A086574 -id:A086574 - cp's OEIS frontend

A074992 a(n) = (10^(2*n) + 10^n + 1)/3.

1, 37, 3367, 333667, 33336667, 3333366667, 333333666667, 33333336666667, 3333333366666667, 333333333666666667, 33333333336666666667, 3333333333366666666667, 333333333333666666666667, 33333333333336666666666667, 3333333333333366666666666667, 333333333333333666666666666667

Offset: 0

Amarnath Murthy, Aug 31 2002

Apart from the initial 1, common difference of the arithmetic progression pertaining to the sequence A074991.
This is also a root sequence pertaining to the patterned perfect square sequence 1369, 11336689,111333666889,... i.e., k ones, k threes and k sixes followed by (k-1) 8's and a 9. (37^2 = 1369).
This is a self-complementing sequence: each term has even number of digits (the first one has to be read 01, the leading zero is important). If you add the first half to the second half of any term, you get the sequence A011557, powers of 10. Furthermore, the reciprocals of the sequence terms, except the first one, give a sequence of periodic terms with period sequence as in A008585, a(n) = 3*n, and value given by A086574, a(n)=k where R(k+3)=3. - Rodolfo A. Fiorini, Jul 14 2016

Rodolfo A. Fiorini, Computerized tomography noise reduction by CICT optimized exponential cyclic sequences (OECS) co-domain, Fundamenta Informaticae, 141(2015), pp. 115-134.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A008585, A011557, A066138, A074991, A086574, A274766.

A074992 := proc(n)
    (10^(2*n)+10^n+1)/3 ;
end proc:
seq(A074992(n),n=0..15) ; # R. J. Mathar, May 06 2017

{01}~Join~Table[FromDigits@ Flatten@ Map[IntegerDigits, {#, 10^n - #}] &@ Floor[10^n/3], {n, 12}] (* Michael De Vlieger, Jul 22 2016 *)

a(n) = (10^(2*n) + 10^n + 1)/3; \\ Michel Marcus, Sep 14 2013

Vec(-x*(1000*x^2-740*x+37)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100)) \\ Colin Barker, Sep 23 2013

a(n)=my(x=10^n); (x^2+x+1)/3 \\ Charles R Greathouse IV, Jul 22 2016

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3), for n > 2, a(0)=1, a(1)=37, a(2)=3367.
G.f.: (1 - 74*x + 370*x^2)/((1-x)*(1-10*x)*(1-100*x)). - Colin Barker, Sep 23 2013 and Robert Israel, Jul 22 2016
From Elmo R. Oliveira, Sep 12 2024: (Start)
E.g.f.: exp(x)*(exp(99*x) + exp(9*x) + 1)/3.
a(n) = A066138(n)/3. (End)

Entry revised (new definition, new offset, new initial term, etc.) by N. J. A. Sloane, Jul 27 2016 (Some of the old programs may need slight modifications.)

A086573 a(n) = 2*(10^n - 1).

Original entry on oeis.org

0, 18, 198, 1998, 19998, 199998, 1999998, 19999998, 199999998, 1999999998, 19999999998, 199999999998, 1999999999998, 19999999999998, 199999999999998, 1999999999999998, 19999999999999998, 199999999999999998, 1999999999999999998, 19999999999999999998, 199999999999999999998

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+2) = 2.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)), A083812.

One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

a(n) = 2*9*A002275(n) = 2*A002283(n).
R(a(n)) = A086580(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 18*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 2*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Edited by Jinyuan Wang, Aug 04 2021

A086578 a(n) = 7*(10^n - 1).

Original entry on oeis.org

0, 63, 693, 6993, 69993, 699993, 6999993, 69999993, 699999993, 6999999993, 69999999993, 699999999993, 6999999999993, 69999999999993, 699999999999993, 6999999999999993, 69999999999999993, 699999999999999993, 6999999999999999993, 69999999999999999993, 699999999999999999993

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+7) = 7.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)).
One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.
Sequences of the form m*10^n - 7: 3*A033175 (m=1, 10), A086943 (m=3), 3*A185127 (m=4), this sequence (m=7), A100412 (m=8).

[7*(10^n -1): n in [0..20]]; // G. C. Greubel, Apr 14 2023

LinearRecurrence[{11,-10}, {0,63}, 31] (* G. C. Greubel, Apr 14 2023 *)

[7*(10^n -1) for n in range(21)] # G. C. Greubel, Apr 14 2023

a(n) = 7*9*A002275(n) = 7*A002283(n).
R(a(n)) = A086575(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 63*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 7*(exp(10*x) - exp(x)). - G. C. Greubel, Apr 14 2023

Edited by Jinyuan Wang, Aug 04 2021

A086580 a(n) = 9*(10^n - 1).

Original entry on oeis.org

0, 81, 891, 8991, 89991, 899991, 8999991, 89999991, 899999991, 8999999991, 89999999991, 899999999991, 8999999999991, 89999999999991, 899999999999991, 8999999999999991, 89999999999999991, 899999999999999991, 8999999999999999991, 89999999999999999991, 899999999999999999991

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+9) = 9.

G. C. Greubel, Table of n, a(n) for n = 0..990
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)), A086573.

One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

[9*(10^n -1): n in [0..30]]; // G. C. Greubel, Jul 07 2023

Table[9*(10^n-1), {n,0,30}] (* G. C. Greubel, Jul 07 2023 *)

A086580=BinaryRecurrenceSequence(11,-10,0,81)
[A086580(n) for n in range(30)] # G. C. Greubel, Jul 07 2023

a(n) = 9*9*A002275(n) = 9*A002283(n).
R(a(n)) = A086573(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 81*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 9*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Name edited by Jinyuan Wang, Aug 04 2021

A086575 a(n) = 4*(10^n - 1).

Original entry on oeis.org

0, 36, 396, 3996, 39996, 399996, 3999996, 39999996, 399999996, 3999999996, 39999999996, 399999999996, 3999999999996, 39999999999996, 399999999999996, 3999999999999996, 39999999999999996, 399999999999999996, 3999999999999999996, 39999999999999999996, 399999999999999999996

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+4) = 4.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)), A083811.

One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

LinearRecurrence[{11,-10},{0,36},20] (* Harvey P. Dale, May 14 2017 *)

a(n) = 4*9*A002275(n) = 4*A002283(n).
R(a(n)) = A086578(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 36*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 4*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Edited by Jinyuan Wang, Aug 04 2021

A086576 a(n) = 5*(10^n - 1).

Original entry on oeis.org

0, 45, 495, 4995, 49995, 499995, 4999995, 49999995, 499999995, 4999999995, 49999999995, 499999999995, 4999999999995, 49999999999995, 499999999999995, 4999999999999995, 49999999999999995, 499999999999999995, 4999999999999999995, 49999999999999999995, 499999999999999999995

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+5) = 5.

			From _John Elias_, Jun 23 2021: (Start)
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45;
11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 = 495;
111 + 222 + 333 + 444 + 555 + 666 + 777 + 888 + 999 = 4995;
1111 + 2222 + 3333 + 4444 + 5555 + 6666 + 7777 + 8888 + 9999 = 49995;
11111 + 22222 + 33333 + 44444 + 55555 + 66666 + 77777 + 88888 + 99999 = 499995; etc. (End)

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)).

Cf. One of family of sequences of form a(n)=k, where R(k+m)=m, m = 1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

Join[{0}, LinearRecurrence[{11,-10},{45,495},50]] (* G. C. Greubel, Jul 08 2016 *)

a(n) = 5*9*A002275(n) = 5*A002283(n).
R(a(n)) = A086577(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 45*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 5*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Name edited by Jinyuan Wang, Aug 04 2021

A086577 a(n) = 6*(10^n - 1).

Original entry on oeis.org

0, 54, 594, 5994, 59994, 599994, 5999994, 59999994, 599999994, 5999999994, 59999999994, 599999999994, 5999999999994, 59999999999994, 599999999999994, 5999999999999994, 59999999999999994, 599999999999999994, 5999999999999999994, 59999999999999999994, 599999999999999999994

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+6) = 6.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)).

One of a family of sequences of the form "Numbers k such that reverse(k+m) = m", m = 1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

LinearRecurrence[{11,-10},{0,54},30] (* Harvey P. Dale, Nov 27 2022 *)

a(n) = 6*9*A002275(n) = 6*A002283(n).
R(a(n)) = A086576(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 54*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 6*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Name edited by Jinyuan Wang, Aug 04 2021

A086579 a(n) = 8*(10^n - 1).

Original entry on oeis.org

0, 72, 792, 7992, 79992, 799992, 7999992, 79999992, 799999992, 7999999992, 79999999992, 799999999992, 7999999999992, 79999999999992, 799999999999992, 7999999999999992, 79999999999999992, 799999999999999992, 7999999999999999992, 79999999999999999992, 799999999999999999992

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+8) = 8.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)).

One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

LinearRecurrence[{11,-10},{0,72},30] (* Harvey P. Dale, Mar 22 2025 *)

a(n) = 8*9*A002275(n) = 8*A002283(n).
R(a(n)) = A086574(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 72*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 8*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Edited by Jinyuan Wang, Aug 04 2021

A173766 a(n) = (10^n+11)/3.

Original entry on oeis.org

7, 37, 337, 3337, 33337, 333337, 3333337, 33333337, 333333337, 3333333337, 33333333337, 333333333337, 3333333333337, 33333333333337, 333333333333337, 3333333333333337, 33333333333333337, 333333333333333337, 3333333333333333337, 33333333333333333337

Offset: 1

Vincenzo Librandi, Feb 24 2010

			For n=2, a(2)=10*7-33=37; n=3, a(3)=10*37-33=337; n=4, a(4)=10*337-33=3337.

Bruno Berselli, Table of n, a(n) for n = 1..1000.
Bruno Berselli, Examples with a(n)*A086574(n-1)+11 = A163449(n).
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A093168.

Cf. A086574, A163449. - Bruno Berselli, Jun 09 2010

NestList[10#-33&,7,20] (* Harvey P. Dale, Aug 01 2022 *)

a(n) = 10*a(n-1)-33 (with a(1) = 7).
From Bruno Berselli, Jun 09 2010: (Start)
G.f.: x*(7-40*x)/((1-x)*(1-10*x)).
a(n)-11*a(n-1)+10*a(n-2) = 0 for n>2. (End)

I reduced the fraction in the definition to "(10^n+11)/3". The factor 3 was simply irrelevant. - Ivan Panchenko, Jun 05 2010

A083813 a(n) = 3*(10^n-1).

Original entry on oeis.org

27, 297, 2997, 29997, 299997, 2999997, 29999997, 299999997, 2999999997, 29999999997, 299999999997, 2999999999997, 29999999999997, 299999999999997

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 08 2003

nonn

Original definition: 3n+1 is the digit reversal of n+1.
1. a(n) = 27 + 270 + 2700 + ... up to n terms = sum of n terms of the geometric progression with the first term 27 and common ratio 10.
2. a(n) = 27*A000042(n) (the unary sequence).
Equals A086574 restricted to positive indices. See that entry for many more comments, formulas and references. - M. F. Hasler, Jul 29 2016

Essentially a duplicate of A086574.

Cf. A000042, A083811, A083812.

3(10^Range[20]-1) (* or *) Table[10 FromDigits[PadRight[{2},n,9]]+7,{n,20}] (* Harvey P. Dale, Jan 25 2020 *)

Edited by M. F. Hasler, Jul 29 2016

cp's OEIS Frontend

A074992 a(n) = (10^(2*n) + 10^n + 1)/3.

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A086573 a(n) = 2*(10^n - 1).

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A086578 a(n) = 7*(10^n - 1).

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A086580 a(n) = 9*(10^n - 1).

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A086575 a(n) = 4*(10^n - 1).

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A086576 a(n) = 5*(10^n - 1).

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A086577 a(n) = 6*(10^n - 1).

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