A017198 -id:A017198 - cp's OEIS frontend

A017305 a(n) = 10*n + 3.

3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, 113, 123, 133, 143, 153, 163, 173, 183, 193, 203, 213, 223, 233, 243, 253, 263, 273, 283, 293, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 403, 413, 423, 433, 443, 453, 463, 473, 483, 493, 503, 513, 523, 533

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(38).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008592, A016885, A017198, A017281, A017293, A156677.

[10*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011

Range[3, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n) = A017198(n) - A156677(n+2). - Reinhard Zumkeller, Jul 13 2010
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 28 2011
G.f.: (3+7*x)/(x-1)^2. - R. J. Mathar, Apr 11 2016
E.g.f.: exp(x)*(3 + 10*x). - Stefano Spezia, Aug 22 2023
a(n) = A016885(2*n). - Elmo R. Oliveira, Apr 10 2025

A017197 a(n) = 9*n + 3.

Original entry on oeis.org

3, 12, 21, 30, 39, 48, 57, 66, 75, 84, 93, 102, 111, 120, 129, 138, 147, 156, 165, 174, 183, 192, 201, 210, 219, 228, 237, 246, 255, 264, 273, 282, 291, 300, 309, 318, 327, 336, 345, 354, 363, 372, 381, 390, 399, 408, 417, 426, 435, 444, 453, 462, 471, 480

Offset: 0

N. J. A. Sloane

Numbers whose digital root is 3. - Cino Hilliard, Dec 26 2006

a(n)^2 = A017198(n). - Reinhard Zumkeller, Jul 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
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. A092292, A092293, A092296.
Cf. sequences with general form q*(q*n+1): A016825 (q=2), this sequence (q=3), A119413 (q=4), ... - Vladimir Joseph Stephan Orlovsky, Feb 16 2009
Cf. A016777.

List([0..60], n-> 9*n+3); # G. C. Greubel, Dec 03 2019

a017197 = (+ 3) . (* 9)
a017197_list = [3, 12 ..]  -- Reinhard Zumkeller, Jun 04 2015

[9*n+3: n in [0..60]]; // G. C. Greubel, Dec 03 2019

seq(9*n+3, n=0..60); # G. C. Greubel, Dec 03 2019

3*(3*Range[60] -2) (* G. C. Greubel, Dec 03 2019 *)
LinearRecurrence[{2,-1},{3,12},80] (* or *) NestList[#+9&,3,80] (* Harvey P. Dale, Jan 22 2023 *)

vector(60, n, 3*(3*n-2) ) \\ G. C. Greubel, Dec 03 2019

[i+3 for i in range(480) if gcd(i,9) == 9] # Zerinvary Lajos, May 20 2009

a(n) = a(n-1) + 9.
a(n) = 3*A016777(n).
a(n) = A092292(n) + A092293(n) + A092296(n).
From Philippe Deléham, Mar 10 2004: (Start)
Sum_{n>=0} (-1)^n / a(n) = (Pi / sqrt(3) + log(2))/9.
G.f.: 3*(1+2*x)/(1-x)^2. (End)
a(n) = 3*(6*n-1) - a(n-1) with a(0)=3. - Vincenzo Librandi, Nov 20 2010
E.g.f.: 3*(1 + 3*x)*exp(x). - G. C. Greubel, Dec 03 2019

More terms from Cino Hilliard, Dec 26 2006

A147650 First trisection of A061040.

Original entry on oeis.org

1, 12, 81, 48, 75, 324, 147, 64, 729, 100, 363, 1296, 507, 588, 2025, 768, 289, 2916, 361, 1200, 3969, 1452, 1587, 5184, 1875, 676, 6561, 784, 2523, 8100, 2883, 3072, 9801, 3468, 1225, 11664, 1369, 4332, 13689, 4800, 5043

Offset: 1

Paul Curtz, Nov 09 2008

a(n) gives the denominator of (n-1)*(n+1)/(9*n^2), for n >= 1. The numerator is given by A144454(n). - Wolfdieter Lang, Mar 16 2018

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,-3,0,0,0,0,0,0,0,0,1).

Cf. A061040, A017198 (2nd trisection), A017234 (3d trisection).

Table[Which[MemberQ[{1, 8}, Mod[n, 9]], n^2, Mod[n, 3] != 0, 3 n^2, True, 9 n^2], {n, 41}] (* Michael De Vlieger, Mar 16 2018 *)

a(n) = denominator((n-1)*(n+1)/(9*n^2)); \\ Michel Marcus, Mar 17 2018

For n >= 1: a(n) = n^2 if n == 1 (mod 9) or == 8 (mod 9). For other n: a(n) = 3*n^2 if n == 1 (mod 3) or == 2 (mod 3), and a(n) = 9*n^2 if n == 0 (mod 3). From the denominator comment above. - Wolfdieter Lang, Mar 16 2018

Offset changed from 0 to 1, and extended by Wolfdieter Lang, Mar 16 2018

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A017305 a(n) = 10*n + 3.

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A017197 a(n) = 9*n + 3.

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A147650 First trisection of A061040.

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