A319127 -id:A319127 - cp's OEIS frontend

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

R. H. Hardin, Oct 06 2014

nonn

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

			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

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).

Row 1 of A248433.

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

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

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

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)

Name simplified by Andrew Howroyd, Aug 14 2020

A330082 a(n) = 5*A064038(n+1).

Original entry on oeis.org

0, 5, 15, 15, 25, 75, 105, 70, 90, 225, 275, 165, 195, 455, 525, 300, 340, 765, 855, 475, 525, 1155, 1265, 690, 750, 1625, 1755, 945, 1015, 2175, 2325, 1240, 1320, 2805, 2975, 1575, 1665, 3515, 3705, 1950, 2050, 4305, 4515, 2365, 2475, 5175, 5405, 2820, 2940

Offset: 0

Paul Curtz, Dec 01 2019

Main column of a pentagonal spiral for A026741:
                       (25)
                    49 (15) 31
                24  29 (15)  8  16
            47  14   7 ( 5)  3  17  33
        23  27  13   2 ( 0)  1   7   9  17
          45  13   6   3   1   4  19  35
            22  25  11   5   9  10  18
              43  12  23  11  21  37
                21  41  20  39  19
a(n) = 5 * A064038(n+1) from a pentagonal spiral.
Compare to A319127 =  6 * A002620 in the hexagonal spiral:
                22  23  23  22 (24)
              20  12  13  13 (12) 25
            21  10   5   4 ( 6) 14  25
          21  11   5   1 ( 0)  7  15  24
        20  11   4   1 ( 0)  2   7  15  26
          18  10   2   3   3   6  14  27
            19   8   9   9   8  16  27
              19  18  16  17  17  26
                30  28  29  29  28

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A026741, A028895, A064038. A033429, A062786, A087348, A147874, A158447, A168668 are in the spiral.

A330082[n_]:=5Numerator[n(n+1)/4];Array[A330082,100,0] (* Paolo Xausa, Dec 04 2023 *)

concat(0, Vec(5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3) + O(x^50))) \\ Colin Barker, Dec 08 2019

a(n) = A026741(A028895(n)).
From Colin Barker, Dec 08 2019: (Start)
G.f.: 5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3).
a(n) = 3*a(n-1) - 6*a(n-2) + 10*a(n-3) - 12*a(n-4) + 12*a(n-5) - 10*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) for n>8.
a(n) = (-5/16 + (5*i)/16)*(((-3-3*i) + (-i)^n + i^(1+n))*n*(1+n)) where i=sqrt(-1).
(End)

More terms from Colin Barker, Dec 22 2019

Name corrected by Paolo Xausa, Dec 04 2023

A345735 A prime-generating quasipolynomial: a(n) = 6*floor(n^2/4) + 17.

Original entry on oeis.org

17, 17, 23, 29, 41, 53, 71, 89, 113, 137, 167, 197, 233, 269, 311, 353, 401, 449, 503, 557, 617, 677, 743, 809, 881, 953, 1031, 1109, 1193, 1277, 1367, 1457, 1553, 1649, 1751, 1853, 1961, 2069, 2183, 2297, 2417, 2537, 2663, 2789, 2921, 3053, 3191, 3329, 3473, 3617

Offset: 0

Charles R Greathouse IV, Jun 30 2021

Fontebasso only claims that the terms are prime from 0 to 22, but in fact a(23)..a(30) are all prime as well. The first composite term is a(31) = 1457 = 31*47.

P. A. Fontebasso, Une'altra formula che dà una serie limitata di numeri primi, Supplemento al Periodico di matematica (1899), p. 130.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A002620, A319127.

LinearRecurrence[{2,0,-2,1},{17,17,23,29},50] (* Harvey P. Dale, Jul 17 2025 *)

PARI
```
a(n)=n^2\4*6+17
```

a(n) = A319127(n+1) + 17 = 6*A002620(n) + 17. - Omar E. Pol, Jul 12 2021
G.f.: (17*x^3-11*x^2-17*x+17)/((x+1)*(1-x)^3). - Alois P. Heinz, Jul 12 2021
E.g.f.: ((34 + 3*x + 3*x^2)*cosh(x) + (31 + 3*x + 3*x^2)*sinh(x))/2. - Stefano Spezia, Jul 13 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A330082 a(n) = 5*A064038(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A345735 A prime-generating quasipolynomial: a(n) = 6*floor(n^2/4) + 17.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula