A008593 -id:A008593 - cp's OEIS frontend

A145568 Characteristic function of numbers relatively prime to 11.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1

Offset: 0

Wolfdieter Lang Feb 05 2009

The x-powers appearing in the numerator polynomial of the o.g.f., given below, give the numbers from 0,1,...,10 which survive the sieve of Eratosthenes for multiples of 11, namely 1,2,...10.
Contribution from Reinhard Zumkeller, Nov 30 2009: (Start)
a(n)=A000007(A010880(n)); a(A160542(n))=1; a(A008593(n))=0;
A033443(n) = SUM(a(k)*(n-k): 0<=k<=n). (End)

Index entries for characteristic functions
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

A000035, A011655, A011558, A109720 for coprimality with 2,3,5,7, respectively.

Cf. A168185, A168184, A168182, A168181, A097325, A166486.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},105] (* Ray Chandler, Aug 26 2015 *)

a(n)=gcd(n,11)==1 \\ Charles R Greathouse IV, Jun 28 2015

a(n)=1 if gcd(n,11)=1, else 0. Periodic with period 11: a(n+11)=a(11).
O.g.f.: x*sum(x^k,k=0..9)/(1-x^11).
Completely multiplicative with a(p) = (if p=11 then 0 else 1), p prime. [From Reinhard Zumkeller, Nov 30 2009]
Dirichlet g.f. (1-11^(-s))*zeta(s). - R. J. Mathar, Mar 06 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

A216995 Multiples of 11 whose digit sum is a multiple of 11.

Original entry on oeis.org

209, 308, 407, 506, 605, 704, 803, 902, 2090, 2299, 2398, 2497, 2596, 2695, 2794, 2893, 2992, 3080, 3289, 3388, 3487, 3586, 3685, 3784, 3883, 3982, 4070, 4279, 4378, 4477, 4576, 4675, 4774, 4873, 4972, 5060, 5269, 5368, 5467, 5566, 5665, 5764, 5863, 5962, 6050

Offset: 1

Jon Perry, Sep 22 2012

Nothing between 1000 and 2000.

Also, there are no a(n) from 10902 to 12198 (this interval contains 117 multiples of 11). [Bruno Berselli, Oct 26 2012]

			3487 = 11*317 and 3+4+8+7 = 22 = 11*2.

Bruno Berselli, Table of n, a(n) for n = 1..1000

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

Cf. A216994, A216996, A216997, A216998, A217009.

function sumarray(arr) {
t=0;
for (i=0;i

Select[11*Range[1000], Mod[Total[IntegerDigits[#]], 11] == 0 &] (* T. D. Noe, Sep 24 2012 *)

def sd(n): return sum(map(int, str(n)))
def ok(n): return n%11 == 0 and sd(n)%11 == 0
print(list(filter(ok, range(1, 6051)))) # Michael S. Branicky, Jul 11 2021

A033584 a(n) = 11*n^2.

Original entry on oeis.org

0, 11, 44, 99, 176, 275, 396, 539, 704, 891, 1100, 1331, 1584, 1859, 2156, 2475, 2816, 3179, 3564, 3971, 4400, 4851, 5324, 5819, 6336, 6875, 7436, 8019, 8624, 9251, 9900, 10571, 11264, 11979, 12716

Offset: 0

N. J. A. Sloane

From Roberto E. Martinez II, Jan 07 2002: (Start)
Number of edges of the complete tripartite graph of order 7n, K_n,n,5n.
Number of edges of the complete tripartite graph of order 6n, K_n,2n,3n. (End)
11 times the squares. - Omar E. Pol, Dec 13 2008

			a(1)=22*1+0-11=11; a(2)=22*2+11-11=44; a(3)=22*3+44-11=99 - _Vincenzo Librandi_, Aug 05 2010

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008593, A195043.

Table[11*n^2, {n, 0, 35}] (* Amiram Eldar, Feb 03 2021 *)

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

a(n) = 11*A000290(n). - Omar E. Pol, Dec 13 2008
a(n) = 22*n + a(n-1) - 11 (with a(0)=0). - Vincenzo Librandi, Aug 05 2010
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/66.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/132.
Product_{n>=1} (1 + 1/a(n)) = sqrt(11)*sinh(Pi/sqrt(11))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(11)*sin(Pi/sqrt(11))/Pi. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 11*x*(1 + x)/(1-x)^3.
E.g.f.: 11*x*(1 + x)*exp(x).
a(n) = n*A008593(n) = A195043(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 3, 4, 3, 0, 4, 6, 6, 4, 0, 5, 8, 9, 8, 5, 0, 6, 10, 12, 12, 10, 6, 0, 7, 12, 15, 16, 15, 12, 7, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 12

Offset: 1

Reinhard Zumkeller, May 31 2004

T(n,k) = A003991(n-1,k) for 1 <= k < n;
T(n,k) = T(n,n-1-k) for k < n;
T(n,1) = n-1; T(n,n) = 0;          T(n,2) = A005843(n-2) for n > 1;
T(n,3) = A008585(n-3) for n>2;     T(n,4) = A008586(n-4) for n > 3;
T(n,5) = A008587(n-5) for n>4;     T(n,6) = A008588(n-6) for n > 5;
T(n,7) = A008589(n-7) for n>6;     T(n,8) = A008590(n-8) for n > 7;
T(n,9) = A008591(n-9) for n>8;     T(n,10) = A008592(n-10) for n > 9;
T(n,11) = A008593(n-11) for n>10;  T(n,12) = A008594(n-12) for n > 11;
T(n,13) = A008595(n-13) for n>12;  T(n,14) = A008596(n-14) for n > 13;
T(n,15) = A008597(n-15) for n>14;  T(n,16) = A008598(n-16) for n > 15;
T(n,17) = A008599(n-17) for n>16;  T(n,18) = A008600(n-18) for n > 17;
T(n,19) = A008601(n-19) for n>18;  T(n,20) = A008602(n-20) for n > 19;
Row sums give A000292; triangle sums give A000332;
All numbers m > 0 occur A000005(m) times;
A002378(n) = T(A005408(n),n+1) = n*(n+1).
k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) =  =  = i  = -i . Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016
T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020

			From _M. F. Hasler_, Feb 02 2013: (Start)
Triangle begins:
  0;
  1, 0;
  2, 2, 0;
  3, 4, 3, 0;
  4, 6, 6, 4, 0;
  5, 8, 9, 8, 5, 0;
  (...)
If an additional 0 was added at the beginning, this would become:
  0;
  0, 1;
  0, 2, 2;
  0, 3, 4; 3;
  0, 4, 6, 6, 4;
  0, 5, 8, 9, 8, 5;
  ... (End)

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

J_3: A114327; J_1^2, J_2^2: A141387, A268759.
Cf. A000005, A002378, A003991, A005408.
Cf. A000292 (row sums), A000332 (triangle sums).
T(n,k) for values of k:
  A005843 (k=2), A008585 (k=3), A008586 (k=4), A008587 (k=5), A008588 (k=6), A008589 (k=7), A008590 (k=8), A008591 (k=9), A008592 (k=10), A008593 (k=11), A008594 (k=12), A008595 (k=13), A008596 (k=14), A008597 (k=15), A008598 (k=16), A008599 (k=17), A008600 (k=18), A008601 (k=19), A008602 (k=20).

/* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016

Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

{for(n=1, 13, for(k=1, n, print1(k*(n - k)," ");); print(););} \\ Indranil Ghosh, Mar 12 2017

A135499 Numbers for which Sum_digits(odd positions) = Sum_digits(even positions).

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 220, 231, 242, 253, 264, 275, 286, 297, 330, 341, 352, 363, 374, 385, 396, 440, 451, 462, 473, 484, 495, 550, 561, 572, 583, 594, 660, 671, 682, 693, 770, 781, 792, 880, 891, 990

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Feb 08 2008

Conjecture: this is a subsequence of A008593 (verified for the first 50 thousand terms). - R. J. Mathar, Feb 10 2008
Subsequence of A008593. - Zak Seidov Feb 11 2008
If k is present, so is 10*k. - Robert G. Wilson v, Jul 13 2014
As Seidov said, a subsequence of multiples of 11. That follows trivially from the divisibility rule for 11. - Jens Kruse Andersen, Jul 13 2014
A225693(a(n)) = 0. - Reinhard Zumkeller, Aug 08 2014

			594, 1023, and 1397 are terms:
   594 -> 4 + 5 = 9;
  1023 -> 3 + 0 = 2 + 1;
  1397 -> 7 + 3 = 9 + 1.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 119 terms from _Paolo P. Lava_ and _Giorgio Balzarotti_)

Cf. A067042, A214527.
Cf. A060979.
Cf. A225693.

a135499 n = a135499_list !! (n-1)
a135499_list = filter ((== 0) . a225693) [1..]
-- Reinhard Zumkeller, Aug 08 2014, Jul 05 2014

P:=proc(n) local i,k,w,x; for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; x:=0; k:=i; while k>0 do x:=x+(k-(trunc(k/10)*10)); k:=trunc(k/100); od; if w=2*x then print(i); fi; od; end: P(3000);
# Alternative:
filter:= proc(n)
local L,d;
L:= convert(n,base,10);
d:= nops(L);
add(L[2*i],i=1..floor(d/2)) = add(L[2*i-1],i=1..floor((d+1)/2))
end proc:
select(filter,[ 11*j $ j= 1 .. 10^4 ]); # Robert Israel, May 28 2014

dQ[n_]:=Module[{p=Transpose[Partition[IntegerDigits[n],2,2,1,0]]},Total[First[p]]== Total[Last[p]]]; Select[Range[1000],dQ] (* Harvey P. Dale, May 26 2011 *)

A017413 a(n) = 11*n + 2.

Original entry on oeis.org

2, 13, 24, 35, 46, 57, 68, 79, 90, 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, 409, 420, 431, 442, 453, 464, 475, 486, 497, 508, 519, 530, 541, 552, 563, 574, 585

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008593, A017401, A017425.

[11*n+2: n in [0..60]]; // Vincenzo Librandi, Sep 02 2011

Range[2, 1000, 11] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{2,13},60] (* Harvey P. Dale, Sep 26 2020 *)

a(n)=11*n+2 \\ Charles R Greathouse IV, Jul 10 2016

From G. C. Greubel, Nov 11 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (2 + 9*x)/(1 - x)^2.
E.g.f.: (2 + 11*x)*exp(x). (End)

A123867 a(n) = n^10 - 1.

Original entry on oeis.org

0, 1023, 59048, 1048575, 9765624, 60466175, 282475248, 1073741823, 3486784400, 9999999999, 25937424600, 61917364223, 137858491848, 289254654975, 576650390624, 1099511627775, 2015993900448, 3570467226623, 6131066257800, 10239999999999, 16679880978200

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 11 = 0 iff n mod 11 > 0; a(A008593(n)) = 10.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A024008, A008454, A005563, A123865, A123866, A123868.

List([1..25], n-> n^10 -1); # G. C. Greubel, Aug 08 2019

[n^10 - 1:n in [1..40]]; // Vincenzo Librandi, Dec 27 2010

A123867:=n->n^10-1; seq(A123867(n), n=1..40); # Wesley Ivan Hurt, Jan 22 2014

Table[n^10-1, {n,40}] (* Wesley Ivan Hurt, Jan 22 2014 *)

vector(25, n, n^10 -1) \\ G. C. Greubel, Aug 08 2019

[n^10 -1 for n in (1..25)] # G. C. Greubel, Aug 08 2019

From G. C. Greubel, Aug 08 2019: (Start)
G.f.: x^2*(1023 + 47795*x + 455312*x^2 + 1310144*x^3 + 1310606*x^4 + 454982*x^5 + 47960*x^6 + 968*x^7 + 11*x^8 + x^9)/(1-x)^11.
E.g.f.: 1 +(-1 + x + 511*x^2 + 9330*x^3 + 34105*x^4 + 42525*x^5 + 22827*x^6 + 5880*x^7 + 750*x^8 + 45*x^9 + x^10)*exp(x). (End)

A225361 Partition numbers of the form 11k.

Original entry on oeis.org

11, 22, 77, 176, 231, 297, 385, 627, 792, 1958, 3718, 4565, 6842, 8349, 14883, 21637, 26015, 31185, 44583, 53174, 63261, 173525, 204226, 239943, 281589, 386155, 526823, 715220, 831820, 1121505, 1300156, 1741630, 5392783, 7089500, 8118264, 12132164, 18004327

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008593 and A000041.

			22 is in the sequence because 11*2 = 22 and 22 is a partition number: p(8) = A000041(8) = 22.

Cf. A000041, A008593, A052001, A072871, A087183, A225324, A225325, A225326, A225327, A225358, A225360, A127544.

Select[PartitionsP[Range[300]], Mod[#, 11] == 0 &]

a(n) = 11*A225323(n).

A017425 a(n) = 11*n + 3.

Original entry on oeis.org

3, 14, 25, 36, 47, 58, 69, 80, 91, 102, 113, 124, 135, 146, 157, 168, 179, 190, 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, 509, 520, 531, 542, 553, 564, 575, 586

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008593, A017401, A017413, A226492.

[11*n+3: n in [0..60]]; // Vincenzo Librandi, Sep 02 2011

11Range[0,50]+3  (* Harvey P. Dale, Apr 21 2011 *)

a(n)=11*n+3 \\ Charles R Greathouse IV, Jul 10 2016

[i+3 for i in range(475) if gcd(i,11) == 11] # Zerinvary Lajos, May 21 2009

From Elmo R. Oliveira, Apr 03 2024: (Start)
G.f.: (3+8*x)/(1-x)^2.
E.g.f.: exp(x)*(3 + 11*x).
a(n) = A226492(n+1) - A226492(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

Terms corrected by Vincenzo Librandi, Sep 02 2011

A317311 Multiples of 11 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 11, 3, 22, 5, 33, 7, 44, 9, 55, 11, 66, 13, 77, 15, 88, 17, 99, 19, 110, 21, 121, 23, 132, 25, 143, 27, 154, 29, 165, 31, 176, 33, 187, 35, 198, 37, 209, 39, 220, 41, 231, 43, 242, 45, 253, 47, 264, 49, 275, 51, 286, 53, 297, 55, 308, 57, 319, 59, 330, 61, 341, 63, 352, 65, 363, 67, 374, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 15-gonal numbers (A277082).

a(n) is also the length of the n-th line segment of the rectangular spiral wh0se vertices are the generalized 15-gonal numbers.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008593 and A005408 interleaved.
Column 11 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A277082.

{0}~Join~Riffle[2 Range@ # - 1, 11 Range@ #] &@ 35 (* or *)
CoefficientList[Series[x (1 + 11 x + x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 69}], x] (* Michael De Vlieger, Jul 26 2018 *)
LinearRecurrence[{0,2,0,-1},{0,1,11,3},90] (* Harvey P. Dale, Aug 28 2022 *)

concat(0, Vec(x*(1 + 11*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jul 26 2018

a(2n) = 11*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 26 2018: (Start)
G.f.: x*(1 + 11*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 11*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 9/2^s). - Amiram Eldar, Oct 25 2023
a(n) = (13 + 9*(-1)^n)*n/4. - Aaron J Grech, Aug 20 2024

cp's OEIS Frontend

A145568 Characteristic function of numbers relatively prime to 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A216995 Multiples of 11 whose digit sum is a multiple of 11.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

JavaScript

Mathematica

Python

A033584 a(n) = 11*n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A135499 Numbers for which Sum_digits(odd positions) = Sum_digits(even positions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A017413 a(n) = 11*n + 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A123867 a(n) = n^10 - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP