A329753 Doubly square pyramidal numbers.

0, 1, 55, 1015, 9455, 56980, 255346, 924490, 2850730, 7757035, 19096385, 43312841, 91753025, 183453270, 349074740, 636310340, 1117143236, 1897397285, 3129084635, 5026125195, 7884086595, 12104671656, 18225763270, 26957923950, 39228339150, 56233289775, 79500340101, 110961532605

Offset

0,3

Links

Table of n, a(n) for n=0..27.

Eric Weisstein's World of Mathematics, Square Pyramidal Number

Index to sequences related to pyramidal numbers

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

G.f.: x*(1 + 45*x + 510*x^2 + 1660*x^3 + 1715*x^4 + 519*x^5 + 30*x^6)/(1 - x)^10.

a(n) = A000330(A000330(n)).

a(n) = Sum_{k=0..A000330(n)} A000290(k).

a(n) = n *(2*n+1) *(n+2) *(n+1) *(2*n^2-n+3) *(2*n^3+3*n^2+n+3) /648. - R. J. Mathar, Nov 28 2019

Mathematica

A000330[n_] := n (n + 1) (2 n + 1)/6; a[n_] := A000330[A000330[n]]; Table[a[n], {n, 0, 27}]
Table[Sum[k^2, {k, 0, n (n + 1) (2 n + 1)/6}], {n, 0, 27}]
nmax = 27; CoefficientList[Series[x (1 + 45 x + 510 x^2 + 1660 x^3 + 1715 x^4 + 519 x^5 + 30 x^6)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 55, 1015, 9455, 56980, 255346, 924490, 2850730, 7757035}, 28]

Crossrefs

Cf. A000290, A000330, A000583, A140236, A329754, A329755, A329756, A329757.

Sequence in context: A241634 A008402 A281073 * A203872 A297753 A134291

Adjacent sequences: A329750 A329751 A329752 * A329754 A329755 A329756

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Nov 20 2019

Status

approved