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

0,3

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.

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).

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A328016 A335796 A260477 * A111733 A127193 A262453

Adjacent sequences: A275540 A275541 A275542 * A275544 A275545 A275546

Keyword

nonn,easy

Author

Daniel Poveda Parrilla, Aug 01 2016

Status

approved