A158057 First differences of A051870: 16*n + 1.

1, 17, 33, 49, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 465, 481, 497, 513, 529, 545, 561, 577, 593, 609, 625, 641, 657, 673, 689, 705, 721, 737, 753, 769, 785, 801, 817, 833, 849

Offset

0,2

Comments

The identity (16*n+1)^2 - (16*n^2+2*n)*(4)^2 = 1 can be written as a(n+1)^2 - A158056(n)*(4)^2 = 1. - Vincenzo Librandi, Feb 09 2012

This sequence gives the 18-gonal (or octadecagonal) gnomonic numbers. Name suggested by Todd Silvestri, Nov 22 2014

All elements are odd and contains subsequence A249356. - Todd Silvestri, Nov 22 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

OEIS Wiki, Gnomonic numbers

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 16*n + 1.

a(n) = 2*a(n-1) - a(n-2), a(0) = 1, a(1) = 17.

G.f.: (1+15*x)/(1-x)^2. - Vincenzo Librandi, Nov 23 2014

E.g.f.: (1 + 18*x)*exp(x). - G. C. Greubel, Sep 18 2019

Maple

seq(16*n+1, n=0..60); # G. C. Greubel, Sep 18 2019

Mathematica

LinearRecurrence[{2, -1}, {1, 17}, 60]
Table[16*n+1, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)
a[n_Integer/; n>=0]:=16 n+1 (* Todd Silvestri, Nov 22 2014 *)
CoefficientList[Series[(1+15x)/(1-x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Nov 23 2014 *)

Prog

(Magma) I:=[1, 17]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];
(PARI) a(n)=n<<4+1 \\ Charles R Greathouse IV, Dec 23 2011
(Sage) [16*n+1 for n in (0..60)] # G. C. Greubel, Sep 18 2019
(GAP) List([0..60], n-> 16*n+1); # G. C. Greubel, Sep 18 2019

Crossrefs

Cf. A051870, A158056.

Sequence in context: A340000 A044062 A044443 * A249356 A329919 A346528

Adjacent sequences: A158054 A158055 A158056 * A158058 A158059 A158060

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 12 2009

Extensions

Name clarified and offset changed by Todd Silvestri, Nov 22 2014

Edited by Vincenzo Librandi Nov 23 2014

Edited: Offset changed to 0 according to the

Todd Silvestri proposal. Name changed. - Wolfdieter Lang, Nov 29 2014

Status

approved