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A237516 Pyramidal centered square numbers. 1
1, 15, 91, 325, 861, 1891, 3655, 6441, 10585, 16471, 24531, 35245, 49141, 66795, 88831, 115921, 148785, 188191, 234955, 289941, 354061, 428275, 513591, 611065, 721801, 846951, 987715, 1145341, 1321125, 1516411, 1732591, 1971105, 2233441, 2521135, 2835771, 3178981, 3552445, 3957891, 4397095, 4871881 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is sum of natural numbers filled in order-n diamond.

First differences give A173962.

LINKS

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

Kival Ngaokrajang, Illustration for n = 1..6

Eric Weisstein's World of Mathematics, Diamond

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

a(n) = 2n^4 - 4n^3 + 5n^2 - 3n + 1.

a(n) = Sum_{i = 1..(2n(n + 1) + 1)} i.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Colin Barker, Jan 17 2015

G.f.: -x*(x^2+4*x+1)*(x^2+6*x+1) / (x-1)^5. - Colin Barker, Jan 17 2015

a(n) = A000217(A001844(n-1)). - Ivan N. Ianakiev, Jun 14 2015

a(n) = A002061(n) * A001844(n-1). - Bruce J. Nicholson, May 14 2017

MATHEMATICA

Table[Sum[i, {i, 2n(n + 1) + 1}], {n, 0, 29}] (* Alonso del Arte, Feb 09 2014 *)

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 15, 91, 325, 861}, 60] (* Harvey P. Dale, Apr 21 2018 *)

PROG

(Small Basic)

For n = 0 to 50

a = 0

For n1 = 1 to 2*n*(n+1)+1

  a = a + n1

EndFor

TextWindow.Write(a+", ")

EndFor

(PARI) Vec(-x*(x^2+4*x+1)*(x^2+6*x+1)/(x-1)^5 + O(x^100)) \\ Colin Barker, Jan 17 2015

CROSSREFS

Cf. A001844, A173962.

Cf. A002061, A000217.

Sequence in context: A048630 A326148 A035163 * A020242 A020255 A180248

Adjacent sequences:  A237513 A237514 A237515 * A237517 A237518 A237519

KEYWORD

nonn,easy

AUTHOR

Kival Ngaokrajang, Feb 08 2014

STATUS

approved

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Last modified May 8 07:02 EDT 2021. Contains 343653 sequences. (Running on oeis4.)