A002276 -id:A002276 - cp's OEIS frontend

A332127 a(n) = 2(10^(2n+1)-1)/9 + 510^n.

7, 272, 22722, 2227222, 222272222, 22222722222, 2222227222222, 222222272222222, 22222222722222222, 2222222227222222222, 222222222272222222222, 22222222222722222222222, 2222222222227222222222222, 222222222222272222222222222, 22222222222222722222222222222, 2222222222222227222222222222222

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. A332117 .. A332197 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

A332127 := n -> 2*(10^(2*n+1)-1)/9+5*10^n;

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

apply( {A332127(n)=10^(n*2+1)\9*2+5*10^n}, [0..15])

def A332127(n): return 10**(n*2+1)//9*2+5*10**n

a(n) = 2*A138148(n) + 7*10^n = A002276(2n+1) + 5*10^n.
G.f.: (7 - 505*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.

A332128 a(n) = 2(10^(2n+1)-1)/9 + 610^n.

Original entry on oeis.org

8, 282, 22822, 2228222, 222282222, 22222822222, 2222228222222, 222222282222222, 22222222822222222, 2222222228222222222, 222222222282222222222, 22222222222822222222222, 2222222222228222222222222, 222222222222282222222222222, 22222222222222822222222222222, 2222222222222228222222222222222

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. A332118 .. A332178, A181965 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

A332128 := n -> 2*(10^(2*n+1)-1)/9+6*10^n;

Array[2 (10^(2 # + 1)-1)/9 + 6*10^# &, 15, 0]

apply( {A332128(n)=10^(n*2+1)\9*2+6*10^n}, [0..15])

def A332128(n): return 10**(n*2+1)//9*2+6*10**n

a(n) = 2*A138148(n) + 8*10^n = A002276(2n+1) + 6*10^n = 2*A332114(n).
G.f.: (8 - 606*x + 400*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.

A125725 Numbers whose base-7 representation is 222....2.

Original entry on oeis.org

0, 2, 16, 114, 800, 5602, 39216, 274514, 1921600, 13451202, 94158416, 659108914, 4613762400, 32296336802, 226074357616, 1582520503314, 11077643523200, 77543504662402, 542804532636816, 3799631728457714, 26597422099204000

Offset: 1

Zerinvary Lajos, Feb 02 2007

			base 7.......decimal
0..................0
2..................2
22................16
222..............114
2222.............800
22222...........5602
222222.........39216
2222222.......274514
22222222.....1921600
222222222...13451202
etc...........etc.

Davis Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A007310, A023000, A047522, A165759.

Cf. also A002276, A005610, A020988, A024023, A125831, A125835, A125857 for related or similarly constructed sequences.

List([1..25], n-> (7^(n-1) -1)/3); # G. C. Greubel, May 23 2019

[0] cat [n:n in [1..15000000]| Set(Intseq(n,7)) subset [2]]; // Marius A. Burtea, May 06 2019

[(7^(n-1)-1)/3: n in [1..25]]; // Marius A. Burtea, May 06 2019

Maple
```
seq(2*(7^n-1)/6, n=0..25);
```

FromDigits[#,7]&/@Table[PadLeft[{2},n,2],{n,0,25}]  (* Harvey P. Dale, Apr 13 2011 *)
(7^(Range[25]-1) - 1)/3 (* G. C. Greubel, May 23 2019 *)

vector(25, n, (7^(n-1)-1)/3) \\ Davis Smith, Apr 04 2019

[(7^(n-1) -1)/3 for n in (1..25)] # G. C. Greubel, May 23 2019

a(n) = (7^(n-1) - 1)/3 = 2*A023000(n-1).
a(n) = 7*a(n-1) + 2, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
G.f.: 2*x^2 / ( (1-x)*(1-7*x) ). - R. J. Mathar, Sep 30 2013
From Davis Smith, Apr 04 2019: (Start)
A007310(a(n) + 1) = 7^(n - 1).
A047522(a(n + 1)) = -1*A165759(n). (End)
E.g.f.: (exp(7*x) - 7*exp(x) + 6)/21. - Stefano Spezia, Jan 12 2025

Offset corrected by N. J. A. Sloane, Oct 02 2010

A274126 Numbers with digits larger than 1 sorted by product of digits minus sum of digits, then by size.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 222, 25, 33, 26, 27, 34, 223, 28, 29, 35, 44, 224, 2222, 36, 233, 37, 45, 225, 38, 46, 226, 39, 55, 234, 2223, 47, 227, 333, 56, 48, 228, 235, 244, 2224, 22222, 49, 57, 229, 66, 236, 334, 2233, 58, 67, 245, 2225, 237, 59, 68, 335

Offset: 1

David A. Corneth, Jun 10 2016

Let PS(n) be product of digits of n minus sum of digits of n (=-A062329(n)). Then a(n) is PS(A037344(m)) ordered by PS(n) for values of m such that A037344 has its digits in nondecreasing order. If PS(m) some nonzero term m of A002276 exceed some bound, all positive integers t larger than that term without zeros and ones exceed have a larger value for PS(t).
Prepending -A062329(a(n)) or more ones before a(n) gives terms of A274125.
Permuting digits of A274125 gives A254621. Permutations of digits can be found in A261370. The union of A254621 and A011540 is A062996. The b-file lists terms having PS(n) <= 10^6.

			Suppose we want to order the nondecreasing integers without zeros and ones up to PS(m) = 50. We see that 222222 has PS(222222) = 52, so we only have to check such nondecreasing integers up to 222222. Not all of those must be checked, which is used in the program.
25 is a term. Prepending PS(25) = -A062329(25) = 3 ones before 25 gives 11125, which is a term of A274125. Permuting digits of 11125 gives for example 12511, which is a term of A254621.

David A. Corneth, Table of n, a(n) for n = 1..10696
David A. Corneth, PARI program

Cf. A011540, A037344, A062329, A062996, A254621, A261370, A274125.

PARI
```
See program in link "PARI program".
```

A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 11, 2, 0, 0, 111, 22, 3, 0, 0, 1111, 222, 33, 4, 0, 0, 11111, 2222, 333, 44, 5, 0, 0, 111111, 22222, 3333, 444, 55, 6, 0, 0, 1111111, 222222, 33333, 4444, 555, 66, 7, 0, 0, 11111111, 2222222, 333333, 44444, 5555, 666, 77, 8, 0

Offset: 0

Stefano Spezia, Sep 14 2023

			The array begins:
  0,     0,     0,     0,     0,     0, ...
  0,     1,     2,     3,     4,     5, ...
  0,    11,    22,    33,    44,    55, ...
  0,   111,   222,   333,   444,   555, ...
  0,  1111,  2222,  3333,  4444,  5555, ...
  0, 11111, 22222, 33333, 44444, 55555, ...
  ...

Cf. A000004 (n=0 or k=0), A001477 (n=1), A002275 (k=1), A002276 (k=2), A002277 (k=3), A002278 (k=4), A002279 (k=5), A002280 (k=6), A002281 (k=7), A002282 (k=8), A002283 (k=9), A008593 (n=2), A053422 (main diagonal), A105279 (k=10), A132583, A177769 (n=3), A365645 (antidiagonal sums), A365646.

A[n_,k_]:=k(10^n-1)/9; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten

O.g.f.: x*y/((1 - x)*(1 - 10*x)*(1 - y)^2).
E.g.f.: y*exp(x+y)*(exp(9*x) - 1)/9.
A(n, 11) = A132583(n-1) for n > 0.
A(n, 12) = A073551(n+1) for n > 0.

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

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

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A137318 Concatenation of segments of the digit sequence 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3....

Original entry on oeis.org

1, 31, 313, 1313, 13131, 313131, 3131313, 13131313, 131313131, 3131313131, 31313131313, 131313131313, 1313131313131, 31313131313131, 313131313131313, 1313131313131313, 13131313131313131, 313131313131313131

Offset: 1

Ctibor O. Zizka, Apr 06 2008

A000042 is 1,11,111,1111,11111,... concatenation of 111111111111111....
A002276 is 2,22,222,2222,22222,... concatenation of 222222222222222....
A133013 is 2,35,71113,... concatenation of 2 3 5 7 11 13 17 19 23 29,...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000042.

Module[{nn=20},FromDigits/@TakeList[PadRight[{},(nn(nn+1))/2,{1,3}],Range[ nn]]] (* Harvey P. Dale, Aug 04 2021 *)

O.g.f.: x*(100x^4 + 200x^3 + 83x^2 + 20x + 1)/((10x-1)(100x^2+1)(x-1)(x^2+1)). - R. J. Mathar, Apr 09 2008

More terms from R. J. Mathar, Apr 09 2008

A176892 Decimal representation of the reverted binary representation of n followed by digits substitution 0->2, 1->3.

Original entry on oeis.org

2, 3, 23, 33, 223, 323, 233, 333, 2223, 3223, 2323, 3323, 2233, 3233, 2333, 3333, 22223, 32223, 23223, 33223, 22323, 32323, 23323, 33323, 22233, 32233, 23233, 33233, 22333, 32333, 23333, 33333, 222223, 322223, 232223, 332223, 223223

Offset: 0

Roger L. Bagula, Apr 28 2010

Revert the digits of A007088(n), preserving zeros, and increase each digit by 2 (add the repunit A002276 with the same number of digits).

			n=10 is A007088(10)= 1010 in binary, reverted 0101. Adding 2222 generates a(10)=2323.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007088, A002276, A032810.

import Data.List (unfoldr); import Data.Tuple (swap)
a176892 0 = 2a176892 n = foldl (\v d -> 10 * v + d + 2) 0 $
   unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) n
-- Reinhard Zumkeller, Jul 16 2015

Table[Sum[Table[((IntegerDigits[ n, 2]) /. 0 -> 2) /. 1 -> 3, {n, 0, 50}][[n]][[m]]*10^(m - 1),
{m, 1, Length[Table[((IntegerDigits[n, 2]) /. 0 -> 2) /. 1 -> 3, {n, 0, 50}][[n]]]}], {n, 1, 51}]

A271528 a(n) = 2*(10^n - 1)^2/27.

Original entry on oeis.org

0, 6, 726, 73926, 7405926, 740725926, 74073925926, 7407405925926, 740740725925926, 74074073925925926, 7407407405925925926, 740740740725925925926, 74074074073925925925926, 7407407407405925925925926, 740740740740725925925925926, 74074074074073925925925925926

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

All terms are multiple of 6.
Converges in a 10-adic sense to ...925925925926.
A transformation of the Wonderful Demlo numbers (A002477).
More generally, the ordinary generating function for the transformation of the Wonderful Demlo numbers, is k*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).

			n=1:                  6 = 2 * 3;
n=2:                726 = 22 * 33;
n=3:              73926 = 222 * 333;
n=4:            7405926 = 2222 * 3333;
n=5:          740725926 = 22222 * 33333;
n=6:        74073925926 = 222222 * 333333;
n=7:      7407405925926 = 2222222 * 3333333;
n=8:    740740725925926 = 22222222 * 33333333;
n=9:  74074073925925926 = 222222222 * 333333333, etc.

Ilya Gutkovskiy, Transformation of the Wonderful Demlo numbers
Eric Weisstein's World of Mathematics, Demlo Number
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275, A002276, A002277, A002477.

Cf. similar sequences of the form k*((10^n - 1)/9)^2: A075411 (k=4), this sequence (k=6), A075412 (k=9), A075413 (k=16), A178630 (k=18), A075414 (k=25), A178631 (k=27), A075415 (k=36), A178632 (k=45), A075416 (k=49), A178633 (k=54), A178634 (k=63), A075417 (k=64), A178635 (k=72), A059988 (k=81).

Table[2 ((10^n - 1)^2/27), {n, 0, 15}]
LinearRecurrence[{111, -1110, 1000}, {0, 6, 726}, 16]

x='x+O('x^99); concat(0, Vec(6*x*(1+10*x)/(1-111*x+1110*x^2-1000*x^3))) \\ Altug Alkan, Apr 09 2016

for n in range(0,10**1):print((int)((2*(10**n-1)**2)/27))
# Soumil Mandal, Apr 10 2016

O.g.f.: 6*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).
E.g.f.: 2 (exp(x) - 2*exp(10*x) + exp(100*x))/27.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
a(n) = 6*A002477(n) = 6*A002275(n)^2 = A002276(n)*A002277(n) = sqrt(A075411(n)*A075412(n)).
Sum_{n>=1} 1/a(n) = 0.1680577405662077350849154881928636039793563...
Lim_{n -> infinity} a(n + 1)/a(n) = 100.

A322925 Expansion of x(1 + 2x + 10x^2)/((1 - x^2)(1 - 10*x^2)).

Original entry on oeis.org

0, 1, 2, 21, 22, 221, 222, 2221, 2222, 22221, 22222, 222221, 222222, 2222221, 2222222, 22222221, 22222222, 222222221, 222222222, 2222222221, 2222222222, 22222222221, 22222222222, 222222222221, 222222222222, 2222222222221, 2222222222222, 22222222222221

Offset: 0

Vincenzo Librandi, Mar 16 2019

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,11,0,-10).

Bisections give: A002276 (even part), A165402 (odd part).

a:=[0,1,2,21];; for n in [5..30] do a[n]:=11*a[n-2]-10*a[n-4]; od; Print(a); # Muniru A Asiru, Apr 10 2019

I:=[0,1,2,21]; [n le 4 select I[n] else 11*Self(n-2)-10*Self(n-4): n in [1..30]];

seq(coeff(series(x*(1+2*x+10*x^2)/((1-x^2)*(1-10*x^2)),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Apr 10 2019

CoefficientList[Series[x (1 + 2 x + 10 x^2)/((1 - x^2) (1 - 10 x^2)), {x, 0, 33}], x]
LinearRecurrence[{0,11,0,-10},{0,1,2,21},30] (* Harvey P. Dale, Mar 02 2021 *)

G.f.: x*(1 + 2*x + 10*x^2)/((1 - x^2)*(1 - 10*x^2)).
a(n) = 11*a(n-2) - 10* a(n-4).
a(n) = 2*(10^n - 1)/9 for n even; a(n) = (2*10^n - 11)/9 otherwise.
a(n) = (2/9)*10^floor((n + 1)/2) + (-1)^n/2 - 13/18. - Bruno Berselli, Mar 16 2019

cp's OEIS Frontend

A332127 a(n) = 2*(10^(2n+1)-1)/9 + 5*10^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A332128 a(n) = 2*(10^(2n+1)-1)/9 + 6*10^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A125725 Numbers whose base-7 representation is 222....2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A274126 Numbers with digits larger than 1 sorted by product of digits minus sum of digits, then by size.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A137215 a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

Extensions

A137318 Concatenation of segments of the digit sequence 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3....

Views

Author

Keywords

Comments

A332127 a(n) = 2(10^(2n+1)-1)/9 + 510^n.

A332128 a(n) = 2(10^(2n+1)-1)/9 + 610^n.

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

A322925 Expansion of x(1 + 2x + 10x^2)/((1 - x^2)(1 - 10*x^2)).