A051868 16-gonal (or hexadecagonal) numbers: a(n) = n*(7*n-6).

0, 1, 16, 45, 88, 145, 216, 301, 400, 513, 640, 781, 936, 1105, 1288, 1485, 1696, 1921, 2160, 2413, 2680, 2961, 3256, 3565, 3888, 4225, 4576, 4941, 5320, 5713, 6120, 6541, 6976, 7425, 7888, 8365, 8856, 9361, 9880, 10413, 10960, 11521

Offset: 0

Author

N. J. A. Sloane, Dec 15 1999

Keywords

• nonn
• easy

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 to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A000566, A051682, A051874, A255187, A282852.

Programs

Mathematica

Table[n(7n-6),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,1,16}, 51] (*  Harvey P. Dale, May 07 2011 *)

PARI

a(n)=n*(7*n-6) \\ Charles R Greathouse IV, Jan 24 2014

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