A081435 Diagonal in array of n-gonal numbers A081422.

1, 5, 18, 46, 95, 171, 280, 428, 621, 865, 1166, 1530, 1963, 2471, 3060, 3736, 4505, 5373, 6346, 7430, 8631, 9955, 11408, 12996, 14725, 16601, 18630, 20818, 23171, 25695, 28396, 31280, 34353, 37621, 41090, 44766, 48655, 52763, 57096, 61660

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 +3*n^2 +3*n +2)/2.

G.f.: (1 +3*x^2 -4*x^3)/(1-x)^5.

E.g.f.: (2 +8*x +9*x^2 +2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

Maple

a := n-> (n+1)*(2*(n+1)^2-3*n)/2; seq(a(n), n = 0..40); # G. C. Greubel, Aug 14 2019

Mathematica

Table[(n^3 +(n+1)^3 -1)/2 +1, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)
CoefficientList[Series[(1 +3x^2 -4x^3)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 5, 18, 46}, 40] (* Harvey P. Dale, Dec 28 2024 *)

Prog

(Magma) [(2*n^3+3*n^2+3*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Aug 08 2013
(PARI) vector(40, n, n--; (n+1)*(2*(n+1)^2-3*n)/2) \\ G. C. Greubel, Aug 14 2019
(SageMath) [(n+1)*(2*(n+1)^2-3*n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019
(GAP) List([0..40], n-> (n+1)*(2*(n+1)^2-3*n)/2); # G. C. Greubel, Aug 14 2019

Crossrefs

Cf. A081434, A081436, A081437.

Sequence in context: A270944 A272457 A340974 * A272748 A273559 A272792

Adjacent sequences: A081432 A081433 A081434 * A081436 A081437 A081438

Keyword

nonn,easy

Author

Paul Barry, Mar 21 2003

Status

approved