A002276 -id:A002276 - cp's OEIS frontend

A002281 a(n) = 7*(10^n - 1)/9.

0, 7, 77, 777, 7777, 77777, 777777, 7777777, 77777777, 777777777, 7777777777, 77777777777, 777777777777, 7777777777777, 77777777777777, 777777777777777, 7777777777777777, 77777777777777777, 777777777777777777, 7777777777777777777, 77777777777777777777, 777777777777777777777

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002276, A002277, A002278, A002279, A002280, A002282, A002283, A178630, A178634.

Cf. A010785, A017245, A099915.

[7*(10^n-1)/9 : n in [0..30]]; // Wesley Ivan Hurt, Mar 24 2015

A002281:=n->7*(10^n-1)/9: seq(A002281(n), n=0..30); # Wesley Ivan Hurt, Mar 24 2015

LinearRecurrence[{11,-10},{0,7},25] (* Robert G. Wilson v, Jul 06 2013 *)

a(n)=7*(10^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A178634(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
From Vincenzo Librandi, Jul 22 2010: (Start)
a(n) = a(n-1) + 7*10^(n-1) with n>0, a(0)=0.
a(n) = 11*a(n-1) - 10*a(n-2) with n>1, a(0)=0, a(1)=7. (End)
G.f.: 7*x/((x-1)*(10*x-1)). - Colin Barker, Jan 24 2013
a(n) = 7*A002275(n). - Wesley Ivan Hurt, Mar 24 2015
E.g.f.: 7*exp(x)*(exp(9*x) - 1)/9. - Stefano Spezia, Sep 13 2023
From Elmo R. Oliveira, Jul 20 2025: (Start)
a(n) = (A099915(n) - 1)/2.
a(n) = A010785(A017245(n-1)) for n >= 1. (End)

A002278 a(n) = 4*(10^n - 1)/9.

Original entry on oeis.org

0, 4, 44, 444, 4444, 44444, 444444, 4444444, 44444444, 444444444, 4444444444, 44444444444, 444444444444, 4444444444444, 44444444444444, 444444444444444, 4444444444444444, 44444444444444444, 444444444444444444, 4444444444444444444, 44444444444444444444, 444444444444444444444

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002276, A002277, A002279, A002280, A002281, A002282, A002283, A075415, A178632.

Cf. A007091, A010785, A017209, A024049.

LinearRecurrence[{11, -10}, {0, 4}, 20] (* Robert G. Wilson v, Jul 06 2013 *)

a(n)=4*(10^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A075415(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
From Vincenzo Librandi, Jul 22 2010: (Start)
a(n) = a(n-1) + 4*10^(n-1) with a(0)=0;
a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=4. (End)
G.f.: 4*x/((1 - x)*(1 - 10*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 4*exp(x)*(exp(9*x) - 1)/9. - Stefano Spezia, Sep 13 2023
a(n) = A007091(A024049(n)). - Michel Marcus, Jun 16 2024
From Elmo R. Oliveira, Jul 19 2025: (Start)
a(n) = 4*A002275(n).
a(n) = A010785(A017209(n-1)) for n >= 1. (End)

A032834 Numbers with digits 3 and 4 only.

Original entry on oeis.org

3, 4, 33, 34, 43, 44, 333, 334, 343, 344, 433, 434, 443, 444, 3333, 3334, 3343, 3344, 3433, 3434, 3443, 3444, 4333, 4334, 4343, 4344, 4433, 4434, 4443, 4444, 33333, 33334, 33343, 33344, 33433, 33434, 33443, 33444, 34333, 34334

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A032829-A032833 (in other bases), A102659 (Lyndon words in this sequence), A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..35000] | Set(IntegerToSequence(n, 10)) subset {3, 4}]; // Vincenzo Librandi, May 30 2012

S[1]:= [3,4]:
for d from 2 to 5 do S[d]:= map(t -> (10*t+3,10*t+4), S[d-1]) od:
seq(op(S[d]),d=1..5); # Robert Israel, Apr 03 2017

Flatten[Table[FromDigits[#,10]&/@Tuples[{3,4},n],{n,5}]] (* Vincenzo Librandi, May 30 2012 *)

A032834(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\3 \\ M. F. Hasler, Mar 27 2015

a(n) = A007931(n) + A002276(A000523(n+1)) = A032810(n) + A256077(n) etc. - M. F. Hasler, Mar 27 2015
From Robert Israel, Apr 03 2017: (Start)
a(2*n+1) = 10*a(n)+3.
a(2*n+2) = 10*a(n)+4.
G.f. g(x) satisfies g(x) = 10*(x+x^2)*g(x^2) + x*(3+4*x)/(1-x^2). (End)

Crossrefs added by M. F. Hasler, Mar 27 2015

Name corrected by Robert Israel, Apr 03 2017

A332120 a(n) = 2(10^(2n+1)-1)/9 - 210^n.

Original entry on oeis.org

0, 202, 22022, 2220222, 222202222, 22222022222, 2222220222222, 222222202222222, 22222222022222222, 2222222220222222222, 222222222202222222222, 22222222222022222222222, 2222222222220222222222222, 222222222222202222222222222, 22222222222222022222222222222, 2222222222222220222222222222222

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332130 .. A332190 (variants with different repeated digit 3, ..., 9).
Cf. A332121 .. A332129 (variants with different middle digit 1, ..., 9).

A332120 := n -> 2*((10^(2*n+1)-1)/9-10^n);

Array[2 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]

apply( {A332120(n)=(10^(n*2+1)\9-10^n)*2}, [0..15])

def A332120(n): return (10**(n*2+1)//9-10**n)*2

a(n) = 2*A138148(n) = A002276(2n+1) - 2*10^n.
G.f.: 2*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: 2*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A178630 a(n) = 18*((10^n - 1)/9)^2.

Original entry on oeis.org

18, 2178, 221778, 22217778, 2222177778, 222221777778, 22222217777778, 2222222177777778, 222222221777777778, 22222222217777777778, 2222222222177777777778, 222222222221777777777778, 22222222222217777777777778, 2222222222222177777777777778, 222222222222221777777777777778

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 18 = 9 * 2;
n=2: ................... 2178 = 99 * 22;
n=3: ................. 221778 = 999 * 222;
n=4: ............... 22217778 = 9999 * 2222;
n=5: ............. 2222177778 = 99999 * 22222;
n=6: ........... 222221777778 = 999999 * 222222;
n=7: ......... 22222217777778 = 9999999 * 2222222;
n=8: ....... 2222222177777778 = 99999999 * 22222222;
n=9: ..... 222222221777777778 = 999999999 * 222222222.

G. C. Greubel, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178631, A178632, A178633, A178634, A178635.

Cf. A002276, A002281, A002283, A002477.

List([1..20], n -> 18*((10^n-1)/9)^2); # G. C. Greubel, Jan 28 2019

[18*((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Dec 28 2010

Table[18*((10^n-1)/9)^2, {n, 1, 20}] (* G. C. Greubel, Jan 28 2019 *)

a(n)=18*(10^n\9)^2 \\ Charles R Greathouse IV, Aug 21 2011

[18*((10^n-1)/9)^2 for n in (1..20)] # G. C. Greubel, Jan 28 2019

a(n) = 18*A002477(n) = A002283(n)*A002276(n).
a(n)=((A002276(n-1)*10 + 1)*10^(n-1) + A002281(n-1))*10 + 8.
G.f.: 18*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
From Elmo R. Oliveira, Jul 30 2025: (Start)
E.g.f.: 2*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. (End)

A178634 a(n) = 63*((10^n - 1)/9)^2.

Original entry on oeis.org

63, 7623, 776223, 77762223, 7777622223, 777776222223, 77777762222223, 7777777622222223, 777777776222222223, 77777777762222222223, 7777777777622222222223, 777777777776222222222223, 77777777777762222222222223, 7777777777777622222222222223, 777777777777776222222222222223

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 63 = 9 * 7;
n=2: ................... 7623 = 99 * 77;
n=3: ................. 776223 = 999 * 777;
n=4: ............... 77762223 = 9999 * 7777;
n=5: ............. 7777622223 = 99999 * 77777;
n=6: ........... 777776222223 = 999999 * 777777;
n=7: ......... 77777762222223 = 9999999 * 7777777;
n=8: ....... 7777777622222223 = 99999999 * 77777777;
n=9: ..... 777777776222222223 = 999999999 * 777777777.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 33 at p. 62.
Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

G. C. Greubel, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178630, A178631, A178632, A178633, A178635.

Cf. A002276, A002281, A002283, A002477.

List([1..20], n -> 63*((10^n - 1)/9)^2); # G. C. Greubel, Jan 28 2019

[63*((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Dec 28 2010

63((10^Range[15]-1)/9)^2 (* or *) Table[FromDigits[Join[PadRight[{},n,7],{6},PadRight[{},n,2],{3}]],{n,0,15}] (* Harvey P. Dale, Apr 23 2012 *)

a(n)=63*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

[63*((10^n - 1)/9)^2 for n in (1..20)] # G. C. Greubel, Jan 28 2019

a(n) = 63*A002477(n) = A002283(n)*A002281(n).
a(n) = ((A002281(n-1)*10 + 6)*10^(n-1) + A002276(n-1))*10 + 3.
G.f.: 63*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 7*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9. - Stefano Spezia, Jul 31 2024
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. - Elmo R. Oliveira, Aug 01 2025

A332121 a(n) = 2*(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

1, 212, 22122, 2221222, 222212222, 22222122222, 2222221222222, 222222212222222, 22222222122222222, 2222222221222222222, 222222222212222222222, 22222222222122222222222, 2222222222221222222222222, 222222222222212222222222222, 22222222222222122222222222222, 2222222222222221222222222222222

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).
Cf. A332131 .. A332191 (variants with different repeated digit 3, ..., 9).

A332121 := n -> 2*(10^(2*n+1)-1)/9-10^n;

Array[2 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]

apply( {A332121(n)=10^(n*2+1)\9*2-10^n}, [0..15])

def A332121(n): return 10**(n*2+1)//9*2-10**n

a(n) = 2*A138148(n) + 1*10^n = A002276(2n+1) - 10^n.
G.f.: (1 + 101*x - 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332129 a(n) = 2(10^(2n+1)-1)/9 + 710^n.

Original entry on oeis.org

9, 292, 22922, 2229222, 222292222, 22222922222, 2222229222222, 222222292222222, 22222222922222222, 2222222229222222222, 222222222292222222222, 22222222222922222222222, 2222222222229222222222222, 222222222222292222222222222, 22222222222222922222222222222, 2222222222222229222222222222222

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332119 .. A332189 (variants with different repeated digit 1, ..., 8).
Cf. A332120 .. A332128 (variants with different middle digit 0, ..., 8).

A332129 := n -> 2*(10^(2*n+1)-1)/9+7*10^n;

Array[2 (10^(2 # + 1)-1)/9 + 7*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{9,292,22922},20] (* Harvey P. Dale, Jun 25 2020 *)

apply( {A332129(n)=10^(n*2+1)\9*2+7*10^n}, [0..15])

def A332129(n): return 10**(n*2+1)//9*2+7*10**n

a(n) = 2*A138148(n) + 9*10^n = A002276(2n+1) + 7*10^n.
G.f.: (9 - 707*x + 500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A086066 a(n) = Sum_{d in D(n)} 2^d, where D(n) = set of digits of n in decimal representation.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 3, 2, 6, 10, 18, 34, 66, 130, 258, 514, 5, 6, 4, 12, 20, 36, 68, 132, 260, 516, 9, 10, 12, 8, 24, 40, 72, 136, 264, 520, 17, 18, 20, 24, 16, 48, 80, 144, 272, 528, 33, 34, 36, 40, 48, 32, 96, 160, 288, 544, 65, 66, 68, 72, 80

Offset: 0

Reinhard Zumkeller, Jul 08 2003

For bitwise logical operations AND and OR:
a(m) = (a(m) AND a(n)) iff D(m) is a subset of D(n),
(a(m) AND a(n)) = 0 iff D(m) and D(n) are disjoint,
a(m) = (a(m) OR a(n)) iff D(n) is a subset of D(m),
a(m) = a(n) iff D(m) = D(n);
A086067(n) = A007088(a(n)).
From Reinhard Zumkeller, Sep 18 2009: (Start)
a(A052382(n)) mod 2 = 0; a(A011540(n)) mod 2 = 1;
for n > 0: a(A000004(n))=1, a(A000042(n))=2, a(A011557(n))=3, a(A002276(n))=4, a(A111066(n))=6, a(A002277(n))=8, a(A002278(n))=16, a(A002279(n))=32, a(A002280(n))=64, a(A002281(n))=128, a(A002282(n))=256, a(A002283(n))=512;
a(n) <= 1023. (End)

			n=242, D(242) = {2,4}: a(242) = 2^2 + 2^4 = 20.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

A086066 := proc(n) local d: if(n=0)then return 1: fi: d:=convert(convert(n,base,10),set): return add(2^d[j],j=1..nops(d)): end: seq(A086066(n),n=0..64); # Nathaniel Johnston, May 31 2011

A180592 Digital root of 2n.

Original entry on oeis.org

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

Offset: 0

Odimar Fabeny, Sep 10 2010

Period 9. - Robert G. Wilson v, Sep 20 2010

Also digital root of A002276(n). - Enrique Pérez Herrero, Nov 05 2022

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

Cf. A007953, A010888, A180593, A180594, A180595, A180596, A180597, A180598, A180599.

f[n_] := Mod[2 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)

a(n)=if(n,(2*n-1)%9+1,0) \\ Charles R Greathouse IV, Aug 25 2014

From R. J. Mathar, Nov 02 2010: (Start)
a(n) = A010888(2*n).
a(n) = a(n-9), n > 9.
G.f.: -x*(2 + 4*x + 6*x^2 + 8*x^3 + x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 9*x^8) / ( (x-1)*(1 + x + x^2)*(x^6 + x^3 + 1) ). (End)

More terms from Robert G. Wilson v, Sep 20 2010

Keyword:base and formulas from R. J. Mathar, Nov 02 2010

cp's OEIS Frontend

A002281 a(n) = 7*(10^n - 1)/9.

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A002278 a(n) = 4*(10^n - 1)/9.

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A032834 Numbers with digits 3 and 4 only.

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A332120 a(n) = 2*(10^(2n+1)-1)/9 - 2*10^n.

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A178630 a(n) = 18*((10^n - 1)/9)^2.

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A178634 a(n) = 63*((10^n - 1)/9)^2.

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A332121 a(n) = 2*(10^(2n+1)-1)/9 - 10^n.

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A332120 a(n) = 2(10^(2n+1)-1)/9 - 210^n.

A332129 a(n) = 2(10^(2n+1)-1)/9 + 710^n.