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A016921 a(n) = 6n + 1. 78
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

Table of n, a(n) for n=0..55.

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

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

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 xrange(0, 10**5):print(6*n+1) # Soumil Mandal, Apr 14 2016

CROSSREFS

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

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

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

Sequence in context: A059335 A070419 A080199 * A260682 A184521 A123843

Adjacent sequences:  A016918 A016919 A016920 * A016922 A016923 A016924

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 22 13:18 EDT 2017. Contains 286872 sequences.