A002279 -id:A002279 - cp's OEIS frontend

A358599 Number of genetic relatives of a person M in a genealogical tree extending back n generations and where everyone has 5 children down to the generation of M.

1, 7, 59, 563, 5571, 55587, 555619, 5555683, 55555811, 555556067, 5555556579, 55555557603, 555555559651, 5555555563747, 55555555571939, 555555555588323, 5555555555621091, 55555555555686627, 555555555555817699, 5555555555556079843, 55555555555556604131

Offset: 0

Hans Braxmeier, Nov 23 2022

M has 2 parents, 4 grandparents, and so on up to 2^n top ancestors at the top of the tree.
The genetic relatives of M are all descendants of those ancestors.
M is a genetic relative of himself or herself.

Hans Braxmeier, Calculating the number of genetic relative people in a genealogical tree.
Index entries for linear recurrences with constant coefficients, signature (13,-32,20).

Other numbers of children: A076024 (2), A358504 (3), A358598 (4), A358600 (6), A358601 (7).

Cf. A000079, A002279.

LinearRecurrence[{13, -32, 20}, {1, 7, 59}, 21] (* Hugo Pfoertner, Dec 05 2022 *)

print([2**n+5*(10**n-1)//9 for n in range(10)])

a(n) = 2^n + 5*(10^n - 1)/9.
a(n) = A000079(n) + A002279(n).
G.f.: (6*x-1)/((x-1)*(2*x-1)*(10*x-1)). - Alois P. Heinz, Dec 05 2022
a(n) = 13*a(n-1) - 32*a(n-2) + 20*a(n-3). - Wesley Ivan Hurt, Jun 19 2025

A093135 Expansion of g.f. (1-8x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 3, 23, 223, 2223, 22223, 222223, 2222223, 22222223, 222222223, 2222222223, 22222222223, 222222222223, 2222222222223, 22222222222223, 222222222222223, 2222222222222223, 22222222222222223, 222222222222222223, 2222222222222222223, 22222222222222222223, 222222222222222222223

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 2*A001045(3*n)/3 + (-1)^n.
Partial sums of A093136.
A convex combination of 10^n and 1.
In general the second binomial transform of k*Jacobsthal(3*n)/3 + (-1)^n is 1, 1+k, 1+11*k, 1+111*k, ... This is the case for k=2.
Essentially the same as A091628 (cf. 2nd formula). - Georg Fischer, Oct 06 2018
a(n) is 3^n represented in bijective base-3 numeration. - Alois P. Heinz, Aug 26 2019

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

Cf. A001045, A002279, A047855, A062397, A091628, A093136.

a(n) = (2*10^n + 7)/9.
a(n) = 10*a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 02 2010
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(7 + 2*exp(9*x))/9.
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = (A062397(n) - A002279(n))/2. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A305322 Repdigit numbers that are divisible by 3.

Original entry on oeis.org

0, 3, 6, 9, 33, 66, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 3333, 6666, 9999, 33333, 66666, 99999, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 3333333, 6666666, 9999999, 33333333, 66666666, 99999999, 111111111, 222222222

Offset: 1

Kritsada Moomuang, May 30 2018

The terms > 0 are (10^d-1)*k/9 for k=1..9 if d is divisible by 3, and for k=3,6,9 otherwise. - Robert Israel, Jun 01 2018

Repdigit remainders A010785(k) mod 3 have period 27. - Karl-Heinz Hofmann, Nov 11 2023

			111 / 3 = 37;
222 / 3 = 74;
333 / 3 = 111;
444 / 3 = 148;
555 / 3 = 185.

Robert Israel, Table of n, a(n) for n = 1..4996
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1001,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1000).

Cf. A008585, A010785.

Cf. A002279 (divisor 5), A366596 (divisor 7), A083118 (the impossible divisors).

L:= proc(d) if d mod 3 = 0 then [$1..9] else [3,6,9] fi end proc:
0,seq(seq((10^d-1)/9*k,k=L(d)),d=1..9); # Robert Israel, Jun 01 2018

def A010785(n): return (n - 9*((n-1)//9))*(10**((n+8)//9) - 1)//9
def A305322(n):
    d0, d1 = divmod(n-1,15)
    if d1 < 7: return A010785(d0 * 27 + d1 * 3)
    return A010785(d0 * 27 + d1 + 12) # Karl-Heinz Hofmann, Nov 26 2023

From Alois P. Heinz, May 30 2018: (Start)
{ A008585 } intersect { A010785 }.
G.f.: 3*(300*x^20 + 200*x^19 + 100*x^18 + 330*x^17 + 220*x^16 + 110*x^15 + 333*x^14 + 296*x^13 + 259*x^12 + 222*x^11 + 185*x^10 + 148*x^9 + 111*x^8 + 74*x^7 + 37*x^6 + 33*x^5 + 22*x^4 + 11*x^3 + 3*x^2 + 2*x + 1)*x^2 / ((x-1) *(x^2 + x + 1) *(x^4 + x^3 + x^2 + x + 1) *(10*x^5-1) *(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) *(100*x^10 + 10*x^5 + 1)).
a(n) = 1001*a(n-15) - 1000*a(n-30). (End)
From Karl-Heinz Hofmann, Nov 11 2023: (Start)
a(n) = A010785(floor((n-1)/15)*27 + ((n-1) mod 15)*3)    iff (n-1 <= 6 (mod 15)).
a(n) = A010785(floor((n-1)/15)*27 + ((n-1) mod 15) + 12) iff (n-1 > 6 (mod 15)).
(End)

Name clarified by Felix Fröhlich, Jun 01 2018

A366596 Repdigit numbers that are divisible by 7.

Original entry on oeis.org

0, 7, 77, 777, 7777, 77777, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 7777777, 77777777, 777777777, 7777777777, 77777777777, 111111111111, 222222222222, 333333333333, 444444444444, 555555555555, 666666666666, 777777777777

Offset: 1

Kritsada Moomuang, Oct 14 2023

7 divides a repdigit iff it consists of only digit 7, or has length 6*k (for any digit).

Repdigit remainders A010785(k) mod 7 have period 54. - Karl-Heinz Hofmann, Dec 04 2023

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..2329
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1000001,0,0,0,0,0,0,0,0,0,0,0,0,0,-1000000).

Intersection of A008589 and A010785.
Cf. A002281 (a subsequence).
Cf. A305322 (divisor 3), A002279 (divisor 5), A083118 (the impossible divisors).

r(n) = 10^((n+8)\9)\9*((n-1)%9+1); \\ A010785
lista(nn) = select(x->!(x%7), vector(nn, k, r(k-1))); \\ Michel Marcus, Oct 26 2023

def A366596(n):
    digitlen, digit = (n+12)//14*6, (n+12)%14-4
    if digit < 1: digitlen += digit - 1; digit = 7
    return 10**digitlen // 9 * digit # Karl-Heinz Hofmann, Dec 04 2023

From Karl-Heinz Hofmann, Dec 04 2023: (Start)
a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14) + 41),   for (n-2) mod 14 >  4.
a(n) = (10^(6*floor((n-2)/14) + 6)-1)/9*(((n-2) mod 14)-4), for (n-2) mod 14 >  4.
a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14)*9 + 7),  for (n-2) mod 14 <= 4.
a(n) = (10^(6*floor((n-2)/14) + 1 + ((n-2) mod 14))-1)/9*7, for (n-2) mod 14 <= 4.
(End)

A069881 Numbers n such that n and 2n+1 are both palindromes.

Original entry on oeis.org

1, 2, 3, 4, 5, 55, 151, 161, 171, 181, 191, 252, 262, 272, 282, 292, 353, 363, 373, 383, 393, 454, 464, 474, 484, 494, 555, 5555, 15051, 15151, 15251, 15351, 15451, 16061, 16161, 16261, 16361, 16461, 17071, 17171, 17271, 17371, 17471, 18081, 18181

Offset: 1

Amarnath Murthy, Apr 30 2002

Note that any number with all digits = 5 (A002279) is part of this sequence. - Jim McCann (jmccann(AT)umich.edu), Jul 16 2002

			151 is a member as 2*151 + 1 = 303 is also a palindrome.

Ivan Neretin, Table of n, a(n) for n = 1..15636 (all terms up to 10^12)

Subsequence of A002113.

isPalin[n_]:=(n==FromDigits[Reverse[IntegerDigits[n]]]); Do[m = 2 n + 1; If[isPalin[n]&&isPalin[m], Print[n]], {n, 1, 10^5}] (* Vincenzo Librandi, Jan 22 2018 *)

isok(n) = (d=digits(n)) && (Vecrev(d)==d) && (dd=digits(2*n+1)) && (Vecrev(dd)==dd); \\ Michel Marcus, Jan 22 2018

$a = 1; while((@b = split("|",$a) and @c = split("|",2*$a+1) and (join("", reverse(@b)) eq join("", @b) and join("", reverse(@c)) eq join("", @c) and eval("print \"\$a \"; return 0;"))) or ++$a) { }

More terms from Jim McCann (jmccann(AT)umich.edu), Jul 16 2002

A332151 a(n) = 5(10^(2n+1)-1)/9 - 4*10^n.

Original entry on oeis.org

1, 515, 55155, 5551555, 555515555, 55555155555, 5555551555555, 555555515555555, 55555555155555555, 5555555551555555555, 555555555515555555555, 55555555555155555555555, 5555555555551555555555555, 555555555555515555555555555, 55555555555555155555555555555, 5555555555555551555555555555555

Offset: 0

M. F. Hasler, Feb 09 2020

Harvey P. Dale, Table of n, a(n) for n = 0..499
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*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. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332151 := n -> 5*(10^(2*n+1)-1)/9-4*10^n;

Array[5 (10^(2 # + 1)-1)/9 - 4*10^# &, 15, 0]
Table[With[{c=PadRight[{},n,5]},FromDigits[Join[c,{1},c]]],{n,0,20}] (* Harvey P. Dale, Mar 16 2021 *)

apply( {A332151(n)=10^(n*2+1)\9*5-4*10^n}, [0..15])

def A332151(n): return 10**(n*2+1)//9*5-4*10**n

a(n) = 5*A138148(n) + 10^n = A002279(2n+1) - 4*10^n.
G.f.: (1 + 404*x - 900*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.

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.

A173735 a(n) = (10^n + 26)/9.

Original entry on oeis.org

3, 4, 14, 114, 1114, 11114, 111114, 1111114, 11111114, 111111114, 1111111114, 11111111114, 111111111114, 1111111111114, 11111111111114, 111111111111114, 1111111111111114, 11111111111111114, 111111111111111114, 1111111111111111114, 11111111111111111114, 111111111111111111114

Offset: 0

Vincenzo Librandi, Feb 23 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..100 (term 0 added by Ivan Panchenko)
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002279, A178769.

[(10^n+26)/9: n in [0..20]]; // Ivan Panchenko, Nov 05 2013

CoefficientList[Series[(3-29*x)/((1-x)*(1-10*x)),{x,0,30}],x] (* Ivan Panchenko, Nov 05 2013 *)

A173735(n):=(10^n+26)/9$
makelist(A173735(n),n,0,20); /* Ivan Panchenko, Nov 03 2013 */

a(n) = a(n-1) + 10^(n-1) = 10*a(n-1) - 26, a(0)=3.
a(n) = 11*a(n-1) - 10*a(n-2). - Vincenzo Librandi, Jul 05 2012
a(n) = 2*(A002279(n-1) + 2). - Martin Ettl, Nov 12 2012
G.f.: (3-29*x)/((1-x)*(1-10*x)). - Ivan Panchenko, Nov 05 2013
From Elmo R. Oliveira, Jun 18 2025: (Start)
E.g.f.: exp(x)*(26 + exp(9*x))/9.
a(n) = 2*A178769(n-1) for n >= 1. (End)

a(0) from Ivan Panchenko, Nov 05 2013

A173737 (10^n+44)/9 for n>0.

Original entry on oeis.org

6, 16, 116, 1116, 11116, 111116, 1111116, 11111116, 111111116, 1111111116, 11111111116, 111111111116, 1111111111116, 11111111111116, 111111111111116, 1111111111111116, 11111111111111116, 111111111111111116

Offset: 1

Vincenzo Librandi, Feb 23 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (11,-10).

[(10^n+44)/9: n in [1..20]]; // Vincenzo Librandi, Jul 05 2012

CoefficientList[Series[(6-50*x)/((1-x)*(1-10*x)),{x,0,30}],x] (* Vincenzo Librandi, Jul 05 2012 *)

A173737(n):=(10^n+44)/9$ makelist(A173737(n),n,1,20); /* Martin Ettl, Nov 08 2012 */

a(n) = a(n-1)+10^(n-1) = 10*a(n-1)-44 with n>0, a(0)=5.
G.f.: x*(6-50*x)/((1-x)*(1-10*x)). - Vincenzo Librandi, Jul 05 2012
a(n) = 11*a(n-1) -10*a(n-2) with a(0)=5, a(1)=6. - Vincenzo Librandi, Jul 05 2012
a(n) = (A002279(n-1)+3)*2. - Martin Ettl, Nov 08 2012

A205085 a(n) = n 5's sandwiched between two 1's.

Original entry on oeis.org

11, 151, 1551, 15551, 155551, 1555551, 15555551, 155555551, 1555555551, 15555555551, 155555555551, 1555555555551, 15555555555551, 155555555555551, 1555555555555551, 15555555555555551, 155555555555555551, 1555555555555555551, 15555555555555555551, 155555555555555555551

Offset: 0

José María Grau Ribas, Jan 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002279.

I:=[11, 151]; [n le 2 select I[n] else 11*Self(n-1)-10*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Jan 23 2012

a[0]=11;a[n_]:=a[n-1]*10+41;Table[a[n],{n,0,44}]
LinearRecurrence[{11, -10}, {11, 151}, 50] (* Vincenzo Librandi, Jan 23 2012 *)
Table[10FromDigits[PadRight[{1},n,5]]+1,{n,20}] (* Harvey P. Dale, May 02 2019 *)

a(n)=(140*10^n-41)/9 \\ Charles R Greathouse IV, Jan 23 2012

a(0)=11, a(n) = 10*a(n-1) + 41.
a(n) = (140*10^n - 41)/9 (see PARI code by Charles R Greathouse IV).
a(n) = 11*a(n-1) - 10*a(n-2). - Vincenzo Librandi, Jan 23 2012
From Elmo R. Oliveira, Feb 18 2025: (Start)
G.f.: (11 + 30*x)/((1 - x)*(1 - 10*x)).
E.g.f.: exp(x)*(140*exp(9*x) - 41)/9. (End)

cp's OEIS Frontend

A358599 Number of genetic relatives of a person M in a genealogical tree extending back n generations and where everyone has 5 children down to the generation of M.

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A093135 Expansion of g.f. (1-8*x)/((1-x)*(1-10*x)).

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A305322 Repdigit numbers that are divisible by 3.

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A366596 Repdigit numbers that are divisible by 7.

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A069881 Numbers n such that n and 2n+1 are both palindromes.

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A332151 a(n) = 5*(10^(2*n+1)-1)/9 - 4*10^n.

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

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A173735 a(n) = (10^n + 26)/9.

A093135 Expansion of g.f. (1-8x)/((1-x)(1-10*x)).

A332151 a(n) = 5(10^(2n+1)-1)/9 - 4*10^n.