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

Author

Paul Barry, Mar 21 2003

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A081436, A081437, A081441, A177058.

Programs

GAP

List([0..45], n-> (2*n^3+9*n^2+9*n+2)/2); # G. C. Greubel, Aug 14 2019

Magma

[(2*n^3+9*n^2+9*n+2)/2: n in [0..45]]; // Vincenzo Librandi, Aug 08 2013

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 *)

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

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