A319384 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.

1, 5, 9, 21, 29, 49, 61, 89, 105, 141, 161, 205, 229, 281, 309, 369, 401, 469, 505, 581, 621, 705, 749, 841, 889, 989, 1041, 1149, 1205, 1321, 1381, 1505, 1569, 1701, 1769, 1909, 1981, 2129, 2205, 2361, 2441, 2605, 2689, 2861, 2949, 3129, 3221, 3409, 3505, 3701, 3801, 4005, 4109, 4321, 4429, 4649, 4761, 4989, 5105, 5341, 5461

Offset

0,2

Comments

The two bisections A136392(n+1)=1,9,29,61, ... and A201279(n)=5,21,49, ... are in the hexagonal spiral based on 2*n+1:

.

67--65--63--61

/ \

69 33--31--29 59

/ / \ \

71 35 11---9 27 57

/ / / \ \ \

73 37 13 1 7 25 55

/ / / / / /

39 15 3---5 23 53

\ \ / /

41 17--19--21 51

\ /

43--45--47--49

.

A201279(n) - A136892(n) = 20*n.

Links

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

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(2*n) = A136392(n+1), a(2*n+1) = A201279(n).

a(-n) = a(n).

a(2*n) + a(2*n+1) = 6*A001844(n).

a(n) = (6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4. - Wesley Ivan Hurt, Oct 04 2018

G.f.: (1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2). - Colin Barker, Jun 05 2019

a(n) = A104585(n) + A032766(n+1). - Alex W. Nowak, Jan 08 2021

Mathematica

Table[(6 n^2 + 6 n + 5 - (2 n + 1)*(-1)^n)/4, {n, 0, 80}] (* Wesley Ivan Hurt, Jan 07 2021 *)

Prog

(PARI) Vec((1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jun 05 2019
(Magma) [(6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jan 19 2021
(Python)
def A319384(n): return (n*(3*n+4)+3 if n&1 else n*(3*n+2)+2)>>1 # Chai Wah Wu, Mar 25 2025

Crossrefs

Cf. A001844, A032766, A104585.

In the spiral: A003154(n+1), A080859, A126587, A136392, A201279, A227776.

Partial sums of A382154.

Sequence in context: A273739 A169709 A269523 * A211427 A273410 A273486

Adjacent sequences: A319381 A319382 A319383 * A319385 A319386 A319387

Keyword

nonn,easy

Author

Paul Curtz, Sep 18 2018

Extensions

More terms from N. J. A. Sloane, Mar 23 2025

Status

approved