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A275543 A081585 and A069129 interleaved. 2
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 (list; graph; refs; listen; history; text; internal format)
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)

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, A275496.

Sequence in context: A164887 A328016 A260477 * A111733 A127193 A262453

Adjacent sequences:  A275540 A275541 A275542 * A275544 A275545 A275546

KEYWORD

nonn,easy

AUTHOR

Daniel Poveda Parrilla, Aug 01 2016

STATUS

approved

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Last modified February 23 00:28 EST 2020. Contains 332157 sequences. (Running on oeis4.)