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
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