|
|
A016921
|
|
a(n) = 6*n + 1.
|
|
125
|
|
|
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
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
|
|
|
FORMULA
|
a(n) = 6*n + 1, n >= 0 (see the name).
G.f.: (1+5*x)/(1-x)^2.
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
|
|
MATHEMATICA
|
|
|
PROG
|
(Haskell)
a016921 = (+ 1) . (* 6)
(Python) for n in range(0, 10**5):print(6*n+1) # Soumil Mandal, Apr 14 2016
|
|
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.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|