A248433 -id:A248433 - 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

A248435 Number of length 2+2 0..n arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

2, 9, 20, 45, 70, 105, 140, 189, 242, 301, 360, 437, 514, 597, 684, 785, 886, 997, 1108, 1233, 1362, 1497, 1632, 1785, 1938, 2097, 2260, 2437, 2614, 2801, 2988, 3189, 3394, 3605, 3816, 4045, 4274, 4509, 4748, 5001, 5254, 5517, 5780, 6057, 6338, 6625, 6912

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 2 of A248433.

Empirical: a(n) = a(n-1) + a(n-3) - a(n-5) - a(n-7) + a(n-8).
Empirical for n mod 12 = 0: a(n) = (19/6)*n^2 - (5/3)*n + 1
Empirical for n mod 12 = 1: a(n) = (19/6)*n^2 - (5/3)*n + (1/2)
Empirical for n mod 12 = 2: a(n) = (19/6)*n^2 - (5/3)*n - (1/3)
Empirical for n mod 12 = 3: a(n) = (19/6)*n^2 - (5/3)*n - (7/2)
Empirical for n mod 12 = 4: a(n) = (19/6)*n^2 - (5/3)*n + 1
Empirical for n mod 12 = 5: a(n) = (19/6)*n^2 - (5/3)*n - (5/6)
Empirical for n mod 12 = 6: a(n) = (19/6)*n^2 - (5/3)*n + 1
Empirical for n mod 12 = 7: a(n) = (19/6)*n^2 - (5/3)*n - (7/2)
Empirical for n mod 12 = 8: a(n) = (19/6)*n^2 - (5/3)*n - (1/3)
Empirical for n mod 12 = 9: a(n) = (19/6)*n^2 - (5/3)*n + (1/2)
Empirical for n mod 12 = 10: a(n) = (19/6)*n^2 - (5/3)*n + 1
Empirical for n mod 12 = 11: a(n) = (19/6)*n^2 - (5/3)*n - (29/6).
Empirical g.f.: x*(2 + 7*x + 11*x^2 + 23*x^3 + 16*x^4 + 17*x^5 - x^6 + x^7) / ((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x + x^2)). - Colin Barker, Nov 08 2018

A248428 Number of length n+2 0..3 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

16, 20, 24, 28, 36, 44, 52, 68, 84, 100, 132, 164, 196, 260, 324, 388, 516, 644, 772, 1028, 1284, 1540, 2052, 2564, 3076, 4100, 5124, 6148, 8196, 10244, 12292, 16388, 20484, 24580, 32772, 40964, 49156, 65540, 81924, 98308, 131076, 163844, 196612, 262148

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 3 of A248433.

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

Empirical g.f.: 4*x*(4 + x + x^2 - 7*x^3) / ((1 - x)*(1 - 2*x^3)). - Colin Barker, Nov 08 2018

A248429 Number of length n+2 0..4 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

29, 45, 69, 101, 165, 261, 389, 645, 1029, 1541, 2565, 4101, 6149, 10245, 16389, 24581, 40965, 65541, 98309, 163845, 262149, 393221, 655365, 1048581, 1572869, 2621445, 4194309, 6291461, 10485765, 16777221, 25165829, 41943045, 67108869, 100663301

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 4 of A248433.

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

Empirical g.f.: x*(29 + 16*x + 24*x^2 - 84*x^3) / ((1 - x)*(1 - 4*x^3)). - Colin Barker, Nov 08 2018

A248430 Number of length n+2 0..5 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

42, 70, 118, 198, 342, 590, 1014, 1766, 3062, 5286, 9222, 16006, 27654, 48262, 83782, 144774, 252678, 438662, 758022, 1323014, 2296838, 3969030, 6927366, 12026374, 20782086, 36272134, 62970886, 108816390, 189923334, 329719814, 569769990

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 5 of A248433.

Empirical: a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - 4*a(n-6) + 4*a(n-7).

Empirical g.f.: 2*x*(21 + 14*x + 24*x^2 - 86*x^3 - 12*x^4 - 20*x^5 + 56*x^6) / ((1 - x)*(1 - 6*x^3 + 4*x^6)). - Colin Barker, Nov 08 2018

A248431 Number of length n+2 0..6 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

61, 105, 185, 327, 601, 1105, 2021, 3761, 6969, 12815, 23897, 44337, 81597, 152209, 282457, 519895, 969849, 1799825, 3312853, 6180081, 11468921, 21110367, 39381145, 73083121, 134521133, 250947665, 465706137, 857206439, 1599109049, 2967610449

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 6 of A248433

Empirical: a(n) = 8*a(n-3) - 11*a(n-6) + 4*a(n-9).

Empirical g.f.: x*(61 + 105*x + 185*x^2 - 161*x^3 - 239*x^4 - 375*x^5 + 76*x^6 + 108*x^7 + 164*x^8) / ((1 - x)*(1 + x + x^2)*(1 - 7*x^3 + 4*x^6)). - Colin Barker, Nov 08 2018

A248432 Number of length n+2 0..7 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

80, 140, 252, 462, 884, 1684, 3200, 6216, 11944, 22810, 44396, 85402, 163204, 317716, 611248, 1168198, 2274196, 4375320, 8362052, 16278784, 31318664, 59855842, 116523764, 224179214, 428448488, 834077068, 1604674164, 3066832822

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

			Some solutions for n=6:
..6....6....0....4....2....3....4....6....4....3....2....4....5....2....3....4
..4....2....3....2....3....1....6....4....2....5....3....2....3....4....5....5
..5....4....6....0....4....2....5....5....3....4....1....6....7....3....7....3
..3....0....0....4....5....0....4....3....1....6....5....4....5....5....6....7
..7....2....3....2....3....1....3....1....2....2....3....5....3....4....5....5
..5....4....6....3....7....2....2....2....0....4....7....3....4....3....7....6
..3....6....0....1....5....0....4....0....1....3....5....1....5....5....3....4
..1....5....3....5....6....1....3....1....2....2....3....5....6....4....5....2

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 7 of A248433.

Empirical: a(n) = a(n-1) + 10*a(n-3) - 10*a(n-4) - 22*a(n-6) + 22*a(n-7) + 12*a(n-9) - 12*a(n-10) - a(n-12) + a(n-13).

Empirical g.f.: 2*x*(40 + 30*x + 56*x^2 - 295*x^3 - 89*x^4 - 160*x^5 + 588*x^6 + 58*x^7 + 96*x^8 - 317*x^9 - 5*x^10 - 9*x^11 + 27*x^12) / ((1 - x)*(1 - 10*x^3 + 22*x^6 - 12*x^9 + x^12)). - Colin Barker, Nov 08 2018

A248436 Number of length 3+2 0..n arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

2, 9, 24, 69, 118, 185, 252, 357, 470, 593, 716, 881, 1046, 1217, 1400, 1621, 1842, 2081, 2320, 2593, 2874, 3161, 3448, 3785, 4126, 4473, 4828, 5213, 5598, 6005, 6412, 6857, 7310, 7769, 8232, 8737, 9242, 9753, 10272, 10833, 11394, 11973, 12552, 13161, 13782

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 3 of A248433.

Empirical: a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + 2*a(n-5) - a(n-6) + a(n-7) - a(n-11) + a(n-12) - 2*a(n-13) + a(n-14) - a(n-15) + a(n-16) - a(n-17) + a(n-18).
Also a quadratic polynomial plus a constant quasipolynomial with period 840, the first 12 being:
Empirical for n mod 840 = 0: a(n) = (106/15)*n^2 - (178/15)*n + 1
Empirical for n mod 840 = 1: a(n) = (106/15)*n^2 - (178/15)*n + (34/5)
Empirical for n mod 840 = 2: a(n) = (106/15)*n^2 - (178/15)*n + (67/15)
Empirical for n mod 840 = 3: a(n) = (106/15)*n^2 - (178/15)*n - 4
Empirical for n mod 840 = 4: a(n) = (106/15)*n^2 - (178/15)*n + (17/5)
Empirical for n mod 840 = 5: a(n) = (106/15)*n^2 - (178/15)*n + (2/3)
Empirical for n mod 840 = 6: a(n) = (106/15)*n^2 - (178/15)*n + (9/5)
Empirical for n mod 840 = 7: a(n) = (106/15)*n^2 - (178/15)*n - (56/5)
Empirical for n mod 840 = 8: a(n) = (106/15)*n^2 - (178/15)*n - (1/3)
Empirical for n mod 840 = 9: a(n) = (106/15)*n^2 - (178/15)*n + (22/5)
Empirical for n mod 840 = 10: a(n) = (106/15)*n^2 - (178/15)*n + 5
Empirical for n mod 840 = 11: a(n) = (106/15)*n^2 - (178/15)*n - (128/15).
Empirical g.f.: x*(2 + 7*x + 17*x^2 + 52*x^3 + 66*x^4 + 117*x^5 + 124*x^6 + 200*x^7 + 175*x^8 + 222*x^9 + 167*x^10 + 210*x^11 + 118*x^12 + 113*x^13 + 52*x^14 + 54*x^15 - x^16 + x^17) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Nov 08 2018

A248437 Number of length 4+2 0..n arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

2, 9, 28, 101, 198, 327, 462, 691, 932, 1203, 1474, 1829, 2184, 2551, 2954, 3459, 3964, 4501, 5038, 5653, 6286, 6925, 7564, 8327, 9114, 9907, 10712, 11577, 12442, 13363, 14284, 15307, 16342, 17383, 18454, 19609, 20764, 21925, 23098, 24389, 25680, 27009

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

Row 4 of A248433

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = -a(n-4) +a(n-5) +a(n-7) -a(n-8) +a(n-9) +a(n-11) -a(n-12) +a(n-13) +a(n-15) -a(n-21) -a(n-23) +a(n-24) -a(n-25) -a(n-27) +a(n-28) -a(n-29) -a(n-31) +a(n-32) +a(n-36)

A248438 Number of length 5+2 0..n arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

Original entry on oeis.org

2, 9, 36, 165, 342, 601, 884, 1381, 1922, 2533, 3144, 3957, 4770, 5613, 6524, 7697, 8870, 10149, 11428, 12873, 14362, 15857, 17352, 19165, 21026, 22893, 24804, 26841, 28878, 31061, 33244, 35697, 38170, 40649, 43200, 45977, 48754, 51537, 54340, 57421

Offset: 1

R. H. Hardin, Oct 06 2014

nonn

Row 5 of A248433

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

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = a(n-1) -a(n-2) -a(n-6) +2*a(n-7) -3*a(n-8) +3*a(n-9) -a(n-10) +2*a(n-13) -3*a(n-14) +4*a(n-15) -4*a(n-16) +3*a(n-17) -a(n-18) -a(n-20) +3*a(n-21) -4*a(n-22) +4*a(n-23) -3*a(n-24) +2*a(n-25) -a(n-28) +3*a(n-29) -3*a(n-30) +2*a(n-31) -a(n-33) +a(n-34) -a(n-36) +a(n-37) -2*a(n-39) +3*a(n-40) -3*a(n-41) +a(n-42) -2*a(n-45) +3*a(n-46) -4*a(n-47) +4*a(n-48) -3*a(n-49) +a(n-50) +a(n-52) -3*a(n-53) +4*a(n-54) -4*a(n-55) +3*a(n-56) -2*a(n-57) +a(n-60) -3*a(n-61) +3*a(n-62) -2*a(n-63) +a(n-64) +a(n-68) -a(n-69) +a(n-70)

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

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A248435 Number of length 2+2 0..n arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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A248428 Number of length n+2 0..3 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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A248429 Number of length n+2 0..4 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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A248430 Number of length n+2 0..5 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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A248431 Number of length n+2 0..6 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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A248432 Number of length n+2 0..7 arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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A248436 Number of length 3+2 0..n arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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A248437 Number of length 4+2 0..n arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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A248438 Number of length 5+2 0..n arrays with every three consecutive terms having the sum of some two elements equal to twice the third.

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