A001972 -id:A001972

A002260

Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.

+10
465

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

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CROSSREFS

Triangle sums (see the comments): A000217 (Row1, Kn11); A004526 (Row2); A000096 (Kn12); A055998 (Kn13); A055999 (Kn14); A056000 (Kn15); A056115 (Kn16); A056119 (Kn17); A056121 (Kn18); A056126 (Kn19); A051942 (Kn110); A101859 (Kn111); A132754 (Kn112); A132755 (Kn113); A132756 (Kn114); A132757 (Kn115); A132758 (Kn116); A002620 (Kn21); A000290 (Kn3); A001840 (Ca2); A000326 (Ca3); A001972 (Gi2); A000384 (Gi3).

A212959

Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

+10
77

1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267

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COMMENTS

A001972 ... w = 4*x + y ...................... G

A130519

a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)

+10
25

0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450

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FORMULA

a(n) = A001972(n-4), n>3. - Franklin T. Adams-Watters, Jul 10 2009

A375981

Number of subsets of {1,2,...,n} such that no two elements differ by 1, 4, or 5.

+10
13

1, 2, 3, 5, 8, 11, 14, 19, 25, 34, 49, 70, 99, 141, 196, 270, 375, 520, 723, 1014, 1420, 1985, 2777, 3874, 5396, 7526, 10496, 14642, 20449, 28555, 39860, 55647, 77660, 108356, 151214, 211028, 294507, 411071, 573763, 800796, 1117679, 1559895, 2177002

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COMMENTS

Other sequences related to restricted combinations along with the sets of disallowed differences between subset elements: A000045 {1}, A011973 {1}, A006498 {2}, A006500 {3}, A031923 {4}, A000930 {1,2}, A102547 {1,2}, A130137 {1,3}, A263710 {1,4}, A374737 {1,5}, A079972 {2,3}, A224809 {2,4}, A351873 {3,4}, A224810 {3,6}, A224815 {4,8}, A003269 {1,2,3}, A180184 {1,2,3}, A317669 {1,2,4}, A351874 {1,3,4}, A177485 {1,3,5}, A121832 {2,3,4}, A375982 {2,3,5}, A375983 {2,4,5}, A224808 {2,4,6}, A224814 {3,6,9}, A003520 {1,2,3,4}, A375185 {1,2,3,5}, A375186 {1,2,4,5}, A259278 {2,3,4,5}, A224811 {2,4,6,8}, A005708 {1,2,3,4,5}, A276106 {2,3,4,5,6}, A224812 {2,4,6,8,10}, A005709 {1,2,3,4,5,6}, A322405 {2,3,4,5,6,7}, A224813 {2,4,6,8,10,12}, A005710 {1,2,3,4,5,6,7}, A368244 {2,3,4,5,6,7,8}, A000027 {1,2,..}, A269445 {1,2,..}\{12,25,..}, A008730 {1,2,..}\{11,23,..}, A008729 {1,2,..}\{10,21,..}, A008728 {1,2,..}\{9,19,..}, A008727 {1,2,..}\{8,17,..}, A008726 {1,2,..}\{7,15,..}, A008725 {1,2,..}\{6,13,..}, A038718 {1,..,5,7,..}, A008724 {1,2,..}\{5,11,..}, A008732 {1,2,..}\{4,9,..}, A179999 {1,2,3,5,7,..}, A001972 {1,2,..}\{3,7,..}, A001840 {1,2,..}\{2,5,..}, A052955 {1,3,..}, A004277 {2,3,..}, A186384 {1,2,..}\{1,6,..}, A186347 {1,2,..}\{1,5,..}, A339573 {1,2,..}\{1,4,..}, A002620 {2,4,..}, A019442 {3,4,..}, A006501 {3,6,..}, A008233 {4,8,..}, A008382 {5,10,..}, A008881 {6,12,..}, A009641 {7,14,..}, A009694 {8,16,..}, A009714 {9,18,..}, A354600 {10,20,..}.

A008728

Molien series for 3-dimensional group [2,n ] = *22n.

+10
9

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 217, 224, 231, 238

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CROSSREFS

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008732.

A122046

Partial sums of floor(n^2/8).

+10
8

0, 0, 0, 1, 3, 6, 10, 16, 24, 34, 46, 61, 79, 100, 124, 152, 184, 220, 260, 305, 355, 410, 470, 536, 608, 686, 770, 861, 959, 1064, 1176, 1296, 1424, 1560, 1704, 1857, 2019, 2190, 2370, 2560, 2760, 2970, 3190, 3421, 3663, 3916, 4180, 4456, 4744, 5044, 5356, 5681, 6019, 6370

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CROSSREFS

Partial sums of A001972.

A008726

Molien series 1/((1-x)^2*(1-x^8)) for 3-dimensional group [2,n] = *22n.

+10
6

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168, 175, 182, 189, 196, 203, 210, 217, 224, 232, 240, 248, 256, 264, 272, 280

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CROSSREFS

Cf. A001840, A001972, A008724, A008725, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A083593

Expansion of 1/((1-2*x)*(1-x^4)).

+10
6

1, 2, 4, 8, 17, 34, 68, 136, 273, 546, 1092, 2184, 4369, 8738, 17476, 34952, 69905, 139810, 279620, 559240, 1118481, 2236962, 4473924, 8947848, 17895697, 35791394, 71582788, 143165576, 286331153, 572662306, 1145324612, 2290649224

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COMMENTS

a(n) is the number of partitions of n into parts 1 and 4 where there are two colors of part 1 and the order of the colors of parts 1 matters. If the order of colors doesn't matter we get A001972. - Joerg Arndt, Jan 18 2024

A008727

Molien series for 3-dimensional group [2,n] = *22n.

+10
5

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252

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

CROSSREFS

Cf. A001840, A001972, A008724, A008725, A008726, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A008729

Molien series for 3-dimensional group [2, n] = *22n.

+10
5

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219

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CROSSREFS

Cf. A001840, A001972, A008724-A008728, A008732, A218530.