A248434 Number of length three 0..n arrays with the sum of two elements equal to twice the third.

2, 9, 16, 29, 42, 61, 80, 105, 130, 161, 192, 229, 266, 309, 352, 401, 450, 505, 560, 621, 682, 749, 816, 889, 962, 1041, 1120, 1205, 1290, 1381, 1472, 1569, 1666, 1769, 1872, 1981, 2090, 2205, 2320, 2441, 2562, 2689, 2816, 2949, 3082, 3221, 3360, 3505, 3650

Offset

1,1

Comments

Number of length three 0..n vectors that contain their arithmetic mean. - Hywel Normington, Aug 15 2020

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 210 terms from R. H. Hardin)

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

Empirical: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4).

Empirical for n mod 2 = 0: a(n) = (3/2)*n^2 + n + 1.

Empirical for n mod 2 = 1: a(n) = (3/2)*n^2 + n - (1/2).

From Hywel Normington, Aug 21 2020: (Start)

a(n) = a(n-1) + 1 + 6*floor(n/2)

a(n) = A319127(n+1) + n + 1 = 6*floor((n+1)/2)*floor(n/2) + n + 1.

(End)

From Colin Barker, Aug 28 2020: (Start)

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

a(n) = (1 + 3*(-1)^n + 4*n + 6*n^2) / 4 for n>0.

(End)

Example

Some solutions for n=6:
..2....3....6....1....3....4....3....1....6....2....4....0....4....5....4....3
..6....1....2....0....2....3....3....2....5....3....0....1....3....6....4....5
..4....5....4....2....1....5....3....3....4....1....2....2....2....4....4....4

Prog

(PARI) a(n) = {my(res = 2); if(n % 2 == 0, res+=(1 + 6*floor(n/2))); n = (n-1)>>1; res+=6*n^2 + 8*n; res} \\ David A. Corneth, Aug 26 2020
(PARI) first(n) = {my(res = vector(n), inc = 7); res[1] = 2; for(i = 2, n, res[i] = res[i-1] + inc; inc += 6 * (i%2 == 1)); res} \\ David A. Corneth, Aug 26 2020

Crossrefs

Row 1 of A248433.

Cf. A168328, A319127, A319127. First differences A168301.

Sequence in context: A338667 A101420 A275498 * A275395 A213389 A304907

Adjacent sequences: A248431 A248432 A248433 * A248435 A248436 A248437

Keyword

nonn

Author

R. H. Hardin, Oct 06 2014

Extensions

Name simplified by Andrew Howroyd, Aug 14 2020

Status

approved