A081438 Diagonal in array of n-gonal numbers A081422.

1, 11, 36, 82, 155, 261, 406, 596, 837, 1135, 1496, 1926, 2431, 3017, 3690, 4456, 5321, 6291, 7372, 8570, 9891, 11341, 12926, 14652, 16525, 18551, 20736, 23086, 25607, 28305, 31186, 34256, 37521, 40987, 44660, 48546, 52651, 56981, 61542, 66340

Offset

0,2

Comments

One of a family of sequences with palindromic generators.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

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

G.f.: (1+6*x-9*x^2+2*x^3)/(1-x)^5.

From Bruno Berselli, Jun 04 2010: (Start)

G.f.: (1+7*x-2*x^2)/(1-x)^4 (simplified).

a(n) = (n+1)*(2*n^2+7*n+2)/2.

a(n) -4*a(n-1) +6*a(n-2) -4*a(n-3) +a(n-4) = 0, with n>3.

a(n) = (A177058(n+3) + A177058(n+2))/2. (End)

E.g.f.: (1/2)*exp(x)*(2 +20*x + 15*x^2 + 2*x^3). - Stefano Spezia, Aug 15 2019

Maple

seq((2*n^3+9*n^2+9*n+2)/2, n=0..45); # G. C. Greubel, Aug 14 2019

Mathematica

CoefficientList[Series[(1 +6x -9x^2 +2x^3)/(1-x)^5, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 11, 36, 82}, 50] (* Harvey P. Dale, Jan 20 2022 *)

Prog

(Magma) [(2*n^3+9*n^2+9*n+2)/2: n in [0..45]]; // Vincenzo Librandi, Aug 08 2013
(PARI) vector(45, n, n--; (2*n^3+9*n^2+9*n+2)/2) \\ G. C. Greubel, Aug 14 2019
(Sage) [(2*n^3+9*n^2+9*n+2)/2 for n in (0..45)] # G. C. Greubel, Aug 14 2019
(GAP) List([0..45], n-> (2*n^3+9*n^2+9*n+2)/2); # G. C. Greubel, Aug 14 2019

Crossrefs

Cf. A081436, A081437, A081441, A177058.

Sequence in context: A044469 A015246 A093500 * A160483 A034309 A005000

Adjacent sequences: A081435 A081436 A081437 * A081439 A081440 A081441

Keyword

nonn,easy

Author

Paul Barry, Mar 21 2003

Status

approved