A002278 -id:A002278 - 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)

A075415 Squares of A002280 or numbers (666...6)^2.

Original entry on oeis.org

0, 36, 4356, 443556, 44435556, 4444355556, 444443555556, 44444435555556, 4444444355555556, 444444443555555556, 44444444435555555556, 4444444444355555555556, 444444444443555555555556, 44444444444435555555555556, 4444444444444355555555555556, 444444444444443555555555555556

Offset: 0

Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 66^2 = 4356.
From _Reinhard Zumkeller_, May 31 2010: (Start)
n=1: ..................... 36 = 9 * 4;
n=2: ................... 4356 = 99 * 44;
n=3: ................. 443556 = 999 * 444;
n=4: ............... 44435556 = 9999 * 4444;
n=5: ............. 4444355556 = 99999 * 44444;
n=6: ........... 444443555556 = 999999 * 444444;
n=7: ......... 44444435555556 = 9999999 * 4444444;
n=8: ....... 4444444355555556 = 99999999 * 44444444;
n=9: ..... 444444443555555556 = 999999999 * 444444444. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..500
Gérard Villemin, Variations sur les carrés.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417.
Cf. A059988, A178630, A178631, A178632, A178633, A178634, A178635. [From Reinhard Zumkeller, May 31 2010]
Cf. A002275, A002278, A002279,  A002280, A002283, A002477, A052041, A102807, A309827.

Table[FromDigits[PadRight[{},n,6]]^2,{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{0,36,4356},20] (* Harvey P. Dale, May 20 2021 *)

a(n) = A002280(n)^2 = (6*A002275(n))^2 = 36*A002275(n)^2.
a(n) = (6*(10^n-1)/9)^2 = (4/9)*(10^(2*n) - 2*10^n + 1), which is n-1 4's, followed by a 3, n-1 5's and a 6. - Ignacio Larrosa Cañestro, Feb 26 2005
From Reinhard Zumkeller, May 31 2010: (Start)
a(n) = ((A002278(n-1)*10 + 3)*10^(n-1) + A002279(n-1))*10 + 6 for n>0.
a(n) = A002283(n)*A002278(n). (End)
G.f.: 36*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Arkadiusz Wesolowski, Dec 26 2011
From Elmo R. Oliveira, Jul 27 2025: (Start)
E.g.f.: 4*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).
a(n) = 36*A002477(n). (End)

Edited by Alois P. Heinz, Aug 21 2019 (merged with A102794, submitted by Richard C. Schroeppel, Feb 26 2005)

A256291 Numbers which have only digits 5 and 6 in base 10.

Original entry on oeis.org

5, 6, 55, 56, 65, 66, 555, 556, 565, 566, 655, 656, 665, 666, 5555, 5556, 5565, 5566, 5655, 5656, 5665, 5666, 6555, 6556, 6565, 6566, 6655, 6656, 6665, 6666, 55555, 55556, 55565, 55566, 55655, 55656, 55665, 55666, 56555, 56556

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..60000] | Set(IntegerToSequence(n, 10)) subset {5, 6}];

[n: n in [1..100000] | Set(Intseq(n)) subset {5,6}]; // :Vincenzo Librandi_, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{5,6},n],{n,5}]]

A256291(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*5

a(n) = A007931(n) + A002278(A000523(n+1)) = A256290(n) + A256077(n) etc.

A178632 a(n) = 45*((10^n - 1)/9)^2.

Original entry on oeis.org

45, 5445, 554445, 55544445, 5555444445, 555554444445, 55555544444445, 5555555444444445, 555555554444444445, 55555555544444444445, 5555555555444444444445, 555555555554444444444445, 55555555555544444444444445, 5555555555555444444444444445, 555555555555554444444444444445

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 45 = 9 * 5;
n=2: ................... 5445 = 99 * 55;
n=3: ................. 554445 = 999 * 555;
n=4: ............... 55544445 = 9999 * 5555;
n=5: ............. 5555444445 = 99999 * 55555;
n=6: ........... 555554444445 = 999999 * 555555;
n=7: ......... 55555544444445 = 9999999 * 5555555;
n=8: ....... 5555555444444445 = 99999999 * 55555555;
n=9: ..... 555555554444444445 = 999999999 * 555555555.

Vincenzo Librandi, 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, A178633, A178634, A178635.

Cf. A002278, A002279, A002283, A002477.

[45*((10^n-1)/9)^2: n in [1..50]]; // Vincenzo Librandi, Dec 28 2010

45 (FromDigits/@Table[PadRight[{}, n, 1], {n, 20}])^2 (* Vincenzo Librandi, Mar 20 2014 *)
LinearRecurrence[{111,-1110,1000},{45,5445,554445},20] (* Harvey P. Dale, Jan 23 2019 *)

A178632(n):=45*((10^n-1)/9)^2$ makelist(A178632(n),n,1,12); /* Martin Ettl, Nov 08 2012 */

a(n)=45*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 45*A002477(n) = A002283(n)*A002279(n).
a(n) = (A002279(n-1)*10^n + A002278(n))*10 + 5.
G.f.: 45*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
From Elmo R. Oliveira, Aug 01 2025: (Start)
E.g.f.: 5*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)

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

A332140 a(n) = 4(10^(2n+1)-1)/9 - 410^n.

Original entry on oeis.org

0, 404, 44044, 4440444, 444404444, 44444044444, 4444440444444, 444444404444444, 44444444044444444, 4444444440444444444, 444444444404444444444, 44444444444044444444444, 4444444444440444444444444, 444444444444404444444444444, 44444444444444044444444444444, 4444444444444440444444444444444

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), A002278 (4*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332141 .. A332149 (variants with different middle digit 1, ..., 9).

A332140 := n -> 4*((10^(2*n+1)-1)/9-10^n);

Array[4 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
LinearRecurrence[{111,-1110,1000},{0,404,44044},20] (* Harvey P. Dale, Jul 06 2021 *)

apply( {A332140(n)=(10^(n*2+1)\9-10^n)*4}, [0..15])

def A332140(n): return (10**(n*2+1)//9-10**n)*4

a(n) = 4*A138148(n) = A002278(2n+1) - 4*10^n.
G.f.: 4*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.

A332149 a(n) = 4(10^(2n+1)-1)/9 + 5*10^n.

Original entry on oeis.org

9, 494, 44944, 4449444, 444494444, 44444944444, 4444449444444, 444444494444444, 44444444944444444, 4444444449444444444, 444444444494444444444, 44444444444944444444444, 4444444444449444444444444, 444444444444494444444444444, 44444444444444944444444444444, 4444444444444449444444444444444

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), A002278 (4*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. A332140 .. A332148 (variants with different middle digit 0, ..., 8).

A332149 := n -> 4*(10^(2*n+1)-1)/9+5*10^n;

Array[4 (10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]

apply( {A332149(n)=10^(n*2+1)\9*4+5*10^n}, [0..15])

def A332149(n): return 10**(n*2+1)//9*4+5*10**n

a(n) = 4*A138148(n) + 9*10^n = A002278(2n+1) + 5*10^n.
G.f.: (9 - 505*x + 100*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.

A180160 (sum of digits) mod (number of digits) of n in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2

Offset: 0

Reinhard Zumkeller, Aug 15 2010

a(n) = A007953(n) mod A055642(n);
a(A061383(n)) = 0; a(A180157(n)) > 0;
a(repdigits)=0: a(A010785(n))=0: a(A002275(n))=0: a(A002276(n))=0: a(A002277(n))=0: a(A002278(n))=0: a(4(n))=0: a(A002279(n))=0: a(A002280(n))=0: a(A002281(n))=0: a(A002282(n))=0: a(A002283(n))=0;
A123522 gives smallest m such that a(m) = n.

Reinhard Zumkeller, Table of n, a(n) for n = 0..25000

Cf. A002275-A002283, A007953, A010785, A055642, A061383, A123522, A180157, A374097.

A180160[n_] := If[n == 0, 0, Mod[Total[#], Length[#]] & [IntegerDigits[n]]];
Array[A180160, 100, 0] (* Paolo Xausa, Jun 30 2024 *)
Join[{0},Table[Mod[Total[IntegerDigits[n]],IntegerLength[n]],{n,110}]] (* Harvey P. Dale, Jul 30 2025 *)

A268620 Numbers whose digital sum is a multiple of 4.

Original entry on oeis.org

0, 4, 8, 13, 17, 22, 26, 31, 35, 39, 40, 44, 48, 53, 57, 62, 66, 71, 75, 79, 80, 84, 88, 93, 97, 103, 107, 112, 116, 121, 125, 129, 130, 134, 138, 143, 147, 152, 156, 161, 165, 169, 170, 174, 178, 183, 187, 192, 196, 202, 206, 211, 215, 219, 220, 224, 228, 233, 237, 242, 246

Offset: 1

Bruno Berselli, Feb 09 2016

a(1498) = 5999 is the smallest term that is congruent to 5 modulo 9.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007953, A061383 (supersequence).
Cf. numbers whose digital sum is a multiple of k: A054683 (k=2), A008585 (k=3), this sequence (k=4), A227793 (k=5).
Subsequences: A002278, A038446, A052218, A052222, A061387, A169967, A235151, A235227, A235229.

[n: n in [0..250] | IsIntegral(&+Intseq(n)/4)];

select(t -> convert(convert(t,base,10),`+`) mod 4 = 0, [$1..1000]); # Robert Israel, Feb 09 2016

Select[Range[0, 250], IntegerQ[Total[IntegerDigits[#]]/4] &]

A332141 a(n) = 4(10^(2n+1)-1)/9 - 3*10^n.

Original entry on oeis.org

1, 414, 44144, 4441444, 444414444, 44444144444, 4444441444444, 444444414444444, 44444444144444444, 4444444441444444444, 444444444414444444444, 44444444444144444444444, 4444444444441444444444444, 444444444444414444444444444, 44444444444444144444444444444, 4444444444444441444444444444444

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), A002278 (4*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

A332141 := n -> 4*(10^(2*n+1)-1)/9-3*10^n;

Array[4 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{1,414,44144},20] (* or *) Table[ FromDigits[Join[PadRight[{},n,4],{1},PadRight[{},n,4]]],{n,0,20}](* Harvey P. Dale, Aug 17 2020 *)

apply( {A332141(n)=10^(n*2+1)\9*4-3*10^n}, [0..15])

def A332141(n): return 10**(n*2+1)//9*4-3*10**n

a(n) = 4*A138148(n) + 1*10^n = A002278(2n+1) - 3*10^n.
G.f.: (1 + 303*x - 700*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.

cp's OEIS Frontend

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

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A075415 Squares of A002280 or numbers (666...6)^2.

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A256291 Numbers which have only digits 5 and 6 in base 10.

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A178632 a(n) = 45*((10^n - 1)/9)^2.

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

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A332140 a(n) = 4*(10^(2n+1)-1)/9 - 4*10^n.

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A332149 a(n) = 4*(10^(2*n+1)-1)/9 + 5*10^n.

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A332140 a(n) = 4(10^(2n+1)-1)/9 - 410^n.

A332149 a(n) = 4(10^(2n+1)-1)/9 + 5*10^n.

A332141 a(n) = 4(10^(2n+1)-1)/9 - 3*10^n.