A074991 -id:A074991 - 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.)

A075000 Smallest number such that n*a(n) is a concatenation of n consecutive integers; or 0 if no such number exists.

Original entry on oeis.org

1, 6, 41, 864, 2469, 20576, 493827, 7098637639, 13717421, 1234567891, 82737383012865106529, 10288065758426, 3513762316247164732, 563643651522439401227280, 8230452606740808761

Offset: 1

Amarnath Murthy, Aug 31 2002

Conjecture: For every n there exists a nonzero a(n).

			a(11) = 82737383012865106529 as 11*82737383012865106529 = 910111213141516171819 is the concatenation of 11 numbers from 9 to 19.

Robert G. Wilson v, Table of n, a(n) for n = 1..100

Cf. A074991, A074992, A074993, A074994, A074995, A074996, A036377, A073086, A074999, A075001.

f[ n_ ] := Block[ {id = Range@n}, While[ k = FromDigits@ Flatten@ IntegerDigits@ id/n; !IntegerQ@k, id++ ]; k ]; Array[ f, 16 ] (* Robert G. Wilson v, Oct 19 2007 *)

a(n) = A077306(n)/n. - Amarnath Murthy, Nov 03 2002

More terms from Rick L. Shepherd, Sep 03 2002

A001703 Decimal concatenation of n, n+1, and n+2.

Original entry on oeis.org

12, 123, 234, 345, 456, 567, 678, 789, 8910, 91011, 101112, 111213, 121314, 131415, 141516, 151617, 161718, 171819, 181920, 192021, 202122, 212223, 222324, 232425, 242526, 252627, 262728, 272829, 282930, 293031, 303132, 313233, 323334, 333435, 343536, 353637, 363738

Offset: 0

mag(AT)laurel.salles.entpe.fr

All terms are divisible by 3. Every third term starting with a(2) is divisible by 9. - Alonso del Arte, May 27 2013

			a(8) = 8910 since the three consecutive numbers starting with 8 are 8, 9, 10, and these concatenate to 8910. (This is the first term that differs from A193431).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A074991.
For concatenations of exactly k consecutive integers see A000027 (k=1), A127421 (k=2), A279204 (k=4). For 2 or more see A035333.
See also A127422, A127423, A127424.

read(transforms) :
A001703 := proc(n)
    digcatL([n,n+1,n+2]) ;
end proc:
seq(A001703(n),n=1..20) ; # R. J. Mathar, Mar 29 2017
# Third Maple program:
a:= n-> parse(cat(n, n+1, n+2)):
seq(a(n), n=0..50); # Alois P. Heinz, Mar 29 2017

concat3Nums[n_] := FromDigits@ Flatten@ IntegerDigits[{n, n + 1, n + 2}]; Array[concat3Nums, 25] (* Robert G. Wilson v *)

a(n)=eval(Str(n,n+1,n+2)) \\ Charles R Greathouse IV, Oct 08 2011

for n in range(100): print(int(str(n)+str(n+1)+str(n+2))) # David F. Marrs, Sep 18 2018

The portion of the sequence with all three numbers having d digits - i.e., n in 10^(d-1)..10^d-3 - is in arithmetic sequence: a(n) = (10^(2*d)+10^d+1)*n + (10^d+2). - Franklin T. Adams-Watters, Oct 07 2011

Initial term 12 added and offset changed to 0 at the suggestion of R. J. Mathar. - N. J. A. Sloane, Mar 29 2017

A074996 Floor of concatenation of n, n+1, n+2, n+3, n+4, n+5 divided by 6.

Original entry on oeis.org

2057, 20576, 39094, 57613, 76131, 946485, 11315168, 131516852, 1485018535, 15168520219, 16852021902, 18535523586, 20219025269, 21902526953, 23586028636, 25269530320, 26953032003, 28636533687, 30320035370

Offset: 0

Amarnath Murthy, Aug 31 2002

Cf. A074991, A074992, A074993, A074994, A074995, A074997, A073086, A074999.

Table[Floor[FromDigits[Flatten[IntegerDigits[Range[n,n+5]]]]/6],{n,0,18}] (* Jayanta Basu, May 22 2013 *)

Edited by Dean Hickerson, Nov 03 2002

A074999 Floor[concatenation of nine consecutive numbers from n to n+8 divided by 9].

Original entry on oeis.org

1371742, 13717421, 260630990, 3840876779, 50754344568, 630990012357, 7543445680146, 87677901347935, 990012357015724, 10112346812683513, 11234681268351302, 12357015724019091, 13479350179686880

Offset: 0

Amarnath Murthy, Aug 31 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A074991, A074992, A074993, A074994, A074995, A074996, A036377, A073086.

Table[FromDigits[Flatten[IntegerDigits/@Range[n,n+8]]]/9,{n,0,20}] (* Harvey P. Dale, Feb 19 2018 *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 18 2003

A073086 Floor[concatenation of eight consecutive numbers from n to n+7 divided by 8].

Original entry on oeis.org

154320, 1543209, 2932098, 43209863, 570986376, 7098637639, 84863763901, 986376390164, 11137639016426, 113763901642689, 126390164268952, 139016426895214, 151642689521477, 164268952147740, 176895214774002

Offset: 0

Amarnath Murthy, Aug 31 2002

Cf. A074991, A074992, A074993, A074994, A074995, A074996, A036377, A074999.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 18 2003

A074993 a(n) = floor((concatenation of n, n+1)/2).

Original entry on oeis.org

0, 6, 11, 17, 22, 28, 33, 39, 44, 455, 505, 556, 606, 657, 707, 758, 808, 859, 909, 960, 1010, 1061, 1111, 1162, 1212, 1263, 1313, 1364, 1414, 1465, 1515, 1566, 1616, 1667, 1717, 1768, 1818, 1869, 1919, 1970, 2020, 2071, 2121, 2172, 2222, 2273, 2323, 2374

Offset: 0

Amarnath Murthy, Aug 31 2002

The first differences follow a pattern. Odd-indexed terms and even-indexed terms form separate A.P.s with the same common difference for all n except n = 10^k -1. The corresponding common differences are the repunits = (10^(d+1)-1)/9 where d = the number of digits in n.

Cf. A001704, A074991, A074992, A074994, A074995, A074996, A074997, A073086, A074999, A127421.

cc[n_]:=Floor[FromDigits[Join[IntegerDigits[n],IntegerDigits[n+1]]]/2]; Array[cc,40,0] (* Harvey P. Dale, Nov 11 2011 *)

A074994 Floor of concatenation of n, n+1, n+2, n+3 divided by 4.

Original entry on oeis.org

30, 308, 586, 864, 1141, 1419, 1697, 19727, 222752, 2275278, 2527803, 2780328, 3032853, 3285379, 3537904, 3790429, 4042954, 4295480, 4548005, 4800530, 5053055, 5305581, 5558106, 5810631, 6063156, 6315682, 6568207, 6820732

Offset: 0

Amarnath Murthy, Aug 31 2002

			a(7) = floor(78910/4) = 19727.

Cf. A074991, A074992, A074993, A074995, A074996, A074997, A073086, A074999.

Edited by Dean Hickerson, Nov 03 2002

A074995 Floor of concatenation of n, n+1, n+2, n+3, n+4 divided by 5.

Original entry on oeis.org

246, 2469, 4691, 6913, 9135, 11357, 135782, 1578202, 17820222, 182022242, 202224262, 222426283, 242628303, 262830323, 283032343, 303234363, 323436384, 343638404, 363840424, 384042444, 404244464, 424446485, 444648505, 464850525

Offset: 0

Amarnath Murthy, Aug 31 2002

Cf. A074991, A074992, A074993, A074994, A074996, A074997, A073086, A074999.

Edited by Dean Hickerson, Nov 03 2002

A075003 Floor[ concatenation of n+1 and n divided by 2 ].

Original entry on oeis.org

5, 10, 16, 21, 27, 32, 38, 43, 49, 54, 555, 605, 656, 706, 757, 807, 858, 908, 959, 1009, 1060, 1110, 1161, 1211, 1262, 1312, 1363, 1413, 1464, 1514, 1565, 1615, 1666, 1716, 1767, 1817, 1868, 1918, 1969, 2019, 2070, 2120, 2171, 2221, 2272, 2322, 2373, 2423

Offset: 0

Amarnath Murthy, Sep 01 2002

			Sequence exhibits similar properties to A074993.

Cf. A074991 to A075000 and A075004 to A075010.

seq(floor((n*10+(n-1))/2),n=1..10),seq(floor((n*100+(n-1))/2),n=11..99);

More terms from Sascha Kurz, Jan 14 2003

cp's OEIS Frontend

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

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A075000 Smallest number such that n*a(n) is a concatenation of n consecutive integers; or 0 if no such number exists.

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A001703 Decimal concatenation of n, n+1, and n+2.

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A074996 Floor of concatenation of n, n+1, n+2, n+3, n+4, n+5 divided by 6.

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A074999 Floor[concatenation of nine consecutive numbers from n to n+8 divided by 9].

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A073086 Floor[concatenation of eight consecutive numbers from n to n+7 divided by 8].

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A074993 a(n) = floor((concatenation of n, n+1)/2).

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A074994 Floor of concatenation of n, n+1, n+2, n+3 divided by 4.

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