A267541 -id:A267541 - cp's OEIS frontend

A017329 a(n) = 10*n + 5.

5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 155, 165, 175, 185, 195, 205, 215, 225, 235, 245, 255, 265, 275, 285, 295, 305, 315, 325, 335, 345, 355, 365, 375, 385, 395, 405, 415, 425, 435, 445, 455, 465, 475, 485, 495, 505, 515, 525, 535

Offset: 0

N. J. A. Sloane

Continued fraction expansion of tanh(1/5). - Benoit Cloitre, Dec 17 2002
n such that 5 divides the numerator of B(2n) where B(2n) = the 2n-th Bernoulli number. - Benoit Cloitre, Jan 01 2004
5 times odd numbers. - Omar E. Pol, May 02 2008
5th transversal numbers (or 5-transversal numbers): Numbers of the 5th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 5th column in the square array A057145. - Omar E. Pol, May 02 2008
Successive sums: 5, 20, 45, 80, 125, ... (see A033429). - Philippe Deléham, Dec 08 2011
3^a(n) + 1 is divisible by 61. - Vincenzo Librandi, Feb 05 2013
If the initial 5 is changed to 1, giving 1,15,25,35,45,..., these are values of m such that A323288(m)/m reaches a new record high value. - N. J. A. Sloane, Jan 23 2019

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000367, A002193, A016957, A057145, A094884, A139600, A139606, A182007.
Subsequence of A034709, together with A017281, A017293, A139222, A139245, A139249, A139264, A139279 and A139280.
Cf. A005408, A008587, A033429, A267541, A323288.

a017329 = (+ 5) . (* 10)
a017329_list = [5, 15 ..]  -- Reinhard Zumkeller, Apr 10 2015

[10*n + 5: n in [0..60]]; // Vincenzo Librandi, Feb 05 2013

Range[5, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{5,15},60] (* Harvey P. Dale, Nov 16 2019 *)

a(n)=10*n+5 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 5*A005408(n). - Omar E. Pol, Oct 19 2008
a(n) = 20*n - a(n-1) (with a(0)=5). - Vincenzo Librandi, Nov 19 2010
G.f.: 5*(x+1)/(x-1)^2. - Colin Barker, Nov 14 2012
a(n) = A057145(n+2,5). - R. J. Mathar, Jul 28 2016
E.g.f.: 5*exp(x)*(1 + 2*x). - Stefano Spezia, Feb 14 2020
Sum_{n>=0} (-1)^n/a(n) = Pi/20. - Amiram Eldar, Dec 12 2021
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(5-sqrt(5))/2 = sqrt(2)*sin(Pi/5) = A182007/A002193.
Product_{n>=0} (1 + (-1)^n/a(n)) = phi/sqrt(2) (A094884). (End)
a(n) = (n+3)^2 - (n-2)^2. - Alexander Yutkin, Mar 16 2025
From Elmo R. Oliveira, Apr 12 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A008587(2*n+1). (End)

A017509 a(n) = 11*n + 10.

Original entry on oeis.org

10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 120, 131, 142, 153, 164, 175, 186, 197, 208, 219, 230, 241, 252, 263, 274, 285, 296, 307, 318, 329, 340, 351, 362, 373, 384, 395, 406, 417, 428, 439, 450, 461, 472, 483, 494, 505, 516, 527, 538, 549, 560, 571, 582

Offset: 0

N. J. A. Sloane

If k is any member of A045572, the sequence lists the numbers n such that (n^k+1)/11 is a nonnegative integer. See also A267541. - Bruno Berselli, Jan 16 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 989
Leo Tavares, Illustration: Triangular Lines
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008593, A017401, A017413, A045572, A267541.
Cf. A211013 (partial sums), A254322 (partial products).
Powers of the form (11*n+10)^m: this sequence (m=1), A017510 (m=2), A017511 (m=3), A017512 (m=4), A017513 (m=5), A017514 (m=6), A017515 (m=7), A017516 (m=8), A017517 (m=9), A017518 (m=10), A017519 (m=11), A017520 (m=12).
Cf. A008591, A005408.

List([0..60], n-> (11*n+10)); # G. C. Greubel, Oct 29 2019

[11*n+10: n in [0..60]]; // Vincenzo Librandi, Sep 18 2011

seq((11*n+10), n=0..60); # G. C. Greubel, Oct 29 2019

Range[10, 1000, 11] (* Vladimir Joseph Stephan Orlovsky, May 29 2011 *)
(11*Range[60] -1) (* G. C. Greubel, Oct 29 2019 *)

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

def a(n): return 11*n + 10
print([a(n) for n in range(53)]) # Michael S. Branicky, Oct 21 2021

[(11*n+10) for n in (0..60)] # G. C. Greubel, Oct 29 2019

From G. C. Greubel, Oct 29 2019: (Start)
G.f.: (10 + x)/(1-x)^2.
E.g.f.: (10 + 11*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2). (End)
a(n) = A008591(n+1) + A005408(n). - Leo Tavares, Oct 25 2022

A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).

Original entry on oeis.org

1, 3, 4, 5, 9, 12, 14, 15, 16, 20, 23, 25, 26, 27, 31, 34, 36, 37, 38, 42, 45, 47, 48, 49, 53, 56, 58, 59, 60, 64, 67, 69, 70, 71, 75, 78, 80, 81, 82, 86, 89, 91, 92, 93, 97, 100, 102, 103, 104, 108, 111, 113, 114, 115, 119, 122, 124, 125, 126, 130, 133, 135, 136, 137

Offset: 0

Bruno Berselli, Jan 20 2016

(m^k-1)/11 is a nonnegative integer when
. m is a member of this sequence and k is an odd multiple of 5 (A017329),
. m is a member of A017401 and k is odd but not multiple of 5 (A045572),
. m is a member of A175885 and k is even but not multiple of 5 (A217562),
. m is a member of A160542 and k is a positive multiple of 10 (A008592),
apart from the trivial case in which k=0.
Also, numbers that are congruent to {1, 3, 4, 5, 9} mod 11. Therefore, the product of two terms belongs to the sequence.
Union of this sequence and A267541 is A160542.
a(n) is prime for n = 1, 3, 10, 14, 17, 21, 24, 27, 30, 33, 40, 44, 47, ...

			From the linear recurrence:
(-A267541) ..., -13, -10, -8, -7, -6, -2, 1, 3, 4, 5, 9, 12, ... (A267755)

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

Cf. A160542, A017329, A267541.

Related sequences (see the first comment): A017401, A160542, A175885.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)));

I:=[1,3,4,5,9,12]; [n le 6 select I[n] else Self(n-1)+Self(n-5)-Self(n-6): n in [1..70]]; // Vincenzo Librandi, Jan 21 2016

gf := (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6): deg := 64: series(gf, x, deg): seq(coeff(%, x, n), n=0..deg-1); # Peter Luschny, Jan 21 2016

CoefficientList[Series[(1 + 2 x + x^2 + x^3 + 4 x^4 + 2 x^5)/(1 - x - x^5 + x^6), {x, 0, 70}], x]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 3, 4, 5, 9, 12}, 70]
Select[Range[140], MemberQ[{1, 3, 4, 5, 9}, Mod[#, 11]]&]

Vec((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)+O(x^70))

gf = (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6)
print(taylor(gf, x, 0, 63).list()) # Peter Luschny, Jan 21 2016

G.f.: (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(-n) = -A267541(n-1).
a(n) = n + 1 + 2*floor(n/5) + 3*floor((n+1)/5) + floor((n+4)/5). - Ridouane Oudra, Sep 06 2023

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A017329 a(n) = 10*n + 5.

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A017509 a(n) = 11*n + 10.

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A267755 Expansion of (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6).

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Magma

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A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).