A262592 a(n) = (3^(n+1) - 2n^2 + 4n + 5) / 8.

1, 2, 4, 10, 29, 88, 268, 812, 2449, 7366, 22124, 66406, 199261, 597836, 1793572, 5380792, 16142465, 48427498, 145282612, 435847970, 1307544061, 3922632352, 11767897244, 35303691940, 105911076049, 317733228398, 953199685468, 2859599056702, 8578797170429, 25736391511636

Offset: 0

Author

N. J. A. Sloane, Oct 21 2015

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• K. Satyanarayana, Sequences whose kth differences form a geometrical progression, Math. Student, 12 (1944), page 109. [Annotated scanned copy. This sequence was formerly A2752 but has now been renumbered]
• Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Crossrefs

Other sequences with generating functions like this: A000340, A052161, A262593, A262594.

Programs

Maple

f1:=(a,b)->(1-a*x)^a/((1-x)^b*(1-b*x));
f2:=(a,b)->seriestolist(series(f1(a,b),x,40));
f2(2,3);

Mathematica

Table[3^(n + 1)/8 + 5/8 - n^2/4 + n/2, {n, 0, 29}] (* Michael De Vlieger, Oct 23 2015 *)
LinearRecurrence[{6,-12,10,-3},{1,2,4,10},30] (* Harvey P. Dale, Jul 18 2025 *)

PARI

a(n) = 3^(n+1)/8+5/8-n^2/4+n/2 \\ Colin Barker, Oct 23 2015

PARI

Vec((1-2*x)^2/((1-x)^3*(1-3*x)) + O(x^40)) \\ Colin Barker, Oct 23 2015

Formula

G.f.: (1-2*x)^2/((1-x)^3*(1-3*x)).

a(n) = 6*a(n-1)-12*a(n-2)+10*a(n-3)-3*a(n-4) for n>3. - Colin Barker, Oct 23 2015