A016921 - OEIS (original) (raw)

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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

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

A (n+1,n) pebbling move involves removing n + 1 pebbles from a vertex in a simple graph and placing n pebbles on an adjacent vertex. A two-player impartial (n+1,n) pebbling game involves two players alternating (n+1,n) pebbling moves. The first player unable to make a move loses. The sequence a(n) is also the minimum number of pebbles such that any assignment of those pebbles on a complete graph with 3 vertices is a next-player winning game in the two player impartial (k+1,k) pebbling game. These games are represented by A347637(3,n). - Joe Miller, Oct 18 2021

REFERENCES

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

LINKS

Kayla Barker, Mia DeStefano, Eugene Fiorini, Michael Gohn, Joe Miller, Jacob Roeder, and Tony W. H. Wong, Generalized Impartial Two-player Pebbling Games on K_3 and C_4, J. Int. Seq. (2024) Vol. 27, Issue 5, Art. No. 24.5.8. See p. 3.

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

Sum_{n>=0} (-1)^n/a(n) = Pi/6 + sqrt(3)*arccoth(sqrt(3))/3. - Amiram Eldar, Dec 10 2021

EXAMPLE

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

PROG

(Haskell)

a016921 = (+ 1) . (* 6)

(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.