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

Author

Vincenzo Librandi, Mar 12 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A016813, A017077, A051870, A158056, A249356.

Programs

GAP

List([0..60], n-> 16*n+1); # G. C. Greubel, Sep 18 2019

Magma

I:=[1, 17]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];

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 *)

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

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 + 16*x)*exp(x). - G. C. Greubel, Sep 18 2019 [corrected by Elmo R. Oliveira, Apr 12 2025]

a(n) = A017077(2*n) = A016813(4*n). - Elmo R. Oliveira, Apr 12 2025

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