A010689 -id:A010689 - cp's OEIS frontend

A174927 Periodic sequence: Repeat 1, 64.

1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64, 1, 64

Offset: 0

Klaus Brockhaus, Apr 02 2010

Interleaving of A000012 and 2*A010871.
Also continued fraction expansion of (4+sqrt(17))/8.
First differences of A174928.

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

Cf. A000012 (all 1's sequence), A010871 (all 32's sequence), A010689 (repeat 1, 8), A174930 (decimal expansion of (4+sqrt(17))/8), A174928.

&cat[ [1, 64]: n in [0..41] ];
[ (65-63*(-1)^n)/2: n in [0..83] ];

PadRight[{},100,{1,64}] (* Harvey P. Dale, Jun 16 2013 *)

a(n) = (65-63*(-1)^n)/2.
a(n) = a(n-2) for n > 1; a(0) = 0, a(1) = 64.
a(n) = -a(n-1)+65 for n > 0; a(0) = 1.
a(n) = ((n+1) mod 2)+64*(n mod 2).
G.f.: (1+64*x)/((1-x)*(1+x)).

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).

A021059 Decimal expansion of 1/55.

Original entry on oeis.org

0, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8

Offset: 0

N. J. A. Sloane

			0.0181818181818181818181818...

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

A010689 shifted right.

RealDigits[1/55, 10, 100, -1][[1]] (* Amiram Eldar, Jun 09 2025 *)

{ default(realprecision); x=1/55*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Felix Fröhlich, Aug 11 2014

Multiplicative with a(2^e) = 8, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005

Dirichlet g.f.: zeta(s) * (1 + 7/2^s). - Amiram Eldar, Jun 09 2025

A174925 Decimal expansion of (2+sqrt(6))/4.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 2, 4, 3, 5, 6, 9, 5, 7, 9, 4, 5, 2, 4, 5, 4, 9, 3, 2, 1, 0, 1, 8, 6, 7, 6, 4, 7, 2, 8, 4, 7, 9, 9, 1, 4, 8, 6, 8, 7, 0, 1, 6, 4, 1, 6, 7, 5, 3, 2, 1, 0, 8, 1, 7, 3, 1, 4, 1, 8, 1, 2, 7, 4, 0, 0, 9, 4, 3, 6, 4, 3, 2, 8, 7, 5, 6, 6, 3, 4, 9, 6, 4, 8, 5, 8, 2, 7, 6, 1, 6, 0, 0, 5, 8, 7, 0, 4, 6, 4

Offset: 1

Klaus Brockhaus, Apr 02 2010

Continued fraction expansion of (2+sqrt(6))/4 is A010689.

			(2+sqrt(6))/4 = 1.11237243569579452454...

Cf. A010464 (decimal expansion of sqrt(6)), A010689 (repeat 1, 8).

RealDigits[N[(2+Sqrt[6])/4,300]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 5, 1, 3, 1, 1, 6, 1, 4, 1, 2, 1, 7, 1, 5, 1, 3, 1, 1, 8, 1, 6, 1, 4, 1, 2, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2

Offset: 2

Paul Curtz, Feb 14 2010

One may define another array B(n,0) = -1, B(n,k) = T(n,k-1) + 2*B(n,k-1), n>=2, which also starts in columns k>=0, as follows:
  -1, -1, 0, 1,  4,  9, 20,  41,  84, 169,  340,  681, 1364 ...: A084639;
  -1, -1, 1, 3,  9, 19, 41,  83, 169, 339,  681, 1363, 2729;
  -1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;
  -1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;
B(n,k) = (n-1)*A001045(k) - T(n,k).
First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).

			The array T(n,k) starts in row n=2 with columns k>=0 as:
  1,  2, 1,  2, 1,  2, 1,  2, 1,  2, 1,  2 ... A000034;
  1,  3, 1,  3, 1,  3, 1,  3, 1,  3, 1,  3 ... A010684;
  1,  4, 1,  4, 1,  4, 1,  4, 1,  4, 1,  4 ... A010685;
  1,  5, 1,  5, 1,  5, 1,  5, 1,  5, 1,  5 ... A010686;
  1,  6, 1,  6, 1,  6, 1,  6, 1,  6, 1,  6 ... A010687;
  1,  7, 1,  7, 1,  7, 1,  7, 1,  7, 1,  7 ... A010688;
  1,  8, 1,  8, 1,  8, 1,  8, 1,  8, 1,  8 ... A010689;
  1,  9, 1,  9, 1,  9, 1,  9, 1,  9, 1,  9 ... A010690;
  1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
Antidiagonal triangle begins as:
  1;
  1,  2;
  1,  3,  1;
  1,  4,  1,  2;
  1,  5,  1,  3,  1;
  1,  6,  1,  4,  1,  2;
  1,  7,  1,  5,  1,  3,  1;
  1,  8,  1,  6,  1,  4,  1,  2;
  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 13,  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 14,  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;

G. C. Greubel, Antidiagonals n = 0..50 of the array, flattened

Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.

Cf. A001045, A024206, A084639, A093178, A173114.

T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* G. C. Greubel, Dec 03 2021 *)

flatten([[(1/2)*((n-k+3) - (n-k+1)*(-1)^k) for k in (2..n)] for n in (2..17)]) # G. C. Greubel, Dec 03 2021

From G. C. Greubel, Dec 03 2021: (Start)
T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).
Sum_{k=0..n} T(n-k, k) = A024206(n).
Sum_{k=0..floor((n+2)/2)} T(n-2*k+2, k) = (1/16)*(2*n^2 4*n -5*(1 +(-1)^n) + 4*sin(n*Pi/2)) (diagonal sums).
T(2*n-2, n) = A093178(n). (End)

A021554 Decimal expansion of 1/550.

Original entry on oeis.org

0, 0, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1

Offset: 0

N. J. A. Sloane

A021059 shifted right, A010689 shifted right twice. [From R. J. Mathar, Oct 25 2008]

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

realDigitsRecip[550] (* The realDigitsRecip program is at A021200 *)  (* or *) PadRight[{0,0},120,{1,8}] (* Harvey P. Dale, Dec 28 2024 *)

{ default(realprecision); x=1/550*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Felix Fröhlich, Aug 11 2014

cp's OEIS Frontend

A174927 Periodic sequence: Repeat 1, 64.

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A381487 Numbers which are a power of their digital root.

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A021059 Decimal expansion of 1/55.

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A174925 Decimal expansion of (2+sqrt(6))/4.

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A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

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A021554 Decimal expansion of 1/550.

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