A008604 -id:A008604 - cp's OEIS frontend

A157276 A157107=220,440,661,881,1105, - 220,440,660,880,1100,=220(n+1)=10A008604(n+1).

0, 0, 1, 1, 5, 5, 6, 6, 11, 11, 12, 12, 16, 16, 17, 17, 31, 31, 32, 32, 36, 36, 37, 37, 42, 42, 43, 43, 47, 47, 48, 48, 66, 66, 67, 67, 71, 71, 72, 72, 77, 77, 78, 78, 82, 82, 83, 83, 97, 97, 98, 98, 102, 102, 103, 103, 108, 108, 109, 109, 113, 113, 114, 114

Offset: 0

Paul Curtz, Feb 26 2009

(From A090822). A157107(0)=220,..,A157107(63)=14194, A157107(64)=14236 ,see 42 in submitted A157206. a(63)=14194-(64*220=14080)=114, a(64)=14236-14300=-64. "Linked" to submitted A156912=1,2,1,5,1,2,1,6,.

A008593 Multiples of 11.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 242, 253, 264, 275, 286, 297, 308, 319, 330, 341, 352, 363, 374, 385, 396, 407, 418, 429, 440, 451, 462, 473, 484, 495, 506, 517, 528, 539, 550, 561, 572, 583

Offset: 0

N. J. A. Sloane

Numbers for which the sum of "digits" in base 100 is divisible by 11. For instance, 193517302 gives 1 + 93 + 51 + 73 + 02 = 220, and 2 + 20 = 22 = 2 * 11. - Daniel Forgues, Feb 22 2016

Numbers in which the sum of the digits in the even positions equals the sum of the digits in the odd positions. - Stefano Spezia, Jan 05 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 323.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A008592, A008604, A059632; union of A135499 and A060979.

a008593 = (* 11)
a008593_list = [0, 11 ..]  -- Reinhard Zumkeller, Jul 05 2014

[11*n: n in [0..60]]; // Vincenzo Librandi, Sep 18 2011

g:=(1+10*z)/((1-z)): gser:=series(g, z=0, 88): seq((coeff(gser, z, n))*n, n=0..77); # Zerinvary Lajos, Feb 25 2009

Range[0, 1000, 11] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

makelist(11*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=11*n \\ Charles R Greathouse IV, Nov 06 2014

a(n) = 11*n.
G.f.: 11*x/(1-x)^2. - David Wilding, Jun 21 2014
E.g.f.: 11*x*exp(x). - Stefano Spezia, Oct 08 2022
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A008604(n)/2. (End)

A008606 Multiples of 24.

Original entry on oeis.org

0, 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360, 384, 408, 432, 456, 480, 504, 528, 552, 576, 600, 624, 648, 672, 696, 720, 744, 768, 792, 816, 840, 864, 888, 912, 936, 960, 984, 1008, 1032, 1056, 1080, 1104, 1128, 1152, 1176, 1200

Offset: 0

N. J. A. Sloane

If n is a multiple of 24, also the sum of the divisors of n-1 is a multiple of 24. - Vincenzo Librandi, Apr 12 2014

This is the sequence of numbers n such that A259748(n)/n = 11/12. - Danny Rorabaugh, Oct 22 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 336.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008604, A008605.

Range[0, 1500, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[24 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 11 2013 *)

a(n)=24*n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 24*x/(1-x)^2. - Vincenzo Librandi, Jun 11 2013
a(n) = 24*A001477(n) - Danny Rorabaugh, Oct 24 2015
E.g.f.: 24*x*exp(x). - Stefano Spezia, Mar 02 2025

A161709 a(n) = 22*n + 1.

Original entry on oeis.org

1, 23, 45, 67, 89, 111, 133, 155, 177, 199, 221, 243, 265, 287, 309, 331, 353, 375, 397, 419, 441, 463, 485, 507, 529, 551, 573, 595, 617, 639, 661, 683, 705, 727, 749, 771, 793, 815, 837, 859, 881, 903, 925, 947, 969, 991, 1013, 1035, 1057, 1079, 1101, 1123

Offset: 0

Reinhard Zumkeller, Jun 17 2009

Italo Ghersi, Matematica dilettevole e curiosa, p. 139, Hoepli, Milano, 1967. [From Vincenzo Librandi, Dec 02 2009]

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

Cf. A005408, A008604, A016813, A016921, A017281, A017533, A128470, A158057, A161700, A161705, A161714.

List([0..60], n-> 1+22*n ); # G. C. Greubel, Sep 18 2019

[1+22*n: n in [0..60]]; // G. C. Greubel, Sep 18 2019

seq(1+22*n, n=0..60); # G. C. Greubel, Sep 18 2019

22*Range[0,60]+1 (* Harvey P. Dale, Jan 09 2011 *)

vector(60, n, 22*n-21) \\ G. C. Greubel, Sep 18 2019

[1+22*n for n in (0..60)] # G. C. Greubel, Sep 18 2019

From G. C. Greubel, Sep 18 2019: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (1 + 21*x)/(1-x)^2.
E.g.f.: (1 + 22*x)*exp(x). (End)

A195323 a(n) = 22*n^2.

Original entry on oeis.org

0, 22, 88, 198, 352, 550, 792, 1078, 1408, 1782, 2200, 2662, 3168, 3718, 4312, 4950, 5632, 6358, 7128, 7942, 8800, 9702, 10648, 11638, 12672, 13750, 14872, 16038, 17248, 18502, 19800, 21142, 22528, 23958, 25432, 26950, 28512, 30118, 31768, 33462, 35200, 36982, 38808

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 22, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195318 in the same spiral.

Surface area of a rectangular prism with dimensions n, 2n and 3n. - Wesley Ivan Hurt, Apr 10 2015

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

Bisection of A195149.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195322.
Cf. A000290, A001105, A008604, A033584, A195313, A195318.

[22*n^2 : n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

A195323:=n->22*n^2: seq(A195323(n), n=0..80); # Wesley Ivan Hurt, Apr 10 2015

22Range[0,50]^2 (* or *) LinearRecurrence[{3,-3,1},{0,22,88},50] (* Harvey P. Dale, Sep 19 2011 *)

vector(50,n,22*(n-1)^2) \\ Derek Orr, Apr 10 2015

a(n) = 22*A000290(n) = 11*A001105(n) = 2*A033584(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 19 2011
G.f.: 22*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Apr 10 2015
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 22*x*(1 + x)*exp(x).
a(n) = n*A008604(n) = A195149(2*n). (End)

A008605 Multiples of 23.

Original entry on oeis.org

0, 23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 690, 713, 736, 759, 782, 805, 828, 851, 874, 897, 920, 943, 966, 989, 1012, 1035, 1058, 1081, 1104, 1127, 1150

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 335.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008602, A008603, A008604, A008606.

Range[0, 1500, 23] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[23 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=23*n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 23*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
E.g.f.: 23*x*exp(x). - Stefano Spezia, Mar 02 2025
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 23*n = (A008604(n) + A008606(n))/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A317322 Multiples of 22 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 22, 3, 44, 5, 66, 7, 88, 9, 110, 11, 132, 13, 154, 15, 176, 17, 198, 19, 220, 21, 242, 23, 264, 25, 286, 27, 308, 29, 330, 31, 352, 33, 374, 35, 396, 37, 418, 39, 440, 41, 462, 43, 484, 45, 506, 47, 528, 49, 550, 51, 572, 53, 594, 55, 616, 57, 638, 59, 660, 61, 682, 63, 704, 65, 726, 67, 748, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 26-gonal numbers (A316724).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 26-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. A008604 and A005408 interleaved.
Column 22 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. A316724.

Module[{nn=40},Riffle[22Range[0,nn],Range[1,2nn,2]]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,22,3},80] (* Harvey P. Dale, Dec 12 2021 *)

concat(0, Vec(x*(1 + 22*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 22*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 22*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, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 5*2^(2-s)). - Amiram Eldar, Oct 26 2023

A141694 a(n) = 22*n + 12.

Original entry on oeis.org

12, 34, 56, 78, 100, 122, 144, 166, 188, 210, 232, 254, 276, 298, 320, 342, 364, 386, 408, 430, 452, 474, 496, 518, 540, 562, 584, 606, 628, 650, 672, 694, 716, 738, 760, 782, 804, 826, 848, 870, 892, 914, 936, 958, 980, 1002, 1024, 1046, 1068, 1090, 1112

Offset: 0

Paul Curtz, Sep 10 2008

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

Cf. A008604, A010861 (first differences), A017461.

[22*n + 12: n in [0..50]]; // Vincenzo Librandi, Aug 08 2011

Range[12, 1500, 22] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
LinearRecurrence[{2,-1},{12,34},60] (* Harvey P. Dale, Sep 06 2024 *)

for(n=0,50, print1(2*(11*n + 6), ", ")) \\ G. C. Greubel, Jun 03 2018

From G. C. Greubel, Jun 03 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: 2*(6 + 5*x)/(1 - x)^2.
E.g.f.: 2*(6 + 11*x)*exp(x). (End)
a(n) = 2*A017461(n). - Elmo R. Oliveira, Apr 11 2025

Edited by R. J. Mathar, Oct 24 2008

Offset changed from 1 to 0 by Vincenzo Librandi, Aug 08 2011

A247128 Positive numbers that are congruent to {0,5,9,13,17} mod 22.

Original entry on oeis.org

5, 9, 13, 17, 22, 27, 31, 35, 39, 44, 49, 53, 57, 61, 66, 71, 75, 79, 83, 88, 93, 97, 101, 105, 110, 115, 119, 123, 127, 132, 137, 141, 145, 149, 154, 159, 163, 167, 171, 176, 181, 185, 189, 193, 198, 203, 207, 211

Offset: 1

Karl V. Keller, Jr., Nov 19 2014

This sequence is the union of 22*n-17, 22*n-13, 22*n-9, and 22*n-5, and A008604(22*n), for n>0.
This sequence is the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; see example.
The sequence numbers with both odd first and last digits are either palindromes or they have corresponding reversed digit numbers, e.g., 105, 501. Prime numbers in this sequence are also in A007500 (reversal primes). Some examples are 13, 17, 31, 71, 79, 97, 101.
The sequence numbers with even first digits and last digits of 2, 4, 6 or 8, are either palindromes or they have corresponding reversed digit numbers in this sequence.
The candidate Lychrel numbers, 295, 493, 691, 1677, 1765, 1857, 1945, 1997, 3493, are in this sequence.

			Sequence consists of the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; e.g.,
for k =  5, sqrt( 20 -  2 + 3 + 1) = sqrt(22)  =  4.6904;
for k =  6, sqrt( 24 -  2 + 3 + 0) = sqrt(25)  =  5;
for k = 21, sqrt( 84 -  7 + 3 + 1) = sqrt(81)  =  9;
for k = 44, sqrt(176 - 15 + 3 + 0) = sqrt(164) = 12.8062;
for k = 45, sqrt(180 - 15 + 3 + 1) = sqrt(169) = 13.
Of these, the only integer values are 5, 9, 13, so they are in the sequence.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, 196 Algorithm.
Eric Weisstein's World of Mathematics, Lychrel Number
Eric Weisstein's World of Mathematics, Palindromic Number Conjecture
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A008604, A002113 (palindromes), A007500 (reversible primes).

Cf. A023108.

a247128[n_Integer] := Select[Range[n], MemberQ[{0, 5, 9, 13, 17}, Mod[#, 22]] &]; a247128[211] (* Michael De Vlieger, Nov 23 2014 *)

isok(n) = m = n % 22; (m==0) || (m==5) || (m==9) || (m==13) || (m==17);
select(x->isok(x), vector(200, i, i)) \\ Michel Marcus, Nov 28 2014

from math import *
for n in range(0,100001):
  if (sqrt(4*n-ceil(n/3)+3+n%2))%1==0:print(int(sqrt(4*n-ceil(n/3)+3+n%2)),end=",")

A247128_list = [n for n in range(1,10**5) if (n % 22) in {0,5,9,13,17}]
# Chai Wah Wu, Dec 31 2014

A247128_list, l = [], [5,9,13,17,22]
for _ in range(10**5):
    A247128_list.extend(l)
    l = [x+22 for x in l] # Chai Wah Wu, Jan 01 2015

a(n) = a(n-1) + a(n-5) - a(n-6). - Colin Barker, Nov 20 2014
G.f.: x*(5*x^4+4*x^3+4*x^2+4*x+5) / ((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Nov 20 2014
Proof that a(n) = a(n-1) + a(n-5) - a(n-6): the sequence a(n) is a concatenation of the sequences [5+22*i, 9+22*i, 13+22*i, 17+22*i, 22+22*i] for i = 0,1,2,..., so it is clear that a(n-1) = a(n-6) + 22 and a(n) = a(n-5) + 22. - Chai Wah Wu, Jan 01 2015

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).

Original entry on oeis.org

0, 0, 2, 0, 6, 4, 0, 10, 12, 6, 0, 14, 20, 18, 8, 0, 18, 28, 30, 24, 10, 0, 22, 36, 42, 40, 30, 12, 0, 26, 44, 54, 56, 50, 36, 14, 0, 30, 52, 66, 72, 70, 60, 42, 16, 0, 34, 60, 78, 88, 90, 84, 70, 48, 18, 0, 38, 68, 90, 104, 110, 108, 98, 80, 54, 20, 0, 42, 76, 102

Offset: 0

Philippe Deléham, Dec 05 2013

Row sums are A006331(n).
Diagonal sums are A212964(n+1).
T(2n,n)=A002943(n).

			Triangle begins:
  0
  0, 2
  0, 6, 4
  0, 10, 12, 6
  0, 14, 20, 18, 8
  0, 18, 28, 30, 24, 10

Cf. columns: A000004, A016825, A017113, A017593, A051062, 10*A005408, A073762.

Cf. diagonals: A005843, A008588, A008592, A008596, A008600, A008604.

T(n+k,k) = A005843(k)*A005408(n).

Sum_{k=0..n} T(n,k) = n*(n+1)*(2*n+1)/3 = A006331(n).

cp's OEIS Frontend

A157276 A157107=220,440,661,881,1105, - 220,440,660,880,1100,=220*(n+1)=10*A008604(n+1).

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A008593 Multiples of 11.

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A008606 Multiples of 24.

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A161709 a(n) = 22*n + 1.

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A195323 a(n) = 22*n^2.

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A008605 Multiples of 23.

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A317322 Multiples of 22 and odd numbers interleaved.

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A157276 A157107=220,440,661,881,1105, - 220,440,660,880,1100,=220(n+1)=10A008604(n+1).

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).