OFFSET
1,3
COMMENTS
Can also be seen as a table with two columns, read by rows: T[n,1] = a(2n-1) = A001652(n), T[n,2] = a(2n) = A001653(n).
The conjectured recurrence formula and g.f. are proved by the formulas for A001652. - M. F. Hasler, Aug 11 2015
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
V. Pletser, Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials, arXiv preprint arXiv:1409.7972 [math.NT], 2014. See Table 3 p. 8.
Index entries for linear recurrences with constant coefficients, signature (0,7,0,-7,0,1).
FORMULA
From Colin Barker, Aug 09 2015: (Start)
a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6) for n>6.
G.f.: -x^2*(x^4-x^3-2*x^2+3*x+1) / ((x-1)*(x+1)*(x^2-2*x-1)*(x^2+2*x-1)).
(End)
EXAMPLE
a(5) = 20 and a(6) = 29, because 20^2 + 21^2 = 29^2.
MATHEMATICA
LinearRecurrence[{0, 7, 0, -7, 0, 1}, {0, 1, 3, 5, 20, 29}, 50] (* Paolo Xausa, Jan 31 2024 *)
PROG
(PARI) concat(0, Vec(-x^2*(x^4-x^3-2*x^2+3*x+1) / ((x-1)*(x+1)*(x^2-2*x-1)*(x^2+2*x-1)) + O(x^50))) \\ Colin Barker, Aug 12 2015
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Marco Ripà, Aug 08 2015
EXTENSIONS
Edited by M. F. Hasler, Aug 11 2015
STATUS
approved