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

Author

Paul Curtz, Dec 27 2019

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

[(3*n^2+n-1+ (-1)^Floor(n/2))/4: n in [0..60]]; // 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 *)

PARI

concat(0, Vec(x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^60))) \\ Colin Barker, Dec 27 2019

Sage

[(3*n^2+n-1+(-1)^floor(n/2))/4 for n in (0..60)] # G. C. Greubel, Dec 30 2019

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