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

Author

Klaus Brockhaus, Aug 10 2009

Keywords

• nonn
• easy

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

Examples

			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
		

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).

Crossrefs

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

Programs

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]];

Mathematica

Accumulate[Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]]] (* Harvey P. Dale, Feb 17 2012 *)

PARI

a(n) = (2+n%2)*3^(n\2)-2 \\ Charles R Greathouse IV, Jul 15 2011

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

Extensions

Incorrect formula removed by Stefano Spezia, Dec 31 2022