A008594 -id:A008594

A370238

a(n) = n*(3*n + 23)/2.

+10
2

0, 13, 29, 48, 70, 95, 123, 154, 188, 225, 265, 308, 354, 403, 455, 510, 568, 629, 693, 760, 830, 903, 979, 1058, 1140, 1225, 1313, 1404, 1498, 1595, 1695, 1798, 1904, 2013, 2125, 2240, 2358, 2479, 2603, 2730, 2860, 2993, 3129, 3268, 3410, 3555, 3703, 3854, 4008

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FORMULA

a(n) = A000326(n) + A008594(n).

CROSSREFS

Cf. A000326, A000566, A008594, A005449, A045943, A059845, A115067, A140090, A140091, A140672, A140673, A140674, A140675, A151542, A277976.

A063669

Hypotenuses of reciprocal Pythagorean triangles: number of solutions to 1/(12n)^2 = 1/b^2 + 1/c^2 [with b >= c > 0]; also number of values of A020885 (with repetitions) which divide n.

+10
1

1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5

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COMMENTS

Primitive reciprocal Pythagorean triangles 1/a^2 = 1/b^2 + 1/c^2 have a=fg, b=ef, c=eg where e^2 = f^2 + g^2; i.e., e,f,g represent the sides of primitive Pythagorean triangles. But the product of the two legs of primitive Pythagorean triangles are multiples of 12 and so the reciprocal of hypotenuses of reciprocal Pythagorean triangles are always multiples of 12 (A008594).

A309118

Number of tiles added at iteration n when successively, layer by layer, building a symmetric patch of a rhombille tiling around a central star of six rhombs.

+10
1

6, 6, 12, 18, 24, 24, 36, 30, 48, 36, 60, 42, 72, 48, 84, 54, 96, 60, 108, 66, 120, 72, 132, 78, 144, 84, 156, 90, 168, 96, 180, 102, 192, 108, 204, 114, 216, 120, 228, 126, 240, 132, 252, 138, 264, 144, 276, 150, 288, 156, 300, 162, 312, 168, 324, 174, 336

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(2*n+1) = A008594(n).

CROSSREFS

Cf. A008588, A008594.

A332019

The number of cells added in the n-th generation of the following procedure: start by coloring any triangle on the snub square tiling, then repeatedly color every cell that shares a vertex with a colored cell.

+10
1

1, 9, 21, 35, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 528, 540, 552, 564, 576, 588, 600, 612, 624, 636

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A008594.

A333814

Multiples of 12 whose sum of digits is 12.

+10
1

48, 84, 156, 192, 228, 264, 336, 372, 408, 444, 480, 516, 552, 624, 660, 732, 804, 840, 912, 1056, 1092, 1128, 1164, 1236, 1272, 1308, 1344, 1380, 1416, 1452, 1524, 1560, 1632, 1704, 1740, 1812, 1920, 2028, 2064, 2136, 2172, 2208, 2244, 2280, 2316, 2352, 2424

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Intersection of A235151 (sum of digits = 12) and A008594 (multiples of 12).

Cf. A008594 (multiples of 12), A235151 (sum of digits = 12).

A062902

Number and its reversal are both multiples of 12.

+10
0

0, 48, 84, 216, 252, 276, 408, 420, 444, 468, 480, 612, 636, 672, 696, 804, 828, 840, 864, 888, 2100, 2112, 2124, 2136, 2148, 2160, 2172, 2184, 2196, 2304, 2316, 2328, 2340, 2352, 2364, 2376, 2388, 2508, 2520, 2532, 2544, 2556, 2568, 2580, 2592, 2700, 2712

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A004086, A008594.

A192491

Molecular topological indices of the complete tripartite graphs K_{n,n,n}.

+10
0

24, 240, 864, 2112, 4200, 7344, 11760, 17664, 25272, 34800, 46464, 60480, 77064, 96432, 118800, 144384, 173400, 206064, 242592, 283200, 328104, 377520, 431664, 490752, 555000, 624624, 699840, 780864, 867912, 961200

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) = 6*A046092(n) + (A008594(n+1) * A140676(n-1)). (End)

CROSSREFS

Cf. A000326, A000330, A000292, A017233, A008594, A140676, A046092.

A242570

a(n) = 252 * n.

+10
0

0, 252, 504, 756, 1008, 1260, 1512, 1764, 2016, 2268, 2520, 2772, 3024, 3276, 3528, 3780, 4032, 4284, 4536, 4788, 5040, 5292, 5544, 5796, 6048, 6300, 6552, 6804, 7056, 7308, 7560, 7812, 8064, 8316, 8568, 8820, 9072, 9324, 9576, 9828, 10080, 10332, 10584, 10836, 11088, 11340

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) = 7*A044102(n) = 9*A135628(n) = 12*A008603(n) = 14*A008600(n) = 18*A008596(n) = 21*A008594(n) = 28*A008591(n) = 36*A008589(n) = 252*A001477(n). (End)

CROSSREFS

Cf. A001477, A008600, A008603, A008589, A008591, A008594, A008596.

A285440

Consider the sums of the numbers < n that share the same greatest common divisor with n. Sequence lists numbers that have only one of those sums equal to n.

+10
0

3, 4, 8, 9, 15, 16, 20, 21, 27, 28, 32, 33, 39, 40, 44, 45, 51, 52, 56, 57, 63, 64, 68, 69, 75, 76, 80, 81, 87, 88, 92, 93, 99, 100, 104, 105, 111, 112, 116, 117, 123, 124, 128, 129, 135, 136, 140, 141, 147, 148, 152, 153, 159, 160, 164, 165, 171, 172, 176, 177

(list; graph; refs; listen; history; text; internal format)

COMMENTS

Numbers with no sum equal to n are listed in A108118, with two sums equal to n are listed in A017593 and with three sums equal to n in A008594.

CROSSREFS

Cf. A008594, A017593, A108118.

A338259

Triangle read by rows: T(n,k) is the coefficient of (1+x)^k in the ZZ polynomial of the hexagonal graphene flake O(3,4,n).

+10
0

1, 12, 18, 41, 24, 120, 200, 120, 24, 11, 36, 306, 996, 1446, 984, 303, 42, 21, 48, 576, 2800, 6525, 7848, 4957, 1644, 274, 22, 11, 60, 930, 6020, 19365, 33600, 32487, 17694, 5336, 858, 71, 21, 72, 1368, 11064, 45435, 103200, 134806, 102912, 45567, 11358, 1510, 86, 1

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Column k=1 is A008594.