OFFSET
1,2
COMMENTS
Apparently a(n) = A062318(n+2) - 1.
The terms beginning with a(2) are the row numbers in Pascal's Triangle where every 3rd element in those rows is divisible by 3 and none of the other elements in those rows are divisible by 3. - Thomas M. Green, Apr 03 2013
REFERENCES
Thomas M. Green, Prime Patterns in Pascal's Triangle, paper in review process, 2013.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Eric Huang, Tanya Khovanova, Timur Kilybayev, Ryan Li, Brandon Ni, Leone Seidel, Samarth Sharma, Nathan Sheffield, Vivek Varanasi, Alice Yin, Boya Yun, and William Zelevinsky, Card Dealing Math, arXiv:2509.11395 [math.NT], 2025. See pp. 18, 28.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).
FORMULA
a(n) = A038754(n+1) - 2.
a(n) = 3*a(n-2) + 4 for n > 2; a(1) = 1, a(2) = 4.
a(n) = (5 - (-1)^n)*3^(1/4*(2*n - 1 + (-1)^n))/2 - 2.
G.f.: x*(1 + 3*x)/((1 - x)*(1 - 3*x^2)).
E.g.f.: 2*(cosh(sqrt(3)*x) - cosh(x)) + sqrt(3)*sinh(sqrt(3)*x) - 2*sinh(x). - Stefano Spezia, Dec 31 2022
EXAMPLE
For n = 3, a(3) = 7. The binomial coefficients of the 7th row of Pascal's Triangle are 1 7 21 35 35 21 7 1 and every 3rd element is a multiple of 3. - Thomas M. Green, Apr 03 2013
MATHEMATICA
Accumulate[Transpose[NestList[{Last[#], 3*First[#]}&, {1, 3}, 40]][[1]]] (* Harvey P. Dale, Feb 17 2012 *)
PROG
(Magma) T:=[ n le 2 select 2*n-1 else 3*Self(n-2): n in [1..33] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];
(PARI) a(n) = (2+n%2)*3^(n\2)-2 \\ Charles R Greathouse IV, Jul 15 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Aug 10 2009
EXTENSIONS
Incorrect formula removed by Stefano Spezia, Dec 31 2022
STATUS
approved