A330987 Alternatively add and half-multiply pairs of the nonnegative integers.

1, 3, 9, 21, 17, 55, 25, 105, 33, 171, 41, 253, 49, 351, 57, 465, 65, 595, 73, 741, 81, 903, 89, 1081, 97, 1275, 105, 1485, 113, 1711, 121, 1953, 129, 2211, 137, 2485, 145, 2775, 153, 3081, 161, 3403, 169, 3741, 177, 4095, 185, 4465, 193, 4851, 201, 5253, 209

Offset

1,2

Comments

In groups of two, add and half-multiply the integers: 0+1, (2*3)/2, 4+5, (6*7)/2, ....

From Bernard Schott, Jan 06 2020: (Start)

The bisection of this sequence gives:

For n odd = 2*k+1, k >= 0: a(2*k+1) = 8*k+1 = A017077(k),

For n even = 2*k, k >= 1: a(2*k) = T(4*k-2) = A000217(4*k-2) = (2*k-1)*(4*k-1) = A033567(k) where T(j) is the j-th triangular number. (End)

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

From Colin Barker, Jan 05 2020: (Start)

G.f.: x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3).

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.

a(n) = -1 + 2*(-1)^n - (1/2)*(-1+7*(-1)^n)*n + (1+(-1)^n)*n^2.

(End)

E.g.f.: (1 + 4*x + 2*x^2)*cosh(x) - (3 + x)*sinh(x) - 1. - Stefano Spezia, Jan 05 2020 after Colin Barker

Mathematica

a[n_]:=If[OddQ[n], 4n-3, (n-1)(2n-1)]; Array[a, 53] (* Stefano Spezia, Jan 05 2020 *)

Prog

(PARI) Vec(x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3) + O(x^50)) \\ Colin Barker, Jan 06 2020

Crossrefs

Cf. A330983.

Interspersion of A017077 and A033567 (excluding first term). - Michel Marcus, Jan 06 2020

Sequence in context: A377947 A006077 A291898 * A365919 A109612 A032668

Adjacent sequences: A330984 A330985 A330986 * A330988 A330989 A330990

Keyword

nonn,easy

Author

George E. Antoniou, Jan 05 2020

Status

approved