A290631 a(n) = (n^2 + 1) * (2*n - 1).

2, 15, 50, 119, 234, 407, 650, 975, 1394, 1919, 2562, 3335, 4250, 5319, 6554, 7967, 9570, 11375, 13394, 15639, 18122, 20855, 23850, 27119, 30674, 34527, 38690, 43175, 47994, 53159, 58682, 64575, 70850, 77519, 84594, 92087, 100010, 108375, 117194, 126479, 136242

Offset

1,1

Comments

Sums of all integers between successive central polygonal numbers: (1) 2 (3) 4,5,6 (7) 8,9,10,11,12 (13) ..., where the sums are taken over the terms not in brackets.

Also for n >= 1, sum of 2n-1 consecutive integers beginning with A(n)+1, where A(n) = n(n-1) + 1.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

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

Formula

From Colin Barker, Aug 09 2017: (Start)

G.f.: x*(2 + 7*x + 2*x^2 + x^3) / (1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. (End)

Example

For n=2, A002061(2)=3, and a(2) = 4 + 5 + 6 = 15.

Mathematica

Array[(#^2 + 1) (2 # - 1) &, 41] (* or *)
Rest@ CoefficientList[Series[x (2 + 7 x + 2 x^2 + x^3)/(1 - x)^4, {x, 0, 41}], x] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {2, 15, 50, 119}, 41] (* Michael De Vlieger, Aug 09 2017 *)

Prog

(PARI) Vec(x*(2 + 7*x + 2*x^2 + x^3) / (1 - x)^4 + O(x^60)) \\ Colin Barker, Aug 09 2017

Crossrefs

Cf. A002061 (central polygonal numbers), A135668 (complement).

Sequence in context: A350383 A025213 A362303 * A116693 A154565 A066562

Adjacent sequences: A290628 A290629 A290630 * A290632 A290633 A290634

Keyword

nonn,easy

Author

Enrique Navarrete, Aug 07 2017

Status

approved