OFFSET
0,3
COMMENTS
Sequence found by reading the line from 0, in the direction 0, 16, ... and the parallel line from 1, in the direction 1, 45, ..., in the square spiral whose vertices are the generalized 16-gonal numbers. - Omar E. Pol, Jul 18 2012
This is also a star octagonal number: a(n) = A000567(n) + 8*A000217(n-1). - Luciano Ancora, Mar 29 2015
Let T(n) = A000217(n), the n-th triangular number. Then a(n) = T(n-1) + T(4n-3) - T(2n-4) + T(n-3). In general, let P(k,n) be the n-th k-gonal number. Then for k>1, P(T(k)+1,n) = T(n-1) + T((k-1)n-(k-2)) - T((k-3)n-2(k-3)) + T((k-4)n-3(k-4)) - ... + (-1)^(k+1)*T(n-(k-2)). - Charlie Marion, Dec 23 2019
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
John Elias, Illustration: 16-gonal Star Configuration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 14*n + a(n-1) - 13, with n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
G.f.: x*(1+13*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(0)=0, a(1)=1, a(2)=16; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 07 2011
a(14*a(n) + 92*n + 1) = a(14*a(n) + 92*n) + a(14*n+1). - Vladimir Shevelev, Jan 24 2014
E.g.f.: exp(x)*x*(1 + 7*x). - Stefano Spezia, Dec 27 2019
a(n) = (4*n-3)^2 - (3*n-3)^2. In general, if we let P(k,n) be the n-th k-gonal number, then P(4k,n) = (k*n-k+1)^2 - ((k-1)*n-k+1)^2. In addition, {P(4k,n)} are the only polygonal number sequences each of whose terms can be written as the difference of two squares. - Charlie Marion, Feb 16 2020
Product_{n>=2} (1 - 1/a(n)) = 7/8. - Amiram Eldar, Jan 22 2021
MATHEMATICA
Table[n(7n-6), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 16}, 51] (* Harvey P. Dale, May 07 2011 *)
PROG
(PARI) a(n)=n*(7*n-6) \\ Charles R Greathouse IV, Jan 24 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 15 1999
STATUS
approved