A164123 Partial sums of A162436.

1, 4, 7, 16, 25, 52, 79, 160, 241, 484, 727, 1456, 2185, 4372, 6559, 13120, 19681, 39364, 59047, 118096, 177145, 354292, 531439, 1062880, 1594321, 3188644, 4782967, 9565936, 14348905, 28697812, 43046719, 86093440, 129140161, 258280324, 387420487, 774840976, 1162261465

Offset

1,2

Comments

Interleaving of A058481 and A100774 without initial term 0.

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.

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

Cf. A162436, A058481 (3^n-2), A100774 (2*(3^n - 1)), A062318, A038754, A038754.

Sequence in context: A245937 A348771 A259653 * A005513 A254323 A254143

Adjacent sequences: A164120 A164121 A164122 * A164124 A164125 A164126

Keyword

nonn,easy

Author

Klaus Brockhaus, Aug 10 2009

Extensions

Incorrect formula removed by Stefano Spezia, Dec 31 2022

Status

approved