A008593 -id:A008593 - cp's OEIS frontend

A283742 Numbers with digit sum 11 that are multiples of 11.

209, 308, 407, 506, 605, 704, 803, 902, 2090, 3080, 4070, 5060, 6050, 7040, 8030, 9020, 10109, 10208, 10307, 10406, 10505, 10604, 10703, 10802, 10901, 20009, 20108, 20207, 20306, 20405, 20504, 20603, 20702, 20801, 20900, 30008, 30107, 30206, 30305, 30404

Offset: 1

Zak Seidov, Mar 15 2017

Subsequence of A166311.
Numbers with digit sum 11 and even-numbered digits all 0. - Robert Israel, Mar 16 2017
Intersection of A008593 and A166311. - Michel Marcus, Mar 17 2017
If m is a term, so is 10*m. - Zak Seidov, Mar 17 2017

			a(1) = A166311(18) = 209 = 19*11,
a(40) = A166311(9881) = 30404 = 2764*11.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A008593 (multiples of 11), A166311 (numbers with digit sum 11).

F:= proc(d,t) option remember;
if d <= 1 then
  if t <= 9 then return [t*10^d] else return [] fi;
fi;
[seq(seq(j*10^d+s, s = procname(d-2, t-j)), j=0..min(9, t))]
end proc:
sort([op(F(4,11)),op(F(5,11))]); # Robert Israel, Mar 16 2017

Reap[Do[If[11==Total[IntegerDigits[m=11*k]],Sow[m]],{k,3000}]][[2,1]]
Select[Range[209,30404,11],11==Total[IntegerDigits[#]]&]

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

A017390 a(n) = (11*n)^2.

Original entry on oeis.org

0, 121, 484, 1089, 1936, 3025, 4356, 5929, 7744, 9801, 12100, 14641, 17424, 20449, 23716, 27225, 30976, 34969, 39204, 43681, 48400, 53361, 58564, 64009, 69696, 75625, 81796, 88209, 94864, 101761, 108900, 116281, 123904, 131769, 139876, 148225, 156816, 165649, 174724

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008593.

[(11*n)^2: n in [0..40]]; // Vincenzo Librandi, Sep 02 2011

(11 Range[0,30])^2 (* or *) LinearRecurrence[{3,-3,1},{0,121,484},30] (* Harvey P. Dale, Nov 04 2021 *)

a(n)=(11*n)^2 \\ Charles R Greathouse IV, Jun 17 2017

From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/726.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/1452.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/11)/(Pi/11).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/11)/(Pi/11). (End)
G.f.: -((121*x*(1+x))/(-1+x)^3). - Harvey P. Dale, Nov 04 2021
From Elmo R. Oliveira, Nov 30 2024: (Start)
E.g.f.: 121*x*(1 + x)*exp(x).
a(n) = 121*A000290(n) = A008593(n)^2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A061471 First (leftmost) digit - second digit + third digit - fourth digit .... = 2.

Original entry on oeis.org

2, 20, 31, 42, 53, 64, 75, 86, 97, 101, 112, 123, 134, 145, 156, 167, 178, 189, 200, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 321, 332, 343, 354, 365, 376, 387, 398, 420, 431, 442, 453, 464, 475, 486, 497, 530, 541, 552, 563, 574, 585, 596, 640, 651

Offset: 1

Amarnath Murthy, May 05 2001

a(n) == 9*(-1)^d (mod 11) if a(n) has d digits. - Robert Israel, Aug 05 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

filter:= proc(n) local d,L,j;
  L:= convert(n,base,10);
  d:= nops(L);
  add(L[j]*(-1)^(d-j),j=1..d)=2
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 05 2020

okQ[n_] := With[{id = IntegerDigits[n]}, id.Array[2 Mod[#, 2] - 1&, Length[id]] == 2]; Select[Range[1000], okQ] (* Jean-François Alcover, Nov 17 2016 *)
Select[Range[700],Total[Times@@@Partition[Riffle[IntegerDigits[#],{1,-1},{2,-1,2}],2]] == 2&] (* Harvey P. Dale, Mar 01 2023 *)

More terms from Robert G. Wilson v, May 10 2001 and from Larry Reeves (larryr(AT)acm.org), May 14 2001

A061472 First (leftmost) digit - second digit + third digit - fourth digit .... = 3.

Original entry on oeis.org

3, 30, 41, 52, 63, 74, 85, 96, 102, 113, 124, 135, 146, 157, 168, 179, 201, 212, 223, 234, 245, 256, 267, 278, 289, 300, 311, 322, 333, 344, 355, 366, 377, 388, 399, 410, 421, 432, 443, 454, 465, 476, 487, 498, 520, 531, 542, 553, 564, 575, 586, 597, 630

Offset: 1

Amarnath Murthy, May 05 2001

			124 is in the sequence since 1 - 2 + 4 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882, A225693.

A225693:= proc(n) local L,m,i;
  L:= convert(n,base,10);
  m:= nops(L);
  add(L[i]*(-1)^(m-i),i=1..m);
end proc:
select(A225693=3, [$1..1000]); # Robert Israel, Jun 12 2019

aQ[n_] := Differences[Total @ Take[IntegerDigits[n], {#, -1, 2}] & /@ {2, 1}][[1]] == 3; Select[Range[1000], aQ] (* Amiram Eldar, Jun 12 2019 *)
Select[Range[1000],Total[Times@@@Partition[Riffle[IntegerDigits[#],{1,-1},{2,-1,2}],2]]==3&] (* Harvey P. Dale, May 16 2020 *)

isok(n) = my(d=digits(n)); sum(k=1, #d, (-1)^(k+1)*d[k]) == 3; \\ Michel Marcus, Jun 12 2019

A225693(n) = 3. - Robert Israel, Jun 12 2019

More terms from Robert G. Wilson v, May 10 2001 and from Larry Reeves (larryr(AT)acm.org), May 14 2001

A080467 Multiples of 11 in which the even positioned digits from left are odd and the odd positioned ones are even.

Original entry on oeis.org

0, 418, 616, 638, 814, 836, 858, 2101, 2123, 2145, 2167, 2189, 2321, 2343, 2365, 2387, 2541, 2563, 2585, 2761, 2783, 2981, 4103, 4125, 4147, 4169, 4301, 4323, 4345, 4367, 4389, 4521, 4543, 4565, 4587, 4741, 4763, 4785, 4961, 4983, 6105, 6127, 6149

Offset: 1

Amarnath Murthy, Mar 02 2003

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Intersection of A008593 and A062285.

Cf. A080466.

Terms corrected by Andrew Howroyd, Sep 29 2024

A083850 Exponent of highest power of 11 dividing the n-th decimal palindrome; a(0) = 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, May 06 2003

Cf. A083851, A083852, A002113, A008593.

IntegerExponent[#,11]&/@Select[Range[0,1000],PalindromeQ]/.\[Infinity]->1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 05 2020 *)

A199799 Totatives of 111111.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 17, 19, 20, 23, 25, 29, 31, 32, 34, 38, 40, 41, 43, 46, 47, 50, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 76, 79, 80, 82, 83, 85, 86, 89, 92, 94, 95, 97, 100, 101, 103, 106, 107, 109, 113, 115, 116, 118, 122, 124, 125, 127, 128, 131, 134

Offset: 1

Reinhard Zumkeller, Nov 11 2011

a(n) and 111111 are coprime, 111111 = 3*7*11*13*37; empty intersections with A008585, A008593, A008595, or A085959; sequence is finite with 51840 terms, A000010(111111) = 51840, last term: a(51840) = 111110.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Totative

Cf. A109492 (divisors of 111111).

a199799 n = a199799_list !! (n-1)
a199799_list = [x | x <- [1..111111], gcd x 111111 == 1]

Select[Range[200],CoprimeQ[#,111111]&] (* Paolo Xausa, Sep 27 2023 *)

A243520 Numbers that are congruent to {0, 8} mod 11.

Original entry on oeis.org

0, 8, 11, 19, 22, 30, 33, 41, 44, 52, 55, 63, 66, 74, 77, 85, 88, 96, 99, 107, 110, 118, 121, 129, 132, 140, 143, 151, 154, 162, 165, 173, 176, 184, 187, 195, 198, 206, 209, 217, 220, 228, 231, 239, 242, 250, 253, 261, 264, 272, 275, 283, 286, 294, 297, 305

Offset: 0

Viet Quoc Le Tran, Jun 14 2014

Union of A008593 and A017485. - Michel Marcus, Jun 15 2014

This sequence mimics in some sense the ceiling function of n/2 (the seq. A110654) relative to variations from a main class of recurrence relations; in order to get the ceiling function of n/2 (see Formula section), the vector v must be [0,1] instead of [3,8]. - R. J. Cano, Jun 15 2014

R. J. Cano, Table of n, a(n) for n = 0..9999
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A010706, A110654, A078707.

&cat [[11*n,11*n+8]: n in [0..30]]; // [Bruno Berselli, Jun 16 2014]

A243520:=n->5*n + 2*(n mod 2) + ceil(n/2); seq(A243520(n), n=0..50); # Wesley Ivan Hurt, Jun 21 2014

Flatten[Table[11 n + {0, 8}, {n, 0, 32}]] (* Alonso del Arte, Jun 15 2014 *)

a(n)=5*n+2*(n%2)+ceil(n/2); \\ R. J. Cano, Jun 15 2014

a(n)=if(!n,0,a(n-1)+[3,8][1+n%2]); \\ R. J. Cano, Jun 15 2014

a(n) = -5/4*(-1)^n + 11*n/2 + 5/4.
From R. J. Cano, Jun 15 2014: (Start)
a(n) = 5*n + 2*(n mod 2) + ceiling(n/2).
If n=0 then a(n) is zero, else a(n) = a(n-1) + v[n mod 2], where v is [3,8]. (End)
G.f.: x*(8 + 3*x) / ((1 + x)*(1 - x)^2). [Bruno Berselli, Jun 16 2014]
a(n) = sum( A010706(i), i=0..n ) - 3. [Bruno Berselli, Jun 16 2014]
E.g.f.: (11*x*exp(x) + 5*sinh(x))/2. - David Lovler, Sep 04 2022

A279895 a(n) = n(5n + 11)/2.

Original entry on oeis.org

0, 8, 21, 39, 62, 90, 123, 161, 204, 252, 305, 363, 426, 494, 567, 645, 728, 816, 909, 1007, 1110, 1218, 1331, 1449, 1572, 1700, 1833, 1971, 2114, 2262, 2415, 2573, 2736, 2904, 3077, 3255, 3438, 3626, 3819, 4017, 4220, 4428, 4641, 4859, 5082, 5310, 5543, 5781, 6024, 6272, 6525

Offset: 0

Bruno Berselli, Dec 22 2016

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566, A008589, A017281.
Second bisection of A165720.
The first differences are in A016885.
Cf. similar sequences provided by P(s,m)+s*m, where P(s,m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number: A008585 (s=2), A055999 (s=3), A028347 (s=4), A140091 (s=5), A033537 (s=6), this sequence (s=7), A067725 (s=8).

Magma
```
[n*(5*n+11)/2: n in [0..60]];
```

Table[n (5 n + 11)/2, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{0,8,21},60] (* Harvey P. Dale, Nov 14 2022 *)

PARI
```
vector(60, n, n--; n*(5*n+11)/2)
```
Python
```
[n*(5*n+11)/2 for n in range(60)]
```
Sage
```
[n*(5*n+11)/2 for n in range(60)]
```

O.g.f.: x*(8 - 3*x)/(1 - x)^3.
E.g.f.: x*(16 + 5*x)*exp(x)/2.
a(n+h) - a(n-h) = h*A017281(n+1), with h>=0. A particular case:
a(n) - a(-n) = 11*n = A008593(n).
a(n+h) + a(n-h) = 2*a(n) + A033429(h), with h>=0. A particular case:
a(n) + a(-n) = A033429(n).
a(n) - a(n-2) = A017281(n) for n>1. Also:
40*a(n) + 121 = A017281(n+1)^2.
a(n) = A000566(n) + 7*n, also a(n) = A000566(n) + A008589(n). - Michel Marcus, Dec 22 2016

cp's OEIS Frontend

A283742 Numbers with digit sum 11 that are multiples of 11.

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A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

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A017390 a(n) = (11*n)^2.

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A061471 First (leftmost) digit - second digit + third digit - fourth digit .... = 2.

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A061472 First (leftmost) digit - second digit + third digit - fourth digit .... = 3.

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Maple

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A080467 Multiples of 11 in which the even positioned digits from left are odd and the odd positioned ones are even.

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A083850 Exponent of highest power of 11 dividing the n-th decimal palindrome; a(0) = 1.

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Mathematica

A199799 Totatives of 111111.

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A279895 a(n) = n(5n + 11)/2.