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A016921 a(n) = 6*n + 1. 101
1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325, 331 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 22 ).

Also solutions to 2^x + 3^x == 5 (mod 7). - Cino Hilliard, May 10 2003

Except for 1, exponents n > 1 such that x^n - x^2 - 1 is reducible. - N. J. A. Sloane, Jul 19 2005

Let M(n) be the n X n matrix m(i,j) = min(i,j); then the trace of M(n)^(-2) is a(n-1) = 6*n - 5. - Benoit Cloitre, Feb 09 2006

If Y is a 3-subset of an (2n+1)-set X then, for n >= 3, a(n-1) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007

A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008

A157176(a(n)) = A013730(n). - Reinhard Zumkeller, Feb 24 2009

All composite terms belong to A269345 as shown in there. - Waldemar Puszkarz, Apr 13 2016

First differences of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016

For b(n) = A103221(n) one has b(a(n)-1) = b(a(n)+1) = b(a(n)+2) = b(a(n)+3) = b(a(n)+4) = n+1 but b(a(n)) = n. So-called "dips" in A103221. See the Avner and Gross remark on p. 178. - Wolfdieter Lang, Sep 16 2016

REFERENCES

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

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).

FORMULA

a(n) = 6*n + 1, n >= 0 (see the name).

G.f.: (1+5*x)/(1-x)^2.

a(n) = 4*(3*n-1) - a(n-1) (with a(0)=1). - Vincenzo Librandi, Nov 20 2010

E.g.f.: (1 + 6*x)*exp(x). - G. C. Greubel, Sep 18 2019

EXAMPLE

From Ilya Gutkovskiy, Apr 15 2016: (Start)

Illustration of initial terms:

                      o

                    o o o

              o     o o o

            o o o   o o o

      o     o o o   o o o

    o o o   o o o   o o o

o   o o o   o o o   o o o

n=0  n=1     n=2     n=3

(End)

MAPLE

a[1]:=1:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..56); # Zerinvary Lajos, Mar 16 2008

MATHEMATICA

Range[1, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

PROG

(Sage) [i+1 for i in range(333) if gcd(i, 6) == 6] # Zerinvary Lajos, May 20 2009

(Haskell)

a016921 = (+ 1) . (* 6)

a016921_list = [1, 7 ..]  -- Reinhard Zumkeller, Jan 15 2013

(PARI) a(n)=6*n+1 \\ Charles R Greathouse IV, Mar 22 2016

(Python) for n in range(0, 10**5):print(6*n+1) # Soumil Mandal, Apr 14 2016

(MAGMA) [6*n+1: n in [0..60]]; // G. C. Greubel, Sep 18 2019

(GAP) List([0..60], n-> 6*n+1); # G. C. Greubel, Sep 18 2019

CROSSREFS

Cf. A093563 ((6, 1) Pascal, column m=1).

Cf. A000567 (partial sums), A002476 (primes), A005408, A008588, A016813, A016933, A016945, A016957, A017281, A017533, A128470, A158057, A161700, A161705, A161709, A161714.

a(n) = A007310(2*(n+1)); complement of A016969 with respect to A007310.

Cf. A287326 (second column).

Sequence in context: A059335 A070419 A080199 * A331425 A260682 A184521

Adjacent sequences:  A016918 A016919 A016920 * A016922 A016923 A016924

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 22 05:01 EDT 2020. Contains 337950 sequences. (Running on oeis4.)