A330707 a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4.

0, 1, 3, 7, 13, 20, 28, 38, 50, 63, 77, 93, 111, 130, 150, 172, 196, 221, 247, 275, 305, 336, 368, 402, 438, 475, 513, 553, 595, 638, 682, 728, 776, 825, 875, 927, 981, 1036, 1092, 1150, 1210, 1271, 1333, 1397, 1463

Offset

0,3

Comments

Essentially four odds followed by four evens.

Last digit is neither 4 nor 9.

Essentially twice or twin sequences in the hexagonal spiral from A002265.

21 21 21 22 22 22 22

21 14 14 14 14 15 15 23

20 13 8 8 8 9 9 15 23

20 13 8 4 4 4 4 9 15 23

20 13 7 3 1 1 1 5 9 16 23

20 13 7 3 1 0 0 2 5 10 16 24

19 12 7 3 0 0 2 5 10 16 24

19 12 7 3 2 2 5 10 16 24

19 12 6 6 6 6 10 17 24

19 12 11 11 11 11 17 25

18 18 18 18 17 17 25

.

There are 12 twin sequences. 6 of them (A001859, A006578, A077043, A231559, A024219, A281026) are in the OEIS. a(n) is the seventh.

0, 1, 3, 7, 13, 20, 28, 38, 50, ...

1, 2, 4, 6, 7, 8, 10, 12, 13, ...

1, 2, 2, 1, 1, 2, 2, 1, 1, ... period 4. See A014695.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Formula

a(n) = A231559(-n).

a(1+2*n) + a(2+2*n) = A033579(n+1).

a(40+n) - a(n) = 1210, 1270, 1330, 1390, 1450, ... . See 10*A016921(n).

From Colin Barker, Dec 27 2019: (Start)

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

a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.

(End)

E.g.f.: (cos(x) + sin(x) + (-1 + 4*x + 3*x^2)*exp(x))/4. - Stefano Spezia, Dec 27 2019

a(n) = ( 3*n^2 + n - 1 + sqrt(2)*sin((2*n+1)*Pi/4) )/4 = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4. - G. C. Greubel, Dec 30 2019

Maple

seq((3*n^2+n-1+sqrt(2)*sin((2*n+1)*Pi/4))/4, n = 0..60); # G. C. Greubel, Dec 30 2019

Mathematica

LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 3, 7, 13}, 60] (* Amiram Eldar, Dec 27 2019 *)

Prog

(PARI) concat(0, Vec(x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^60))) \\ Colin Barker, Dec 27 2019
(Magma) [(3*n^2+n-1+ (-1)^Floor(n/2))/4: n in [0..60]]; // G. C. Greubel, Dec 30 2019
(Sage) [(3*n^2+n-1+(-1)^floor(n/2))/4 for n in (0..60)] # G. C. Greubel, Dec 30 2019

Crossrefs

Cf. A001859, A002265, A006578, A014695, A016921, A024219, A033579, A077043, A231559, A281026.

Sequence in context: A014283 A341299 A294398 * A033551 A350296 A022777

Adjacent sequences: A330704 A330705 A330706 * A330708 A330709 A330710

Keyword

nonn,easy

Author

Paul Curtz, Dec 27 2019

Status

approved