A155883 a(n) = 14*n^3 - 30*n^2 + 24*n - 7.

1, 33, 173, 505, 1113, 2081, 3493, 5433, 7985, 11233, 15261, 20153, 25993, 32865, 40853, 50041, 60513, 72353, 85645, 100473, 116921, 135073, 155013, 176825, 200593, 226401, 254333, 284473, 316905, 351713, 388981, 428793, 471233, 516385, 564333, 615161, 668953

Offset

1,2

Comments

Previous name was: The sequence gives the three-dimensional forms of the centered hexagonal numbers. Two examples: its third term 173 is built 19 + 37 + 61 + 37 + 19 and its fourth term 505 is built 37 + 61 + 91 + 127 + 91 + 61 + 37.

The sequence's digital roots run through 1, 6, 2.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

David Z. Crookes, De Pulchritudine Numerorum Figuratorum (On the Beauty of Figurate Numbers), Mathematics in School (May, 1988), 38-39.

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

Formula

a(n) = 14*n^3 - 30*n^2 + 24*n - 7.

G.f.: x*(1+29*x+47*x^2+7*x^3)/(1-x)^4. [Colin Barker, Jun 16 2012]

a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Vincenzo Librandi, Jun 30 2012

Mathematica

CoefficientList[Series[(1+29*x+47*x^2+7*x^3)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 30 2012 *)

Prog

(Magma) I:=[1, 33, 173, 505]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]];  // Vincenzo Librandi, Jun 30 2012

Crossrefs

Sequence in context: A172077 A180943 A113752 * A071914 A279867 A230650

Adjacent sequences: A155880 A155881 A155882 * A155884 A155885 A155886

Keyword

nonn,easy,less

Author

_David Z. Crookes_, Jan 29 2009

Extensions

More terms from Colin Barker, Jun 16 2012

New name using explicit formula from Joerg Arndt, Jan 15 2021

Status

approved