harvey P. Dale, Nov 08 2017 *)

a(n) - A141425(n) = A131533(n+2).
a(6n+0) + a(6n+5) = a(6n+1) + a(6n+4) = a(6n+2) + a(6n+3) = 9.
G.f.: (1+2*x+5*x^2+4*x^3+7*x^4+8*x^5)/((1-x)*(1+x)*(1+x+x^2)*(x^2-x+1)). [R. J. Mathar, Jan 23 2009]
From Wesley Ivan Hurt, Jun 17 2016: (Start)
a(n) = (27-cos(n*Pi)-8*sqrt(3)*cos((1-4*n)*Pi/6)-16*sin((1+2*n)*Pi/6))/6.
a(n) = a(n-6) for n>5. (End)

Edited by R. J. Mathar, Jan 23 2009

A154870 Period 6: repeat [7, 5, 1, -7, -5, -1].

Original entry on oeis.org

7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1

Offset: 0

Paul Curtz, Jan 16 2009

The sequence b(n) = (-A153130(n)) mod 9 = A153130(n+3) = A146501(n-1) = 8, 7, 5, 1, 2, 4,... has period length 6. This here is a(n)=b(n)-A153130(n).

a(n) is (-1)^(n+1) * numerator of F(n) where F(n) = f(F(n-1)) starting from F(0) = -7/4 and step f(z) = z^2 -29/16. - Nicolas Bělohoubek, Nov 20 2024

Holly Krieger and Brady Haran, A Fascinating Thing about Fractions, Numberphile video, Dec 15 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Cf. A146501, A153130.

&cat [[7, 5, 1, -7, -5, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A154870:=n->[7, 5, 1, -7, -5, -1][(n mod 6)+1]: seq(A154870(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

PadRight[{}, 100, {7, 5, 1, -7, -5, -1}] (* Wesley Ivan Hurt, Jun 20 2016 *)

a(n) = -a(n-3) for n>2; G.f.: (7+5*x+x^2)/((1+x)*(1-x+x^2)). [R. J. Mathar, Jan 23 2009]

a(n) = cos(n*Pi) + 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3). - Wesley Ivan Hurt, Jun 20 2016

Edited and extended by R. J. Mathar, Jan 23 2009

A167784 a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).

Original entry on oeis.org

1, 0, 4, 2, 16, 14, 64, 74, 256, 350, 1024, 1562, 4096, 6734, 16384, 28394, 65536, 117950, 262144, 484922, 1048576, 1979054, 4194304, 8034314, 16777216, 32491550, 67108864, 131029082, 268435456, 527304974, 1073741824, 2118785834, 4294967296, 8503841150

Offset: 0

Paul Curtz, Nov 12 2009

Binomial transform of A077917, the signed variant of A127864.

Robert Israel, Table of n, a(n) for n = 0..3318
Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Cf. A154383.

seq(2^n - (1 - (-1)^n)*3^((n-1)/2), n=0..100); # Robert Israel, Apr 11 2019

LinearRecurrence[{2, 3, -6}, {1, 0, 4}, 40] (* Harvey P. Dale, Nov 29 2011 *)

a(n) = A167936(n+1) - A167936(n).
a(2n) = A000302(n). a(2n+1) = 2*A005061(n).
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3).
G.f.: (x-1)^2/((2*x-1)*(3*x^2-1)).
a(n+4) mod 9 = A153130(n+4) = A146501(n+2), n>=0.
E.g.f.: exp(2*x) - (2/sqrt(3))*sinh(sqrt(3)*x). - G. C. Greubel, Jun 27 2016

Edited and extended by R. J. Mathar, Feb 27 2010

Incorrect b-file corrected by Robert Israel, Apr 11 2019

A381487 Numbers which are a power of their digital root.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 128, 256, 512, 729, 2401, 6561, 8192, 16384, 32768, 59049, 78125, 524288, 531441, 823543, 1048576, 2097152, 4782969, 33554432, 43046721, 67108864, 134217728, 282475249, 387420489, 1220703125, 2147483648, 3486784401, 4294967296

Offset: 1

Stefano Spezia, Feb 25 2025

			a(12) = 128 is a term since 128 = 2^7 = A010888(128)^7.

Michel Marcus, Table of n, a(n) for n = 1..3347

Cf. A010888, A023106, A381491, A381492.

Digital root of k^n: A000012 (1), A153130 (2), A100401 (3), A100402 (4), A070366 (5), A100403 (6), A070403 (7), A010689 (8), A010734 (9).

A010888[n_]:=n - 9*Floor[(n-1)/9]; kmax=5*10^6; a={0,1}; For[k=2, k<=kmax, k++, If[A010888[k]!=1, If[IntegerQ[Log[A010888[k],k]], AppendTo[a,k]]]]; a

isok(k) = if ((k==0) || (k==1), return(1)); my(d=(k-1)%9+1); if (d>1, d^logint(k, d) == k); \\ Michel Marcus, Feb 26 2025

lista(nn) = my(list = List()); listput(list, 0); listput(list, 1); for (n=2, 9, for (k=1, logint(nn, n), if ((n^k-1)%9+1 == n, listput(list, n^k)););); vecsort(Vec(list)); \\ Michel Marcus, Feb 27 2025

a(n) = A381491(n)^A381492(n).

A154815 Period 6: repeat [8, 7, 4, 5, 2, 1].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Jan 15 2009

Obtained through reversion of the period in A153990, or by taking a half period of A154811.
Shares digits with other 6-periodic sequences, see the list in A153130.
Also the decimal expansion of the constant 97169/111111. [R. J. Mathar, Jan 23 2009]

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

Cf. A153130, A153990, A154811.

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

A154815:=n->[8, 7, 4, 5, 2, 1][(n mod 6)+1]: seq(A154815(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

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

a(n) = (8*A153990(n)) mod 9.
G.f.: (8+7*x+4*x^2+5*x^3+2*x^4+x^5)/((1-x)*(1+x)*(1+x+x^2)(x^2-x+1)). [R. J. Mathar, Jan 23 2009]
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (27 + cos(n*Pi) + 8*cos(n*Pi/3) + 12*cos(2*n*Pi/3) + 8*sqrt(3)*sin(n*Pi/3) + 4*sqrt(3)*sin(2*n*Pi/3))/6. (End)

Edited by R. J. Mathar, Jan 23 2009

A156067 a(0)=1. a(n)= -2^(n-1)-3*(-1)^n, n>1.

Original entry on oeis.org

1, 2, -5, -1, -11, -13, -35, -61, -131, -253, -515, -1021, -2051, -4093, -8195, -16381, -32771, -65533, -131075, -262141, -524291, -1048573, -2097155, -4194301, -8388611, -16777213, -33554435, -67108861, -134217731, -268435453, -536870915, -1073741821, -2147483651

Offset: 0

Paul Curtz, Feb 03 2009

The main diagonal of the array of A153130 and its successive differences.

A154589 is the second upper diagonal of the array.

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

Join[{1},LinearRecurrence[{1,2},{2,-5},40]] (* Harvey P. Dale, Dec 11 2011 *)

a(n)= +a(n-1) +2*a(n-2), n>2.
G.f.: x*(-2+7*x) / ( (1+x)*(2*x-1) ).
a(n) == A153130(n) (mod 9).
a(n+1)-2*a(n) = (-1)^n*9, n>0.
a(n) = A154589(n)-3*(-1)^n.
a(n)+a(n+3) = -A005010(n-1) = -9*A131577(n).
a(2*n)+a(2*n+1) = -3*2^(2n-1) = -A002023(n-2).

A166577 Inverse binomial transform of A166517.

Original entry on oeis.org

1, 4, -5, 10, -20, 40, -80, 160, -320, 640, -1280, 2560, -5120, 10240, -20480, 40960, -81920, 163840, -327680, 655360, -1310720, 2621440, -5242880, 10485760, -20971520, 41943040, -83886080, 167772160, -335544320, 671088640, -1342177280, 2684354560, -5368709120

Offset: 0

Paul Curtz, Oct 17 2009

The definition assumes that the offset of A166517 is changed to 0.
A166517 mod 9 yields a periodic sequence with period 1, 5, 4, 8, 7, 2.
This set of numbers in the period appears also in A153130, A146501, and A166304.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A010716, A010692, A010859.

Join[{1,4},NestList[-2#&,-5,40]] (* Harvey P. Dale, Aug 02 2012 *)
Join[{1, 4}, LinearRecurrence[{-2}, {-5}, 48]] (* G. C. Greubel, May 17 2016 *)

a(n) = -2*a(n-1), n>2.
a(n) = (-1)^(n+1)*A020714(n-2), n>1.
From Colin Barker, Jan 07 2013: (Start)
a(n) = -5*(-1)^n*2^(n-2) for n>1.
G.f.: (3*x^2+6*x+1)/(2*x+1). (End)
E.g.f.: (9/4) + (3/2)*x - (5/4)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited, comments not concerning this sequence removed, and extended by R. J. Mathar, Oct 21 2009

cp's OEIS Frontend

A070403 a(n) = 7^n mod 9.

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A166304 Third trisection of A022998.

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A154127 Period 6: repeat [1, 2, 5, 8, 7, 4].

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A153990 Period 6: repeat [1, 2, 5, 4, 7, 8].

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A154870 Period 6: repeat [7, 5, 1, -7, -5, -1].

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A167784 a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).

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A381487 Numbers which are a power of their digital root.

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