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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
OFFSET
1,2
COMMENTS
a(n) is sum of natural numbers filled in order-n diamond.
First differences give A173962.
LINKS
Kival Ngaokrajang, Illustration for n = 1..6
Eric Weisstein's World of Mathematics, Diamond
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
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Feb 08 2014
STATUS
approved