A094951 a(n) = A081038(n) + A077616(n).

6, 31, 144, 621, 2538, 9963, 37908, 140697, 511758, 1830519, 6456024, 22497669, 77590386, 265189059, 899198172, 3027619377, 10130328342, 33705582543, 111577100832, 367662044061, 1206427402746, 3943553157531, 12845313733284

Offset

1,1

Comments

Performing the same operation but using the multiplier [1 0 0] yields [3^n 2*A027471(n+1) A077616(n)]. Example: M^4 * [1 0 0] = [81 216 324] where 324 = A077616(4) and 216/2 = 108 = A027471(5).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Formula

a(n) = A081038(n) + A077616(n).

Let M = the 3 X 3 matrix [3 0 0 / 2 3 0 / 1 2 3]; then M^n * [1 1 1] = [3^n A081038(n) a(n)], where a(n) - A081038(n) = A077616(n).

From Colin Barker, Nov 09 2012: (Start)

a(n) = 3^(n-2)*(9 + 7*n + 2*n^2).

a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3).

G.f.: x*(6 - 23*x + 27*x^2)/(1-3*x)^3. (End)

E.g.f.: -1 + (1 + 3*x + 2*x^2)*exp(3*x). - G. C. Greubel, Jun 06 2019

Example

a(3) = 144 = 81 + 63 = A081038(3) + A077616(3).
a(4) = 621 = 297 + 324 = A081038(4) + A077616(4).
a(4) = 621 since M^4 * [1 1 1] = [81 297 621] = [3^4 A081038(4), a(4)].

Mathematica

a[n_] := (MatrixPower[{{3, 0, 0}, {2, 3, 0}, {1, 2, 3}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 23}] (* Robert G. Wilson v, Jun 05 2004 *)
Table[3^(n-2)*(9+7*n+2*n^2), {n, 1, 30}] (* G. C. Greubel, Jun 06 2019 *)

Prog

(PARI) vector(30, n, 3^(n-2)*(9+7*n+2*n^2)) \\ G. C. Greubel, Jun 06 2019
(Magma) [3^(n-2)*(9+7*n+2*n^2): n in [1..30]]; // G. C. Greubel, Jun 06 2019
(Sage) [3^(n-2)*(9+7*n+2*n^2) for n in (1..30)] # G. C. Greubel, Jun 06 2019
(GAP) List([1..30], n-> 3^(n-2)*(9+7*n+2*n^2)) # G. C. Greubel, Jun 06 2019

Crossrefs

Cf. A081038, A077616, A027471.

Sequence in context: A009076 A291396 A012714 * A099621 A291002 A268401

Adjacent sequences: A094948 A094949 A094950 * A094952 A094953 A094954

Keyword

nonn,easy

Author

Gary W. Adamson, May 26 2004

Extensions

Edited and extended by Robert G. Wilson v, Jun 05 2004

Status

approved