|
| |
|
|
A005914
|
|
Number of points on surface of hexagonal prism: 12*n^2 + 2 for n > 0 (coordination sequence for W(2)).
(Formerly M4931)
|
|
12
|
|
|
|
1, 14, 50, 110, 194, 302, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, 4802, 5294, 5810, 6350, 6914, 7502, 8114, 8750, 9410, 10094, 10802, 11534, 12290, 13070, 13874, 14702, 15554, 16430, 17330, 18254, 19202, 20174, 21170
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
For n >= 1, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n,n+1} such that Im(f) contains 2 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007
Equals binomial transform of [1, 13, 23, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Apr 22 2008
Also sequence found by reading the segment (1, 14) together with the line from 14, in the direction 14, 50, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Nov 02 2012
Unique sequence such that for all n > 0, n*a(1) + (n-1)*a(2) + (n-3)*a(3) + ... + 2*a(2) + a(1) = n^4. - Warren Breslow, Dec 12 2014
|
|
|
REFERENCES
|
Gmelin Handbook of Inorganic and Organometallic Chemistry, 8th Ed., 1994, TYPIX search code (229) cI2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: (1+x)*(1+10*x+x^2)/(1-x)^3. - Simon Plouffe (see MAPLE line)
a(n) = (2n-1)^2 + (2n)^2 + (2n+1)^2 for n > 0. - Bruno Berselli, Jan 30 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=14, a(2)=50, a(3)=110. - Harvey P. Dale, Oct 09 2012
Sum_{n>=0} 1/a(n) = ((Pi/sqrt(6))*coth(Pi/sqrt(6)) + 3)/4.
Sum_{n>=0} (-1)^n/a(n) = ((Pi/sqrt(6))*cosech(Pi/sqrt(6)) + 3)/4. (End)
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
Join[{1}, LinearRecurrence[{3, -3, 1}, {14, 50, 110}, 50]] (* Harvey P. Dale, Oct 09 2012 *)
|
|
|
PROG
|
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
