OFFSET
1,2
COMMENTS
A three-dimensional version of the centered hexagonal numbers (A003215). Two examples: the third term 173 is built up as 19 + 37 + 61 + 37 + 19 and the fourth term 505 is built up as 37 + 61 + 91 + 127 + 91 + 61 + 37.
The sequence's digital roots are 1, 6, 2 (repeat).
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
Comment from N. J. A. Sloane, Feb 16 2025 (Start)
Let h(i) denote the centered hexagonal number A003215(i). Then for n >= 1,
a(n) = h(2*n-2) + 2*Sum_{i=n-1..2*n-3) h(i).
E.g. a(3) = h(2) + h(3) + h(4) + h(3) + h(2), as in the COMMENTS.
This sequence should really have had offset 0, not 1, which would have given a simpler formula. (End)
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
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
Edited by N. J. A. Sloane, Feb 16 2025
STATUS
approved