OFFSET
1,2
COMMENTS
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.
Partial sums of Chebyshev sequence S(n,7) = U(n,7/2) = A004187(n+1). - Wolfdieter Lang, Aug 31 2004
From Klaus Purath, Aug 06 2025: (Start)
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.
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)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1197
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Claudio de Jesús Pita Ruiz Velasco, On s-Fibonomials, J. Int. Seq. 14 (2011), Article 11.3.7.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).
FORMULA
G.f.: x/(1 - 8*x + 8*x^2 - x^3) = x/((1 - x)*(1 - 7*x + x^2)).
a(n) = 7*a(n-1) - a(n-2) + 1, n>=2, a(0):=0, a(1)=1.
a(n) = (S(n, 7)-S(n-1, 7) -1)/5, n>=1, with S(n, 7) = U(n, 7/2) = A004187(n+1).
a(n) = A058038(n)/3.
a(n) = (1/3)*Sum_{k=0..n} Fibonacci(4*k). - Gary Detlefs, Dec 07 2010
a(n) = a(-1-n) for all n in Z. - Michael Somos, Jan 23 2025
From G. C. Greubel, Jun 12 2025: (Start)
a(n) = A081079(n)/15.
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)
Sum_{n>=1} 1/a(n) = 3*(3 - sqrt(5))/2. - Amiram Eldar, Dec 25 2025
EXAMPLE
G.f. = x + 8*x^2 + 56*x^3 + 385*x^4 + 2640*x^5 + 18096*x^6 + ... - Michael Somos, Jan 23 2025
MATHEMATICA
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 *)
Table[(LucasL[4n+2]-3)/15, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)
LinearRecurrence[{8, -8, 1}, {1, 8, 56}, 30] (* Harvey P. Dale, Dec 27 2015 *)
PROG
(PARI) Vec(x/((1-x)*(1-7*x+x^2)) + O(x^100)) \\ Altug Alkan, Oct 29 2015
(Magma)
A092521:= func< n | (Lucas(4*n+2) -3)/15 >; // G. C. Greubel, Jun 12 2025
(SageMath)
def A092521(n): return (lucas_number2(4*n+2, 1, -1) -3)//15 # G. C. Greubel, Jun 12 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
K. S. Bhanu (bhanu_105(AT)yahoo.com) and M. N. Deshpande, Apr 06 2004
EXTENSIONS
Edited and extended by Robert G. Wilson v, Apr 08 2004
STATUS
approved