Mark Dols - cp's OEIS frontend

A180115 A109613(n)-fold concatenation of A008619(n).

1, 111, 222, 22222, 33333, 3333333, 4444444, 444444444, 555555555, 55555555555, 66666666666, 6666666666666, 7777777777777, 777777777777777, 888888888888888

Offset: 1

Mark Dols, Aug 10 2010

Cf. A000217, A109613, A008619

Definition rephrased, kewyord:base,less added - R. J. Mathar, Aug 19 2010

A180153 a(n) = 10*a(n-1) + A109242(n).

Original entry on oeis.org

1, 121, 12421, 1246421, 124696421, 12470296421, 1247037296421, 124703817296421, 12470382717296421, 1247038282717296421, 124703828392717296421, 12470382840592717296421, 1247038284073592717296421, 124703828407513592717296421, 12470382840753013592717296421

Offset: 1

Mark Dols, Aug 13 2010

Colin Barker, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (121,-2220,12100,-10000).

Cf. A109242, A000292, A002620, A059259, A094028.

Vec(x / ((x-1)*(10*x-1)^2*(100*x-1)) + O(x^30)) \\ Colin Barker, Oct 03 2015

From Colin Barker, Oct 03 2015: (Start)
a(n) = 121*a(n-1)-2220*a(n-2)+12100*a(n-3)-10000*a(n-4) for n>4.
G.f.: x / ((x-1)*(10*x-1)^2*(100*x-1)).
(End)

A180116 A008619(n-1)-fold concatenation of A109613(n).

Original entry on oeis.org

1, 3, 33, 55, 555, 777, 7777, 9999, 99999, 1111111111, 111111111111, 131313131313, 13131313131313, 15151515151515, 1515151515151515, 1717171717171717, 171717171717171717, 191919191919191919, 19191919191919191919, 21212121212121212121, 2121212121212121212121

Offset: 1

Mark Dols, Aug 10 2010

Written underneath, the first terms have increasing lengths filling a triangular shape:
1,
3,
33,
55,
555,
777,
...

Cf. A000217, A109613, A008619.

cat2 := proc(a,b) a*10^(max(1,1+ilog10(b)))+b ; end proc:
A008619 := proc(n) 1+floor(n/2) ; end proc:
A109613 := proc(n) 2*floor(n/2)+1 ; end proc:
A180116 := proc(n) a := A109613(n) ; for t from 2 to A008619(n-1) do a := cat2(a,A109613(n)) ; end do: a ; end proc:
seq(A180116(n),n=1..24) ; # R. J. Mathar, Sep 19 2010

Edited by R. J. Mathar, Sep 19 2010

A180052 Partial sums of A180039.

Original entry on oeis.org

1, 113, 111336, 1111333670, 111113333667115, 111111333336667111671, 1111111333333666671111667338, 111111113333333666667111116667334116, 111111111333333336666667111111666673334112005, 1111111111333333333666666671111111666667333341112001005

Offset: 1

Mark Dols, Aug 08 2010

nonn

Row 5 of the Pascal array expressed as triangular numbers in 2 ways (value and number of digits). Could be viewed as part of a group expressing the n-th row of the Pascal array in n ways (see example).

			For the fifth row of the Pascal array, n=5:
1,5,15,..
1,14,150,..
1,113,111336,..
1,1112,1111112223,..
1,11111,111111111111111,..
For n=4:
1,4,10,20,..
1,13,136,1370,..
1,112,111223,1111222334,..
1,1111,1111111111,11111111111111111111,..

Cf. A007318, A052262, A180039.

Accumulate[Accumulate[Table[FromDigits[PadRight[{},n,1]],{n,Accumulate[ Range[10]]}]]] (* Harvey P. Dale, Jun 12 2021 *)

A180027 Partial sums of A100706.

Original entry on oeis.org

1, 112, 11223, 1122334, 112233445, 11223344556, 1122334455667, 112233445566778, 11223344556677889, 1122334455667789000, 112233445566778900111, 11223344556677890011222, 1122334455667789001122333, 112233445566778900112233444, 11223344556677890011223344555, 1122334455667789001122334455666

Offset: 0

Mark Dols, Aug 07 2010

nonn

Up to n=8 the digits of a(n) sum up to n^2.
Similar to this, A014824 (1,12,123,1234,...) is a representation of the triangular numbers; (1,1112,1112223,1112223334,...) of the pentagonal numbers;(1,11112,111122223,...) of the hexagonal numbers, and so on. A nice thing about this sequence(s) is that the (represented) value of the integer matches the partial sums of the number of digits in the sequence.
f(n) = 100*f(n-1) + A100706(n) gives a mirrored version of this sequence, and f(n) = 10*f(n-1) + A100706(n) the symmetrical version (A002477).

Cf. A014824, A100706, A002477.

A100706(n) = (10^(2*n + 1) - 1)/9;
a(n) = sum(k=0, n, A100706(k)); \\ Michel Marcus, Mar 12 2023

a(n) = Sum_{k=0..n} A100706(k). - Michel Marcus, Mar 12 2023

More terms and edited by Michel Marcus, Mar 12 2023

A180236 a(n) = a(n-2)+a(n-4); a(1)=a(4)=101, a(2)=a(3)=10.

Original entry on oeis.org

101, 10, 10, 101, 111, 111, 121, 212, 232, 323, 353, 535, 585, 858, 938, 1393, 1523, 2251, 2461, 3644, 3984, 5895, 6445, 9539, 10429, 15434, 16874, 24973, 27303, 40407, 44177, 65380, 71480, 105787, 115657, 171167, 187137, 276954, 302794, 448121, 489931

Offset: 1

Mark Dols, Aug 18 2010

Generalization of A115339.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A115339, A000045, A000032.

LinearRecurrence[{0,1,0,1},{101,10,10,101},50] (* Paolo Xausa, Jan 04 2024 *)

Vec(-x*(91*x^3-91*x^2+10*x+101)/(x^4+x^2-1) + O(x^100)) \\ Colin Barker, Oct 03 2015

a(n) = if(n==1||n==4, 101, if(n==2||n==3, 10, a(n-2)+a(n-4))); \\ Altug Alkan, Oct 03 2015

G.f.: -x*(91*x^3-91*x^2+10*x+101) / (x^4+x^2-1). Colin Barker, Oct 03 2015

A179980 a(n) = 10a(n-1) - (-1)^n n, a(1) = 1.

Original entry on oeis.org

1, 8, 83, 826, 8265, 82644, 826447, 8264462, 82644629, 826446280, 8264462811, 82644628098, 826446280993, 8264462809916, 82644628099175, 826446280991734, 8264462809917357, 82644628099173552, 826446280991735539, 8264462809917355370, 82644628099173553721

Offset: 1

Mark Dols, Aug 04 2010

Index entries for linear recurrences with constant coefficients, signature (8,19,10).

Cf. A007908, A021125.

LinearRecurrence[{8,19,10},{1,8,83},20] (* Harvey P. Dale, Nov 17 2014 *)

From R. J. Mathar, Aug 06 2010: (Start)
a(n) = +8*a(n-1) +19*a(n-2) +10*a(n-3).
G.f.: -x/((10*x-1)*(1+x)^2).
a(n) = (10^(n+1) -11*n*(-1)^n -10*(-1)^n)/121. (End)

A180175 Diagonal sums of A164844.

Original entry on oeis.org

1, 10, 101, 1011, 10112, 101123, 1011235, 10112358, 101123593, 1011235951, 10112359544, 101123595495, 1011235955039, 10112359550534, 101123595505573, 1011235955056107, 10112359550561680, 101123595505617787, 1011235955056179467, 10112359550561797254

Offset: 1

Mark Dols, Aug 15 2010

Sums are built along inclined lines through the triangle with (1,2)-steps in the (row,column) indices. - R. J. Mathar, Aug 19 2010

			From _R. J. Mathar_, Aug 19 2010: (Start)
One example is a(5), the sum of numbers in parentheses:
  1;
  1, 10;
  (1), 11, 100;
  1, 12, (111) ; 1000;;
  1, 13, 123 ; 1111, (10000); (End)

Index entries for linear recurrences with constant coefficients, signature (11,-9,-10).

Cf. A022100, A164844.

From R. J. Mathar, Aug 19 2010: (Start)
G.f.: ( 1-x ) / ( (10*x-1)*(x^2+x-1) ).
a(n) = +11*a(n-1) -9*a(n-2) -10*a(n-3).
a(n) = (90*10^n -A022100(n))/89. (End)

More terms from R. J. Mathar, Aug 19 2010

A180053 a(1)=1, a(2)=101, a(n) = 1001*a(n-1) for n > 2.

Original entry on oeis.org

1, 101, 101101, 101202101, 101303303101, 101404606404101, 101506011010505101, 101607517021515606101, 101709124538537121707101, 101810833663075658828808101, 101912644496738734487636909101, 102014557141235473222124546010101

Offset: 1

Mark Dols, Aug 08 2010

			In a column:
                1
              101
           101101
        101202101
     101303303101
  101404606404101

Index entries for linear recurrences with constant coefficients, signature (1001).

Cf. A093645.

LinearRecurrence[{1001}, {1, 101}, 15] (* Paolo Xausa, Jan 29 2024 *)
Join[{1},NestList[1001#&,101,10]] (* Harvey P. Dale, Jul 29 2024 *)

a(n)=max(101*1001^(n-2),1) \\ Charles R Greathouse IV, Oct 08 2011

G.f.: x(1-900*x)/(1-1001*x). - Philippe Deléham, Oct 08 2011

A179907 Numerators in the approximation of sqrt(2) satisfying the recurrence: a(n)= [a(n-1)*a(n-2)+2]/[a(n-1)+a(n-2)] with a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 3, 7, 41, 577, 47321, 54608393, 5168247530883, 564459384575477049359

Offset: 1

Mark Dols, Aug 01 2010

Cf. A179823

cp's OEIS Frontend

User: Mark Dols

A180115 A109613(n)-fold concatenation of A008619(n).

Views

Author

Keywords

Crossrefs

Extensions

A180153 a(n) = 10*a(n-1) + A109242(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A180116 A008619(n-1)-fold concatenation of A109613(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Extensions

A180052 Partial sums of A180039.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A180027 Partial sums of A100706.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

Extensions

A180236 a(n) = a(n-2)+a(n-4); a(1)=a(4)=101, a(2)=a(3)=10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A179980 a(n) = 10*a(n-1) - (-1)^n * n, a(1) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A180175 Diagonal sums of A164844.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A180053 a(1)=1, a(2)=101, a(n) = 1001*a(n-1) for n > 2.

Views

Author

Keywords

A179980 a(n) = 10a(n-1) - (-1)^n n, a(1) = 1.