A275543 A081585 and A069129 interleaved.

1, 1, 9, 17, 33, 49, 73, 97, 129, 161, 201, 241, 289, 337, 393, 449, 513, 577, 649, 721, 801, 881, 969, 1057, 1153, 1249, 1353, 1457, 1569, 1681, 1801, 1921, 2049, 2177, 2313, 2449, 2593, 2737, 2889, 3041, 3201, 3361, 3529, 3697, 3873, 4049, 4233, 4417, 4609

Offset: 0

Author

Daniel Poveda Parrilla, Aug 01 2016

Keywords

• nonn
• easy

Comments

a(A000129(n)) is a square.

(n^2)*a(n) = A275496(n) which is a triangular number.

(A000129(n)^2)*a(A000129(n)) = A275496(A000129(n)) = A001110(n) which is a square triangular number.

a(2n+1)/a(2n) is convergent to 1.

Examples

			a(1) = A275496(1) = 1.
a(5) = A275496(5)/25 = 1225/25 = 49.
a(7) = A275496(7)/49 = 4753/49 = 97.
a(12) = A275496(12)/144 = 41616/144 = 289.
		

Links

• Daniel Poveda Parrilla, Table of n, a(n) for n = 0..500000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A081585(n) = a(2n), A069129(n) = a(2n + 1).

Cf. A000129, A000217, A000290, A001110, A077221, A093954, A275496.

Programs

Mathematica

CoefficientList[Series[(1 - x + 7 x^2 + x^3)/((1 - x)^3 (1 + x)), {x, 0, 48}], x] (* or as defined *)
Riffle[LinearRecurrence[{3, -3, 1}, {1, 9, 33}, #], FoldList[#1 + #2 &, 1, 16 Range@ #]] &@ 25 (* Michael De Vlieger, Aug 01 2016, after Vincenzo Librandi at A081585 and Robert G. Wilson v at A069129 *)

PARI

a(n)=(-1)^n + 2*n^2 \\ Charles R Greathouse IV, Aug 03 2016

PARI

Vec((1-x+7*x^2+x^3)/((1-x)^3*(1+x)) + O(x^100)) \\ Colin Barker, Aug 21 2016

Formula

a(0) = 1; a(n) = A275496(n)/(n^2) for n > 0.

From Colin Barker, Aug 01 2016: (Start)

a(n) = (2*n^2 + (-1)^n).

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.

G.f.: (1 -x +7*x^2 +x^3) / ((1 - x)^3*(1 + x)).

(End)

From Daniel Poveda Parrilla, Aug 18 2016: (Start)

a(2n) = A077221(2n) + 1.

a(2n + 1) = A077221(2n + 1). (End)

Sum_{n>=0} 1/a(n) = (1 + (tan(c) + coth(c))*c)/2, where c = Pi/(2*sqrt(2)) is A093954. - Amiram Eldar, Aug 21 2022