A051062 - OEIS

%I #92 Nov 27 2024 07:27:14

%S 8,24,40,56,72,88,104,120,136,152,168,184,200,216,232,248,264,280,296,

%T 312,328,344,360,376,392,408,424,440,456,472,488,504,520,536,552,568,

%U 584,600,616,632,648,664,680,696,712,728,744,760,776,792,808,824,840

%N a(n) = 16*n + 8.

%C Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(97).

%C n such that 32 is the largest power of 2 dividing A003629(k)^n-1 for any k. - _Benoit Cloitre_, Mar 23 2002

%C Continued fraction expansion of tanh(1/8). - _Benoit Cloitre_, Dec 17 2002

%C If Y and Z are 2-blocks of a (4n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - _Milan Janjic_, Oct 28 2007

%C General form: (q*n+x)*q x=+1; q=2=A016825, q=3=A017197, q=4=A119413, ... x=-1; q=3=A017233, q=4=A098502, ... x=+2; q=4=A051062, ... - _Vladimir Joseph Stephan Orlovsky_, Feb 16 2009

%C a(n)*n+1 = (4n+1)^2 and a(n)*(n+1)+1 = (4n+3)^2 are both perfect squares. - _Carmine Suriano_, Jun 01 2014

%C For all positive integers n, there are infinitely many positive integers k such that k*n + 1 and k*(n+1) + 1 are both perfect squares. Except for 8, all the numbers of this sequence are the smallest integers k which are solutions for getting two perfect squares. Example: a(1) = 24 and 24 * 1 + 1 = 25 = 5^2, then 24 * (1+1) + 1 = 49 = 7^2. [Reference AMM] - _Bernard Schott_, Sep 24 2017

%C Numbers k such that 3^k + 1 is divisible by 17*193. - _Bruno Berselli_, Aug 22 2018

%D Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov 11 1999

%H Mihaly Bencze, <a href="http://www.mat.uniroma2.it/~tauraso/AMM/AMM11508.pdf">Problem 11508</a>, The American Mathematical Monthly, Vol. 117, N° 5, May 2010, p. 459.

%H Milan Janjić, <a href="https://web.archive.org/web/20181001110015/https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>. [Wayback Machine link]

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.

%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>.

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = A118413(n+1,4) for n>3. - _Reinhard Zumkeller_, Apr 27 2006

%F a(n) = 32*n - a(n-1) for n>0, a(0)=8. - _Vincenzo Librandi_, Aug 06 2010

%F A003484(a(n)) = 8; A209675(a(n)) = 9. - _Reinhard Zumkeller_, Mar 11 2012

%F A007814(a(n)) = 3; A037227(a(n)) = 7. - _Reinhard Zumkeller_, Jun 30 2012

%F a(-1 - n) = - a(n). - _Michael Somos_, Jun 02 2014

%F Sum_{n>=0} (-1)^n/a(n) = Pi/32 (A244978). - _Amiram Eldar_, Feb 28 2023

%F From _Elmo R. Oliveira_, Apr 16 2024: (Start)

%F G.f.: 8*(1+x)/(1-x)^2.

%F E.g.f.: 8*exp(x)*(1 + 2*x).

%F a(n) = 8*A005408(n) = A008598(n) + 8 = A139098(n+1) - A139098(n).

%F a(n) = 4*A016825(n) = 2*A017113(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

%F From _Amiram Eldar_, Nov 25 2024: (Start)

%F Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)*sin(7*Pi/32).

%F Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2)*cos(7*Pi/32). (End)

%p A051062:=n->16*n+8; seq(A051062(n), n=0..50); # _Wesley Ivan Hurt_, Jun 01 2014

%t Range[8, 1000, 16] (* _Vladimir Joseph Stephan Orlovsky_, May 31 2011 *)

%t Table[16n+8, {n,0,50}] (* _Wesley Ivan Hurt_, Jun 01 2014 *)

%t LinearRecurrence[{2,-1},{8,24},60] (* or *) NestList[#+16&,8,60] (* _Harvey P. Dale_, Aug 18 2019 *)

%o (Magma) [16*n+8: n in [0..50]]; // _Wesley Ivan Hurt_, Jun 01 2014

%o (PARI) a(n)=16*n+8 \\ _Charles R Greathouse IV_, May 09 2016

%Y Cf. A005408, A008598, A017113, A106839, A139098, A244978.

%Y Cf. A003484, A007814, A037227, A118413, A209675.

%Y Cf. A003629, A016825, A017197, A017233, A098502, A119413.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_, _Gary W. Adamson_, Dec 11 1999