This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A092521 #62 Aug 13 2025 00:59:51
%S A092521 1,8,56,385,2640,18096,124033,850136,5826920,39938305,273741216,
%T A092521 1876250208,12860010241,88143821480,604146740120,4140883359361,
%U A092521 28382036775408,194533374068496,1333351581704065,9138927697859960
%N A092521 a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3), with a(1) = 1, a(2) = 8, a(3) = 56.
%C A092521 a(n) such that 9*(T(a(n)-1) + T(a(n+1)-1)) = 7*(T(a(n) + a(n+1) - 1)), where T(i) denotes the i-th triangular number.
%C A092521 Partial sums of Chebyshev sequence S(n,7) = U(n,7/2) = A004187(n+1). - _Wolfdieter Lang_, Aug 31 2004
%C A092521 From _Klaus Purath_, Aug 06 2025: (Start)
%C A092521 Numbers k such that both 3*k + 1 and 15*k + 1 are perfect squares. Also the sum of two consecutive terms is a square.
%C A092521 Take any recurrence (r) of the form (3,-1) with initial value 0 followed by an arbitrary positive integer i. Then the product of two consecutive terms of r divided by 3*i^2 gives the current sequence. (End)
%H A092521 Michael De Vlieger, <a href="/A092521/b092521.txt">Table of n, a(n) for n = 1..1197</a>
%H A092521 Francesca Arici and Jens Kaad, <a href="https://arxiv.org/abs/2012.11186">Gysin sequences and SU(2)-symmetries of C*-algebras</a>, arXiv:2012.11186 [math.OA], 2020.
%H A092521 C. Pita, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pita/pita12.html">On s-Fibonomials</a>, J. Int. Seq. 14 (2011) # 11.3.7.
%H A092521 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A092521 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A092521 G.f.: x/(1 - 8*x + 8*x^2 - x^3) = x/((1 - x)*(1 - 7*x + x^2)).
%F A092521 a(n) = 7*a(n-1) - a(n-2) + 1, n>=2, a(0):=0, a(1)=1.
%F A092521 a(n) = (S(n, 7)-S(n-1, 7) -1)/5, n>=1, with S(n, 7) = U(n, 7/2) = A004187(n+1).
%F A092521 a(n) = A058038(n)/3.
%F A092521 a(n) = (1/3)*Sum_{k=0..n} Fibonacci(4*k). - _Gary Detlefs_, Dec 07 2010
%F A092521 a(n) = a(-1-n) for all n in Z. - _Michael Somos_, Jan 23 2025
%F A092521 From _G. C. Greubel_, Jun 12 2025: (Start)
%F A092521 a(n) = A081079(n)/15.
%F A092521 E.g.f.: (1/15)*( exp(7*x/2)*( 3*cosh(p*x) + sqrt(5)*sinh(p*x) ) - 3*exp(x) ), where p = 3*sqrt(5)/2. (End)
%e A092521 G.f. = x + 8*x^2 + 56*x^3 + 385*x^4 + 2640*x^5 + 18096*x^6 + ... - _Michael Somos_, Jan 23 2025
%t A092521 a[1] = 1; a[2] = 8; a[3] = 56; a[n_] := a[n] = 8 a[n - 1] - 8 a[n - 2] + a[n - 3]; Table[ a[n], {n, 20}] (* _Robert G. Wilson v_, Apr 08 2004 *)
%t A092521 Table[(LucasL[4n+2]-3)/15, {n, 1, 20}] (* _Vladimir Reshetnikov_, Oct 28 2015 *)
%t A092521 LinearRecurrence[{8,-8,1},{1,8,56},30] (* _Harvey P. Dale_, Dec 27 2015 *)
%o A092521 (PARI) Vec(x/((1-x)*(1-7*x+x^2)) + O(x^100)) \\ _Altug Alkan_, Oct 29 2015
%o A092521 (Magma)
%o A092521 A092521:= func< n | (Lucas(4*n+2) -3)/15 >; // _G. C. Greubel_, Jun 12 2025
%o A092521 (SageMath)
%o A092521 def A092521(n): return (lucas_number2(4*n+2,1,-1) -3)//15 # _G. C. Greubel_, Jun 12 2025
%Y A092521 Cf. A212336 for more sequences with g.f. of the type 1/(1 - k*x + k*x^2 - x^3).
%Y A092521 Cf. A000032, A004187, A058038, A081079.
%K A092521 nonn,easy
%O A092521 1,2
%A A092521 K. S. Bhanu (bhanu_105(AT)yahoo.com) and M. N. Deshpande, Apr 06 2004
%E A092521 Edited and extended by _Robert G. Wilson v_, Apr 08 2004