A052150 Partial sums of A000340, second partial sums of A003462.

1, 6, 24, 82, 261, 804, 2440, 7356, 22113, 66394, 199248, 597822, 1793557, 5380776, 16142448, 48427480, 145282593, 435847950, 1307544040, 3922632330, 11767897221, 35303691916, 105911076024, 317733228372, 953199685441

Offset

0,2

Comments

Convolution of the powers of 3 with the triangular numbers [1, 3, 6, 10, ...]. - Enrique Navarrete, Nov 04 2025

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, N.Y., 1991, p. 53.

Links

Table of n, a(n) for n=0..24.

Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Formula

a(n) = (3^(n+3) - (2*n^2 + 12*n + 19))/8.

a(n) = 3*a(n-1)+C(n+2,2); a(0)=1.

a(n) = Sum_{k=0..n} binomial(n+3, k+3)*2^k. - Paul Barry, Aug 20 2004

From Colin Barker, Dec 18 2012: (Start)

a(n) = 6*a(n-1) - 12*a(n-2) + 10*a(n-3) - 3*a(n-4).

G.f.: 1/((x-1)^3*(3*x-1)). (End)

E.g.f.: exp(x)*(27*exp(2*x) - 2*x^2 - 14*x - 19)/8. - Enrique Navarrete, Nov 04 2025

Mathematica

LinearRecurrence[{6, -12, 10, -3}, {1, 6, 24, 82}, 40] (* Harvey P. Dale, Sep 05 2013 *)

Crossrefs

Cf. A003462, A000340.

Cf. A000217, A000244.

Sequence in context: A320856 A047790 A133474 * A118043 A317408 A166060

Adjacent sequences: A052147 A052148 A052149 * A052151 A052152 A052153

Keyword

easy,nonn

Author

Barry E. Williams, Jan 23 2000

Status

approved