A003945 -id:A003945 - cp's OEIS frontend

A275565 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,0) (-1,2) or (0,-2) and new values introduced in order 0..2.

1, 2, 2, 3, 14, 3, 6, 36, 54, 6, 12, 96, 126, 216, 12, 24, 288, 294, 504, 864, 24, 48, 864, 672, 1176, 1872, 3456, 48, 96, 2592, 1536, 3192, 4056, 7200, 13824, 96, 192, 7776, 3552, 8664, 13104, 15000, 27360, 55296, 192, 384, 23328, 8214, 23712, 42336, 57600

Offset: 1

R. H. Hardin, Aug 01 2016

Table starts
...1......2.......3.......6.......12........24.........48..........96
...2.....14......36......96......288.......864.......2592........7776
...3.....54.....126.....294......672......1536.......3552........8214
...6....216.....504....1176.....3192......8664......23712.......64896
..12....864....1872....4056....13104.....42336.....138600......453750
..24...3456....7200...15000....57600....221184.....867456.....3402054
..48..13824...27360...54150...248520...1140576....5360184....25190406
..96..55296..104256..196566..1075140...5880600...33275880...188294424
.192.221184..397440..714150..4663710..30456054..206892990..1405458150
.384.884736.1513728.2589894.20186982.157347846.1286374716.10516571736

			Some solutions for n=4 k=4
..0..1..2..2. .0..1..1..2. .0..1..2..2. .0..1..2..0. .0..1..2..0
..1..0..0..1. .0..1..2..0. .1..0..0..2. .0..1..1..2. .1..1..2..2
..2..2..1..1. .1..2..2..1. .1..0..0..1. .2..0..1..1. .1..0..0..2
..2..2..1..0. .1..0..0..1. .2..2..1..1. .2..2..0..1. .2..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..286

Column 1 is A003945(n-2).
Column 2 is A208428.
Row 1 is A003945(n-2).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>3
k=2: a(n) = 4*a(n-1) for n>3
k=3: a(n) = 2*a(n-1) +8*a(n-2) -16*a(n-4) for n>5
k=4: [order 10] for n>11
k=5: [order 32] for n>34
k=6: [order 35] for n>37
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>3
n=2: a(n) = 3*a(n-1) for n>4
n=3: a(n) = 3*a(n-1) -5*a(n-3) +a(n-4) +7*a(n-5) -5*a(n-6) +2*a(n-8) -a(n-9) for n>10
n=4: [order 25] for n>28
n=5: [order 63] for n>66

A276248 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-1,1) or (0,-2) and new values introduced in order 0..2.

Original entry on oeis.org

1, 2, 2, 3, 9, 5, 6, 24, 36, 14, 12, 72, 85, 144, 41, 24, 216, 279, 347, 576, 122, 48, 648, 900, 1447, 1404, 2304, 365, 96, 1944, 2837, 6372, 7316, 5671, 9216, 1094, 192, 5832, 9148, 26325, 43662, 36744, 23000, 36864, 3281, 384, 17496, 29570, 115682, 234431

Offset: 1

R. H. Hardin, Aug 25 2016

Table starts
....1......2.......3........6........12..........24...........48.............96
....2......9......24.......72.......216.........648.........1944...........5832
....5.....36......85......279.......900........2837.........9148..........29570
...14....144.....347.....1447......6372.......26325.......115682.........509750
...41....576....1404.....7316.....43662......234431......1423062........8496628
..122...2304....5671....36744....291113.....2069454.....17450554......141165944
..365...9216...23000...188696...2003694....18671229....219977330.....2410124377
.1094..36864...93204...966555..13727745...167951009...2780927371....41132001645
.3281.147456..377421..4951790..93489265..1509288801..35144231606...700435484735
.9842.589824.1529844.25428687.640009243.13609728840.446083313365.11973407175492

			Some solutions for n=4 k=4
..0..0..1..1. .0..0..1..2. .0..0..1..1. .0..1..1..2. .0..0..1..2
..1..2..2..1. .2..0..0..1. .2..2..0..1. .2..2..0..0. .2..2..0..1
..0..1..2..2. .1..1..2..2. .0..1..2..2. .1..1..2..2. .1..1..2..2
..2..0..0..1. .0..0..1..2. .0..0..1..1. .2..0..1..1. .2..0..0..2

R. H. Hardin, Table of n, a(n) for n = 1..241

Column 1 is A007051(n-1).
Column 2 is A002063(n-2).
Row 1 is A003945(n-2).

Empirical for column k:
k=1: a(n) = 4*a(n-1) -3*a(n-2)
k=2: a(n) = 4*a(n-1) for n>2
k=3: a(n) = 5*a(n-1) -4*a(n-2) +17*a(n-3) -83*a(n-4) +54*a(n-5) +56*a(n-6) for n>9
k=4: [order 36] for n>37
k=5: [order 41] for n>45
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>3
n=2: a(n) = 3*a(n-1) for n>3
n=3: [order 14] for n>15
n=4: [order 57] for n>60

A301790 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0 or 1 horizontally or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 3, 5, 4, 5, 6, 13, 8, 8, 9, 12, 34, 16, 13, 14, 17, 24, 89, 32, 21, 22, 25, 32, 48, 233, 64, 34, 35, 38, 45, 61, 96, 610, 128, 55, 56, 59, 66, 82, 116, 192, 1597, 256, 89, 90, 93, 100, 116, 150, 221, 384, 4181, 512, 144, 145, 148, 155, 171, 205, 275, 421, 768, 10946

Offset: 1

R. H. Hardin, Mar 26 2018

Table starts
...1....2...3...5...8...13...21...34...55...89..144...233...377...610...987
...2....5...6...9..14...22...35...56...90..145..234...378...611...988..1598
...4...13..12..17..25...38...59...93..148..237..381...614...991..1601..2588
...8...34..24..32..45...66..100..155..244..388..621...998..1608..2595..4192
..16...89..48..61..82..116..171..260..404..637.1014..1624..2611..4208..6792
..32..233..96.116.150..205..294..438..671.1048.1658..2645..4242..6826.11007
..64..610.192.221.275..364..508..741.1118.1728.2715..4312..6896.11077.17842
.128.1597.384.421.505..648..881.1258.1868.2855.4452..7036.11217.17982.28928
.256.4181.768.802.928.1156.1532.2142.3129.4726.7310.11491.18256.29202.46913

			Some solutions for n=5 k=4
..0..0..1..0. .0..1..0..1. .0..1..1..0. .0..1..1..0. .0..0..1..0
..1..0..1..1. .0..1..0..0. .0..0..1..0. .0..0..1..0. .1..0..1..1
..1..0..0..1. .0..1..1..0. .1..0..1..0. .1..0..1..1. .1..0..0..1
..1..1..0..0. .0..0..1..0. .0..1..0..1. .1..0..0..1. .1..1..0..1
..0..1..1..0. .1..0..1..0. .0..1..0..0. .1..1..0..1. .0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1920

Column 1 is A000079(n-1).
Column 2 is A001519(n+1).
Column 3 is A003945.
Row 1 is A000045(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-2)
k=3: a(n) = 2*a(n-1)
k=4: a(n) = a(n-1) +2*a(n-2) -a(n-4)
k=5: a(n) = 2*a(n-1) -a(n-4)
k=6: a(n) = a(n-1) +2*a(n-2) -a(n-4) -a(n-5) -a(n-6)
k=7: a(n) = 2*a(n-1) -a(n-4) -a(n-6)
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-2)
n=2: a(n) = 2*a(n-1) -a(n-3) for n>5
n=3: a(n) = 2*a(n-1) -a(n-3) for n>5
n=4: a(n) = 2*a(n-1) -a(n-3) for n>5
n=5: a(n) = 2*a(n-1) -a(n-3) for n>5
n=6: a(n) = 2*a(n-1) -a(n-3) for n>6
n=7: a(n) = 2*a(n-1) -a(n-3) for n>7

A232920 T(n,k)=Number of nXk 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally, diagonally or antidiagonally.

Original entry on oeis.org

3, 6, 9, 12, 18, 27, 24, 54, 54, 81, 48, 144, 246, 162, 243, 96, 396, 912, 1122, 486, 729, 192, 1080, 3612, 5808, 5118, 1458, 2187, 384, 2952, 13992, 33702, 37008, 23346, 4374, 6561, 768, 8064, 54600, 186720, 316800, 235824, 106494, 13122, 19683, 1536, 22032

Offset: 1

R. H. Hardin, Dec 02 2013

Table starts
.....3......6.......12........24..........48............96............192
.....9.....18.......54.......144.........396..........1080...........2952
....27.....54......246.......912........3612.........13992..........54600
....81....162.....1122......5808.......33702........186720........1054446
...243....486.....5118.....37008......316800.......2515716.......20706696
...729...1458....23346....235824.....2986152......33994188......409408542
..2187...4374...106494...1502736....28178262.....459797904.....8119777890
..6561..13122...485778...9575856...266016264....6221092260...161274860934
.19683..39366..2215902..61020048..2511769872...84180552504..3205524631536
.59049.118098.10107954.388836912.23718269934.1139126465856.63736076920680

			Some solutions for n=4 k=4
..0..0..1..0....0..0..1..0....0..0..1..0....0..0..1..0....1..0..0..1
..0..0..1..2....0..0..1..2....1..0..1..0....1..0..1..0....1..0..0..1
..1..0..1..2....0..0..1..0....1..0..1..2....0..0..1..2....0..0..0..0
..0..0..1..2....0..0..0..0....0..0..1..0....1..0..1..2....1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..391

Column 1 is A000244
Column 2 is A008776
Column 3 is A206144(n-1) for n>2
Column 4 is A223373
Row 1 is A003945

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 3*a(n-1)
k=3: a(n) = 5*a(n-1) -2*a(n-2)
k=4: a(n) = 7*a(n-1) -4*a(n-2)
k=5: a(n) = 14*a(n-1) -45*a(n-2) +15*a(n-3) +36*a(n-4) -19*a(n-5) +2*a(n-6)
k=6: [order 9]
k=7: [order 19]
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 2*a(n-1) +2*a(n-2)
n=3: a(n) = 3*a(n-1) +4*a(n-2) -2*a(n-3) for n>4
n=4: a(n) = 3*a(n-1) +16*a(n-2) -3*a(n-3) -25*a(n-4) +2*a(n-5) +4*a(n-6) for n>7
n=5: [order 10] for n>11
n=6: [order 23] for n>24
n=7: [order 46] for n>47

A274895 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (0,-1) (-1,-1) or (-2,0) and new values introduced in order 0..2.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 12, 7, 6, 8, 36, 16, 14, 12, 16, 108, 37, 38, 26, 24, 32, 324, 86, 104, 84, 50, 48, 64, 972, 200, 290, 275, 192, 95, 96, 128, 2916, 465, 815, 913, 753, 436, 181, 192, 256, 8748, 1081, 2291, 3064, 3017, 2049, 990, 345, 384, 512, 26244, 2513, 6434, 10337

Offset: 1

R. H. Hardin, Jul 10 2016

Table starts
...1...1....2.....4......8......16.......32........64........128.........256
...2...4...12....36....108.....324......972......2916.......8748.......26244
...3...7...16....37.....86.....200......465......1081.......2513........5842
...6..14...38...104....290.....815.....2291......6434......18065.......50729
..12..26...84...275....913....3064....10337.....34921.....117975......398560
..24..50..192...753...3017...12217....49697....202749.....828828.....3391310
..48..95..436..2049...9863...48269...237807...1173787....5803040....28746995
..96.181..990..5602..32539..191974..1143185...6843349...41072451...246859250
.192.345.2253.15305.107369..767905..5539989..40156061..292253909..2133745005
.384.657.5121.41866.354366.3065418.26833885.236220817.2086382703.18485204565

			Some solutions for n=4 k=4
..0..1..0..2. .0..1..2..0. .0..1..0..2. .0..1..2..0. .0..1..2..1
..2..1..2..1. .1..2..0..1. .2..1..0..2. .1..2..0..1. .1..2..0..1
..1..0..2..1. .2..0..1..2. .1..0..2..1. .1..2..1..2. .1..0..1..2
..1..0..1..0. .2..0..1..0. .1..0..2..0. .2..0..1..2. .2..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..420

Column 1 is A003945(n-2).
Column 2 is A052535(n+1).
Row 1 is A000079(n-2).
Row 2 is A003946(n-1).
Row 3 is A010912(n-1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>3
k=2: a(n) = a(n-1) +2*a(n-2) -a(n-4) for n>5
k=3: a(n) = a(n-1) +4*a(n-2) -6*a(n-4) -a(n-5) +4*a(n-6) -a(n-8) for n>10
k=4: [order 16] for n>18
k=5: [order 32] for n>34
k=6: [order 64] for n>66
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>2
n=2: a(n) = 3*a(n-1) for n>2
n=3: a(n) = 3*a(n-1) -2*a(n-2) +a(n-3)
n=4: a(n) = 5*a(n-1) -9*a(n-2) +10*a(n-3) -6*a(n-4) +a(n-5) for n>6
n=5: [order 8] for n>9
n=6: [order 13] for n>14
n=7: [order 21] for n>22

A275504 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,0) or (-1,-1) and new values introduced in order 0..2.

Original entry on oeis.org

1, 2, 2, 5, 9, 3, 14, 54, 16, 6, 41, 324, 80, 28, 12, 122, 1944, 400, 136, 56, 24, 365, 11664, 2000, 656, 232, 104, 48, 1094, 69984, 10000, 3168, 988, 516, 200, 96, 3281, 419904, 50000, 15296, 4180, 2628, 1168, 380, 192, 9842, 2519424, 250000, 73856, 17712

Offset: 1

R. H. Hardin, Jul 30 2016

Table starts
...1....2.....5.....14......41......122.......365.......1094........3281
...2....9....54....324....1944....11664.....69984.....419904.....2519424
...3...16....80....400....2000....10000.....50000.....250000.....1250000
...6...28...136....656....3168....15296.....73856.....356608.....1721856
..12...56...232....988....4180....17712.....75024.....317812.....1346268
..24..104...516...2628...13384....68080....346528....1763408.....8974288
..48..200..1168...7140...43780...268152...1643372...10069540....61703488
..96..380..2660..19368..143784..1063756...7886280...58423188...432942008
.192..724..6024..52864..470352..4220952..37846556..339516412..3045734096
.384.1380.13716.144228.1549756.16808164.182923008.1989999904.21655912500

			Some solutions for n=4 k=4
..0..0..1..1. .0..0..0..0. .0..0..0..0. .0..1..2..0. .0..1..0..0
..0..1..2..0. .1..2..1..1. .1..1..2..2. .0..1..2..0. .1..1..2..2
..1..1..2..0. .1..2..1..2. .1..2..2..1. .1..2..0..1. .1..2..2..1
..2..2..0..1. .0..0..0..2. .2..2..0..0. .1..2..0..1. .0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..312

Column 1 is A003945(n-2).
Row 1 is A007051(n-1).
Row 3 is A055842.
Row 4 is A108051(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>3
k=2: a(n) = a(n-1) +2*a(n-2) -a(n-4) for n>6
k=3: a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -4*a(n-4) +3*a(n-5) +a(n-6) -a(n-7) for n>11
k=4: [order 16] for n>20
k=5: [order 32] for n>36
k=6: [order 64] for n>68
Empirical for row n:
n=1: a(n) = 4*a(n-1) -3*a(n-2)
n=2: a(n) = 6*a(n-1) for n>2
n=3: a(n) = 5*a(n-1) for n>2
n=4: a(n) = 4*a(n-1) +4*a(n-2)
n=5: a(n) = 3*a(n-1) +5*a(n-2) +a(n-3)
n=6: a(n) = 3*a(n-1) +10*a(n-2) +4*a(n-3) -4*a(n-4) for n>5
n=7: a(n) = 3*a(n-1) +18*a(n-2) +11*a(n-3) -23*a(n-4) -4*a(n-5) for n>6

A352825 Number of nonfixed points y(i) != i, where y is the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 5, 2, 3, 1, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 2, 5, 2, 2, 1, 3, 1, 3, 1, 4, 1, 2, 3, 3, 2, 2, 1, 5, 3, 2, 1, 3, 2, 2, 1, 4, 1, 3, 2, 3, 1, 2, 2, 6, 1, 3, 2, 4, 1, 2, 1, 4, 3

Offset: 1

Gus Wiseman, Apr 05 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition (3,2,2,1) has Heinz number 90, so a(90) = 3. The partition (3,3,1,1) has Heinz number 100, so a(100) = 4.

* = unproved
Positions of first appearances are A003945.
The version for standard compositions is A352513, complement A352512.
A corresponding triangle for compositions is A352523, complement A238349.
The reverse complement version is A352822, triangle A238352.
The reverse version is A352823.
The complement version is A352824, triangle version A352833.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point, ranked by A352827.
A056239 adds up prime indices, row sums of A112798 and A296150.
*A064428 counts partitions without a fixed point, ranked by A352826.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914, conjugate rank A238745.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352832 counts reversed partitions with one fixed point, ranked by A352831.
Cf. A065770, A093641, A114088, A252464, A257990, A325163, A325164, A325165, A325169, A342192, A352486-A352491, A352828.

pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
Table[pnq[Reverse[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

A352825(n) = { my(f=factor(n),i=bigomega(n),c=0); for(k=1,#f~,while(f[k,2], f[k,2]--; c += (i!=primepi(f[k,1])); i--)); (c); }; \\ Antti Karttunen, Apr 14 2022

a(n) = A001222(n) - A352824(n).

Data section extended up to 105 terms by Antti Karttunen, Apr 14 2022

A233155 T(n,k) = Number of n X k 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally or antidiagonally.

Original entry on oeis.org

3, 6, 9, 12, 24, 27, 24, 72, 96, 81, 48, 216, 432, 384, 243, 96, 648, 1944, 2592, 1536, 729, 192, 1944, 8856, 17496, 15552, 6144, 2187, 384, 5832, 40392, 121176, 157464, 93312, 24576, 6561, 768, 17496, 184248, 842616, 1658232, 1417176, 559872, 98304, 19683

Offset: 1

R. H. Hardin, Dec 05 2013

Table starts
.....3.......6........12.........24...........48.............96
.....9......24........72........216..........648...........1944
....27......96.......432.......1944.........8856..........40392
....81.....384......2592......17496.......121176.........842616
...243....1536.....15552.....157464......1658232.......17587584
...729....6144.....93312....1417176.....22692312......367125912
..2187...24576....559872...12754584....310536504.....7663517136
..6561...98304...3359232..114791256...4249585944...159971190624
.19683..393216..20155392.1033121304..58154132088..3339300422232
.59049.1572864.120932352.9298091736.795819434328.69705848287656

			Some solutions for n=4, k=4
..1..2..2..1....1..2..2..1....0..0..0..0....1..2..1..0....2..1..0..1
..2..1..2..2....2..1..0..1....1..0..1..2....1..0..0..0....0..1..0..1
..2..1..2..1....2..1..0..1....1..0..1..0....0..0..1..0....2..1..2..1
..0..1..2..2....0..1..0..0....1..0..0..1....0..0..0..0....2..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..390

Column 1 is A000244.
Column 2 is A002023(n-1).
Column 3 is 2*A000400.
Column 4 is 3*A055275.
Row 1 is A003945.
Row 2 is A005051(n-1) for n>1.

Empirical for column k:
k=1: a(n) = 3*a(n-1).
k=2: a(n) = 4*a(n-1).
k=3: a(n) = 6*a(n-1).
k=4: a(n) = 9*a(n-1).
k=5: a(n) = 15*a(n-1) -18*a(n-2).
k=6: a(n) = 25*a(n-1) -90*a(n-2) +81*a(n-3).
k=7: a(n) = 42*a(n-1) -351*a(n-2) +972*a(n-3) -810*a(n-4).
Empirical for row n:
n=1: a(n) = 2*a(n-1).
n=2: a(n) = 3*a(n-1) for n>2.
n=3: a(n) = 5*a(n-1) -2*a(n-2) for n>4.
n=4: a(n) = 9*a(n-1) -15*a(n-2) +6*a(n-3) for n>7.
n=5: [order 7] for n>11.
n=6: [order 9] for n>15.
n=7: [order 27] for n>33.

A110164 Expansion of (1-x^2)/(1+2x).

Original entry on oeis.org

1, -2, 3, -6, 12, -24, 48, -96, 192, -384, 768, -1536, 3072, -6144, 12288, -24576, 49152, -98304, 196608, -393216, 786432, -1572864, 3145728, -6291456, 12582912, -25165824, 50331648, -100663296, 201326592, -402653184, 805306368, -1610612736, 3221225472

Offset: 0

Paul Barry, Jul 14 2005

Diagonal sums of Riordan array ((1-x)/(1+x),x/(1+x)^2), A110162.
The positive sequence with g.f. (1-x^2)/(1-2x) gives the row sums of the Riordan array (1+x,x/(1-x)). - Paul Barry, Jul 18 2005
The inverse g.f. is (1 + 2*x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + ...). - Gary W. Adamson, Jan 07 2011
In absolute value, essentially the same as A007283(n) = A003945(n+1) = A042950(n+1) = A082505(n+1) = A087009(n+3) = A091629(n) = A098011(n+4) = A111286(n+2). - M. F. Hasler, Apr 19 2015

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

Cf. A098011, A122803.

CoefficientList[Series[(1 - x^2)/(1 + 2x), {x, 0, 33}], x] (* Robert G. Wilson v, Jul 08 2006 *)
LinearRecurrence[{-2},{1,-2,3},40] (* Harvey P. Dale, May 10 2023 *)

A110164(n)=if(n>1,3*(-2)^(n-2),1-3*n) \\ M. F. Hasler, Apr 19 2015

a(n) = 3*(-2)^(n-2) = 3*A122803(n-2) for n >= 2. a(n) = -2 a(n-1) for n >= 3. - M. F. Hasler, Apr 19 2015

E.g.f.: (1/4) - (x/2) + (3/4)*exp(-2*x). - Alejandro J. Becerra Jr., Jan 29 2021

A110287 a(n) = 17*2^n.

Original entry on oeis.org

17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 8704, 17408, 34816, 69632, 139264, 278528, 557056, 1114112, 2228224, 4456448, 8912896, 17825792, 35651584, 71303168, 142606336, 285212672, 570425344, 1140850688, 2281701376, 4563402752, 9126805504, 18253611008

Offset: 0

Alexandre Wajnberg, Sep 07 2005

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).

17 times powers of 2. - Omar E. Pol, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..235
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A003945 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), this sequence (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).

Cf. A007283.

[17*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

17*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,17,30] (* Harvey P. Dale, Jul 21 2014 *)

a(n)=17*2^n \\ Charles R Greathouse IV, Oct 07 2015

[17*2^n for n in range(51)] # G. C. Greubel, Jan 05 2023

G.f.: 17/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = 17*A000079(n). - Omar E. Pol, Dec 17 2008
a(n) = 2*a(n-1) (with a(0)=17). - Vincenzo Librandi, Dec 26 2010
a(n) = A173786(n+4, n) for n>3. - Reinhard Zumkeller, Feb 28 2010
E.g.f.: 17*exp(2*x). - G. C. Greubel, Jan 05 2023

Edited by Omar E. Pol, Dec 16 2008

cp's OEIS Frontend

A275565 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,0) (-1,2) or (0,-2) and new values introduced in order 0..2.

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A276248 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-1,1) or (0,-2) and new values introduced in order 0..2.

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A301790 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0 or 1 horizontally or antidiagonally adjacent elements, with upper left element zero.

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A232920 T(n,k)=Number of nXk 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally, diagonally or antidiagonally.

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A274895 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (0,-1) (-1,-1) or (-2,0) and new values introduced in order 0..2.

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A275504 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,0) or (-1,-1) and new values introduced in order 0..2.

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A352825 Number of nonfixed points y(i) != i, where y is the integer partition with Heinz number n.

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A233155 T(n,k) = Number of n X k 0..2 arrays with no element x(i,j) adjacent to value 2-x(i,j) horizontally or antidiagonally.

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A110164 Expansion of (1-x^2)/(1+2x).

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