A152746 -id:A152746 - cp's OEIS frontend

A195320 7 times hexagonal numbers: a(n) = 7n(2*n-1).

0, 7, 42, 105, 196, 315, 462, 637, 840, 1071, 1330, 1617, 1932, 2275, 2646, 3045, 3472, 3927, 4410, 4921, 5460, 6027, 6622, 7245, 7896, 8575, 9282, 10017, 10780, 11571, 12390, 13237, 14112, 15015, 15946, 16905, 17892, 18907, 19950, 21021, 22120, 23247, 24402, 25585

Offset: 0

Omar E. Pol, Sep 18 2011

Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277.

Also sequence found by reading the same line (mentioned above) in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5]. - Omar E. Pol, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A024966.

Cf. A000384, A118277, A152746, A152750, A185019, A193053, A195019, A195020, A198017, A316466.

[7*n*(2*n-1): n in [0..50]]; // Vincenzo Librandi, Sep 28 2011

Table[7*n*(2*n - 1), {n, 0, 50}] (* Paolo Xausa, Jan 29 2025 *)
7*PolygonalNumber[6, Range[0, 50]] (* Paolo Xausa, Jan 29 2025 *)

a(n)=7*n*(2*n-1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 14*n^2 - 7*n = 7*A000384(n).
G.f.: -7*x*(1+3*x)/(x-1)^3. - R. J. Mathar, Sep 27 2011
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 7*exp(x)*x*(2*x + 1).
a(n) = A316466(n) - n = A024966(2*n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A207024 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 13, 81, 90, 64, 10, 18, 169, 252, 168, 100, 12, 25, 324, 624, 558, 270, 144, 14, 34, 625, 1350, 1586, 1035, 396, 196, 16, 46, 1156, 3025, 3726, 3315, 1719, 546, 256, 18, 62, 2116, 6256, 9450, 8280, 6123, 2646, 720, 324, 20, 83, 3844

Offset: 1

R. H. Hardin, Feb 14 2012

Table starts
..2...4...6....9....13....18.....25.....34......46......62.......83......111
..4..16..36...81...169...324....625...1156....2116....3844.....6889....12321
..6..36..90..252...624..1350...3025...6256...12788...25792....50630....99012
..8..64.168..558..1586..3726...9450..21318...47518..104470...220531...464202
.10.100.270.1035..3315..8280..23400..56814..136114..322834...725005..1627260
.12.144.396.1719..6123.16038..49925.129302..329498..836938..1984447..4716834
.14.196.546.2646.10374.28224..95900.263228..707756.1914436..4765030.11929281
.16.256.720.3852.16484.46260.170300.492932.1389476.3984244.10362550.27202659

			Some solutions for n=4, k=3
..1..0..0....0..1..0....0..0..1....1..0..1....0..0..1....1..1..0....1..1..1
..1..0..0....1..1..0....1..1..1....0..0..1....0..0..1....1..0..1....1..1..1
..1..0..0....1..1..0....0..0..1....0..0..1....0..0..1....1..0..1....0..1..0
..1..0..0....1..0..0....0..0..1....0..0..1....0..0..1....0..0..1....0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..1984

Column 2 is A016742.
Column 3 is A152746.
Row 1 is A171861(n+1).

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 12*n^2 - 6*n
k=4: a(n) = 6*n^3 + (27/2)*n^2 - (21/2)*n
k=5: a(n) = (13/6)*n^4 + 13*n^3 + (52/3)*n^2 - (39/2)*n
k=6: a(n) = (33/4)*n^4 + (45/2)*n^3 + (75/4)*n^2 - (63/2)*n
k=7: a(n) = (55/24)*n^5 + (75/4)*n^4 + (275/8)*n^3 + (75/4)*n^2 - (295/6)*n

A207169 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 13, 81, 90, 64, 10, 19, 169, 261, 168, 100, 12, 28, 361, 624, 603, 270, 144, 14, 41, 784, 1482, 1612, 1161, 396, 196, 16, 60, 1681, 3808, 3952, 3445, 1989, 546, 256, 18, 88, 3600, 9512, 11452, 8455, 6513, 3141, 720, 324, 20, 129, 7744

Offset: 1

R. H. Hardin Feb 15 2012

Table starts
..2...4...6....9....13....19.....28.....41......60......88......129......189
..4..16..36...81...169...361....784...1681....3600....7744....16641....35721
..6..36..90..261...624..1482...3808...9512...23280...58080...144996...359100
..8..64.168..603..1612..3952..11452..32021...84300..231616...641775..1736910
.10.100.270.1161..3445..8455..26908..82861..228060..672760..2029041..5846337
.12.144.396.1989..6513.15789..54208.182081..515760.1608288..5222049.15774129
.14.196.546.3141.11284.26866..98224.357356.1032000.3365824.11680176.36617616
.16.256.720.4671.18304.42712.164668.645217.1888380.6392320.23581071.76187790

			Some solutions for n=4 k=3
..1..0..0....0..0..1....0..1..1....1..1..1....0..0..1....1..0..0....1..0..0
..1..0..0....0..1..1....0..0..1....1..1..1....0..0..1....1..1..0....0..0..1
..1..0..0....0..1..0....0..0..1....1..1..1....0..0..1....0..1..0....0..0..1
..1..0..0....0..1..0....0..0..1....1..1..1....0..0..1....0..1..0....0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..1512

Column 2 is A016742
Column 3 is A152746
Row 1 is A000930(n+3)

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 12*n^2 - 6*n
k=4: a(n) = 9*n^3 + 9*n - 9
k=5: a(n) = (13/4)*n^4 + (13/2)*n^3 + (117/4)*n^2 - 26*n
k=6: a(n) = (19/4)*n^4 + (95/2)*n^3 - (57/4)*n^2 - 19*n
k=7: a(n) = 35*n^4 + 42*n^3 + 7*n^2 - 84*n + 28
Empirical for rows:
n=1: a(k)=a(k-1)+a(k-3) for k>4
n=2: a(k)=a(k-1)+a(k-2)+3*a(k-3)+a(k-4)-a(k-5)-a(k-6) for k>7
n=3: a(k)=a(k-1)+9*a(k-3)+2*a(k-4)+2*a(k-5)-12*a(k-6)-8*a(k-7)+8*a(k-9) for k>11
n=4: a(k)=a(k-1)+13*a(k-3)+3*a(k-4)+3*a(k-5)-27*a(k-6)-18*a(k-7)+27*a(k-9) for k>11
n=5: a(k)=a(k-1)+17*a(k-3)+4*a(k-4)+4*a(k-5)-48*a(k-6)-32*a(k-7)+64*a(k-9) for k>11
n=6: a(k)=a(k-1)+21*a(k-3)+5*a(k-4)+5*a(k-5)-75*a(k-6)-50*a(k-7)+125*a(k-9) for k>11
n=7: a(k)=a(k-1)+25*a(k-3)+6*a(k-4)+6*a(k-5)-108*a(k-6)-72*a(k-7)+216*a(k-9) for k>11
n=8: a(k)=a(k-1)+29*a(k-3)+7*a(k-4)+7*a(k-5)-147*a(k-6)-98*a(k-7)+343*a(k-9) for k>11
n=9: a(k)=a(k-1)+33*a(k-3)+8*a(k-4)+8*a(k-5)-192*a(k-6)-128*a(k-7)+512*a(k-9) for k>11
n=10: a(k)=a(k-1)+37*a(k-3)+9*a(k-4)+9*a(k-5)-243*a(k-6)-162*a(k-7)+729*a(k-9) for k>11
n=11: a(k)=a(k-1)+41*a(k-3)+10*a(k-4)+10*a(k-5)-300*a(k-6)-200*a(k-7)+1000*a(k-9) for k>11
n=12: a(k)=a(k-1)+45*a(k-3)+11*a(k-4)+11*a(k-5)-363*a(k-6)-242*a(k-7)+1331*a(k-9) for k>11
n=13: a(k)=a(k-1)+49*a(k-3)+12*a(k-4)+12*a(k-5)-432*a(k-6)-288*a(k-7)+1728*a(k-9) for k>11
n=14: a(k)=a(k-1)+53*a(k-3)+13*a(k-4)+13*a(k-5)-507*a(k-6)-338*a(k-7)+2197*a(k-9) for k>11
n=15: a(k)=a(k-1)+57*a(k-3)+14*a(k-4)+14*a(k-5)-588*a(k-6)-392*a(k-7)+2744*a(k-9) for k>11
apparently a(k)=a(k-1)+(4*n-3)*a(k-3)+(n-1)*a(k-4)+(n-1)*a(k-5)-3*(n-1)^2*a(k-6)-2*(n-1)^2*a(k-7)+(n-1)^3*a(k-9) for n>2 and k>11

A207453 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 8, 16, 100, 90, 64, 10, 26, 256, 330, 168, 100, 12, 42, 676, 1008, 760, 270, 144, 14, 68, 1764, 3354, 2560, 1450, 396, 196, 16, 110, 4624, 10710, 10088, 5200, 2460, 546, 256, 18, 178, 12100, 34884, 36456, 23530, 9216, 3850, 720

Offset: 1

R. H. Hardin, Feb 17 2012

Table starts
..2...4...6...10....16.....26.....42......68......110......178.......288
..4..16..36..100...256....676...1764....4624....12100....31684.....82944
..6..36..90..330..1008...3354..10710...34884...112530...364722...1179360
..8..64.168..760..2560..10088..36456..138176...509960..1910296...7096320
.10.100.270.1450..5200..23530..92610..396100..1610950..6754210..27799200
.12.144.396.2460..9216..46956.196812..932688..4086060.18819228..83939328
.14.196.546.3850.14896..84266.370734.1922564..8935850.44655394.212625504
.16.256.720.5680.22528.139984.640080.3599104.17556880.94358512.474439680

			Some solutions for n=5, k=3
..1..0..1....1..1..0....1..1..1....0..1..1....0..1..0....0..1..0....0..1..1
..1..0..1....1..0..0....0..1..1....0..1..1....0..1..1....1..0..0....1..0..1
..1..0..1....1..0..0....0..1..0....0..1..0....0..1..1....1..0..0....1..0..1
..1..0..1....1..0..0....0..1..0....0..1..0....0..1..0....1..0..0....1..0..1
..1..0..0....1..0..0....0..1..0....0..1..0....0..1..0....1..0..0....1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..1573

Column 2 is A016742.
Column 3 is A152746.
Row 1 is A006355(n+2).
Row 2 is A206981.

Empirical for column k:
k=1: a(n) = 2*n;
k=2: a(n) = 4*n^2;
k=3: a(n) = 12*n^2 - 6*n;
k=4: a(n) = 10*n^3 + 10*n^2 - 10*n;
k=5: a(n) = 48*n^3 - 32*n^2;
k=6: a(n) = 26*n^4 + 78*n^3 - 104*n^2 + 26*n;
k=7: a(n) = 168*n^4 - 84*n^3 - 84*n^2 + 42*n;
k=8: a(n) = 68*n^5 + 408*n^4 - 612*n^3 + 204*n^2;
k=9: a(n) = 550*n^5 - 990*n^3 + 660*n^2 - 110*n;
k=10: a(n) = 178*n^6 + 1780*n^5 - 2670*n^4 + 534*n^3 + 534*n^2 - 178*n;
k=11: a(n) = 1728*n^6 + 1440*n^5 - 6912*n^4 + 5184*n^3 - 1152*n^2;
k=12: a(n) = 466*n^7 + 6990*n^6 - 9320*n^5 - 2796*n^4 + 8388*n^3 - 3728*n^2 + 466*n;
k=13: a(n) = 5278*n^7 + 10556*n^6 - 36946*n^5 + 27144*n^4 - 3016*n^3 - 3016*n^2 + 754*n;
k=14: a(n) = 1220*n^8 + 25620*n^7 - 25620*n^6 - 42700*n^5 + 73200*n^4 - 36600*n^3 + 6100*n^2;
k=15: a(n) = 15792*n^8 + 55272*n^7 - 165816*n^6 + 98700*n^5 + 39480*n^4 - 59220*n^3 + 19740*n^2 - 1974*n.
Empirical for row n:
n=1: a(k)=a(k-1)+a(k-2);
n=2: a(k)=2*a(k-1)+2*a(k-2)-a(k-3);
n=3: a(k)=a(k-1)+7*a(k-2)+2*a(k-3)-4*a(k-4);
n=4: a(k)=a(k-1)+10*a(k-2)+3*a(k-3)-9*a(k-4);
n=5: a(k)=a(k-1)+13*a(k-2)+4*a(k-3)-16*a(k-4);
n=6: a(k)=a(k-1)+16*a(k-2)+5*a(k-3)-25*a(k-4);
n=7: a(k)=a(k-1)+19*a(k-2)+6*a(k-3)-36*a(k-4);
apparently for row n>2: a(k)=a(k-1)+(3*n-2)*a(k-2)+(n-1)*a(k-3)+(n-1)^2*a(k-4).

A207599 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 15, 81, 90, 64, 10, 25, 225, 225, 168, 100, 12, 40, 625, 825, 441, 270, 144, 14, 64, 1600, 3025, 1995, 729, 396, 196, 16, 104, 4096, 9240, 9025, 3915, 1089, 546, 256, 18, 169, 10816, 28224, 30400, 21025, 6765, 1521, 720, 324, 20, 273

Offset: 1

R. H. Hardin Feb 19 2012

Table starts
..2...4...6....9....15.....25.....40......64......104......169.......273
..4..16..36...81...225....625...1600....4096....10816....28561.....74529
..6..36..90..225...825...3025...9240...28224....93912...312481....997815
..8..64.168..441..1995...9025..30400..102400...403520..1590121...5746377
.10.100.270..729..3915..21025..75400..270400..1223560..5536609..21791133
.12.144.396.1089..6765..42025.157440..589824..3005184.15311569..64177113
.14.196.546.1521.10725..75625.292600.1132096..6404216.36228361.159389139
.16.256.720.2025.15975.126025.499840.1982464.12318592.76545001.350003745

			Some solutions for n=4 k=3
..1..0..1....1..0..0....0..0..1....1..0..0....1..1..0....1..0..0....1..1..0
..1..1..0....0..0..1....0..0..1....0..1..1....1..0..1....1..0..0....0..1..1
..1..0..0....0..0..1....0..0..1....0..0..1....1..0..0....1..0..0....0..1..1
..1..0..0....0..0..1....0..0..1....0..0..1....1..0..0....1..0..0....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..1250

Column 2 is A016742.
Column 3 is A152746.
Column 4 is A016946(n-1).
Row 1 is A006498(n+2).
Row 2 is A189145(n+2).

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 12*n^2 - 6*n
k=4: a(n) = 36*n^2 - 36*n + 9
k=5: a(n) = 30*n^3 + 15*n^2 - 45*n + 15
k=6: a(n) = 25*n^4 + 50*n^3 - 25*n^2 - 50*n + 25
k=7: a(n) = 120*n^4 + 40*n^3 - 200*n^2 + 80*n

A287318 Square array A(n,k) = (2n)! [x^n] BesselI(0, 2sqrt(x))^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 36, 20, 0, 1, 8, 90, 400, 70, 0, 1, 10, 168, 1860, 4900, 252, 0, 1, 12, 270, 5120, 44730, 63504, 924, 0, 1, 14, 396, 10900, 190120, 1172556, 853776, 3432, 0, 1, 16, 546, 19920, 551950, 7939008, 32496156, 11778624, 12870, 0

Offset: 0

Peter Luschny, May 23 2017

			Arrays start:
  k\n| 0   1    2      3        4          5           6
  ---|---------------------------------------------------------
  k=0| 1,  0,   0,     0,       0,         0,            0, ... A000007
  k=1| 1,  2,   6,    20,      70,       252,          924, ... A000984
  k=2| 1,  4,  36,   400,    4900,     63504,       853776, ... A002894
  k=3| 1,  6,  90,  1860,   44730,   1172556,     32496156, ... A002896
  k=4| 1,  8, 168,  5120,  190120,   7939008,    357713664, ... A039699
  k=5| 1, 10, 270, 10900,  551950,  32232060,   2070891900, ... A287317
  k=6| 1, 12, 396, 19920, 1281420,  96807312,   8175770064, ... A356258
  k=7| 1, 14, 546, 32900, 2570050, 238935564,  25142196156, ...
  k=8| 1, 16, 720, 50560, 4649680, 514031616,  64941883776, ...
  k=9| 1, 18, 918, 73620, 7792470, 999283068, 147563170524, ...

Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.

Rows: A000007 (k=0), A000984 (k=1), A002894 (k=2), A002896 (k=3), A039699 (k=4), A287317 (k=5), A356258 (k=6).
Columns: A005843 (n=1), A152746 (n=2), 20*A169711 (n=3), 70*A169712 (n=4), 252*A169713 (n=5).
Main diagonal gives A303503.
Cf. A287316.

A287318_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((2*i)!*coeff(ser,x,i), i=0..len-1) end:
for k from 0 to 6 do A287318_row(k, 9) od;

Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (2 n)!, {n, 0, 6}], {k, 0, 6}]

A(n,k) = A287316(n,k) * binomial(2*n,n).

A152745 5 times hexagonal numbers: 5n(2*n-1).

Original entry on oeis.org

0, 5, 30, 75, 140, 225, 330, 455, 600, 765, 950, 1155, 1380, 1625, 1890, 2175, 2480, 2805, 3150, 3515, 3900, 4305, 4730, 5175, 5640, 6125, 6630, 7155, 7700, 8265, 8850, 9455, 10080, 10725, 11390, 12075, 12780, 13505, 14250, 15015

Offset: 0

Omar E. Pol, Dec 12 2008

Sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 18 2011

Also sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is one of the four semi-diagonals of the spiral. - Omar E. Pol, Oct 14 2011

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A085250, A152746.

Bisection of A028895.

[5*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Sep 01 2018

LinearRecurrence[{3,-3,1}, {0, 5, 30}, 50] (* or *) Table[5*n*(2*n-1), {n,0,50}] (* G. C. Greubel, Sep 01 2018 *)

a(n)=5*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 10*n^2 - 5*n = A000384(n)*5.
a(n) = a(n-1) + 20*n-15 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
From G. C. Greubel, Sep 01 2018: (Start)
G.f.: 5*x*(1+ 3*x)/(1-x)^3.
E.g.f.: 5*x*(1+2*x)*exp(x). (End)
From Vaclav Kotesovec, Sep 02 2018: (Start)
Sum_{n>=1} 1/a(n) = 2*log(2)/5.
Sum_{n>=1} (-1)^n/a(n) = log(2)/5 - Pi/10. (End)

A154105 a(n) = 12n^2 + 18n + 7.

Original entry on oeis.org

7, 37, 91, 169, 271, 397, 547, 721, 919, 1141, 1387, 1657, 1951, 2269, 2611, 2977, 3367, 3781, 4219, 4681, 5167, 5677, 6211, 6769, 7351, 7957, 8587, 9241, 9919, 10621, 11347, 12097, 12871, 13669, 14491, 15337, 16207, 17101, 18019, 18961, 19927, 20917, 21931

Offset: 0

Klaus Brockhaus, Jan 04 2009

a(n) is the number of partitions with three integral dissimilar components of the number 12(n+1), e.g for n=0, 12 may be partitioned in the 7 ways (1,2,9), (1,3,8), (1,4,7), (1,5,6), (2,3,7), (2,4,6) and (3,4,5). - Ian Duff, Jan 31 2010

Sequence found by reading the line from 7, in the direction 7, 37, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, May 08 2018

			a(2) = 12*2^2 + 18*2 + 7 = 91 = 6*14 + 7 = 6*A014106(2) + 7.
a(3) = a(2) + 24*3 + 6 = 91 + 72 + 6 = 169.
a(-4) = 12*4^2 - 18*4 + 7 = 127 = 2*64 - 1 = 2*A085473(3) - 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
John Elias, Animated Illustration: Starburst Hexagrams
Leo Tavares, Illustration: Hexagonal Halos
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001082, A014106, A152746, A153286, A085473.

Cf. A003215, A000290, A000217.

Magma
```
[ 12*n^2+18*n+7: n in [0..40] ];
```

Table[12*n^2 + 18*n + 7, {n, 0, 42}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
LinearRecurrence[{3,-3,1}, {7,37,91}, 25] (* G. C. Greubel, Sep 02 2016 *)

a(n)=12*n^2+18*n+7 \\ Charles R Greathouse IV, Sep 02 2016

G.f.: (7 + 16*x + x^2)/(1-x)^3.
a(n) = 6*A014106(n) + 7.
a(0) = 7; for n > 0, a(n) = a(n-1) + 24*n + 6.
a(-n-1) = 2*A085473(n) - 1. - Bruno Berselli, Sep 05 2011
E.g.f.: (7 + 30*x + 12*x^2)*exp(x). - G. C. Greubel, Sep 02 2016
a(n) = 1 + A152746(n+1). - Omar E. Pol, May 08 2018
a(n) = A003215(n) + 6*A000290(n+1) + 6*A000217(n). - Leo Tavares, Sep 12 2022

A259752 a(n) = 24*n - 18.

Original entry on oeis.org

6, 30, 54, 78, 102, 126, 150, 174, 198, 222, 246, 270, 294, 318, 342, 366, 390, 414, 438, 462, 486, 510, 534, 558, 582, 606, 630, 654, 678, 702, 726, 750, 774, 798, 822, 846, 870, 894, 918, 942, 966, 990, 1014, 1038, 1062, 1086, 1110, 1134, 1158, 1182, 1206

Offset: 1

José María Grau Ribas, Jul 12 2015

Original name: Numbers n such that n/A259748(n) = 6.

Partial sums give A152746. - Leo Tavares, Jul 29 2023

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000914, A016813, A063128, A063148, A152746.

Other sequences of numbers n such that A259748(n)/n equals a constant: A008606, A073762, A259749, A259750, A259751, A259754, A259755.

A[n_] := A[n] = Sum[a b, {a, 1,  n}, {b, a + 1, n}] ; Select[Range[600], Mod[A[#], #]/# == 1/6 & ]

A259748(a(n))/a(n) = 1/6.
a(n) = 6*A016813(n-1). - Michel Marcus, Jul 18 2015
G.f.: 6*x*(3*x+1)/(x-1)^2. - Alois P. Heinz, Jul 29 2023
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 6*(exp(x)*(4*x - 3) + 3).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

Better name from Danny Rorabaugh, Oct 22 2015

A143698 12 times hexagonal numbers: 12n(2*n-1).

Original entry on oeis.org

0, 12, 72, 180, 336, 540, 792, 1092, 1440, 1836, 2280, 2772, 3312, 3900, 4536, 5220, 5952, 6732, 7560, 8436, 9360, 10332, 11352, 12420, 13536, 14700, 15912, 17172, 18480, 19836, 21240, 22692, 24192, 25740, 27336, 28980, 30672, 32412

Offset: 0

Omar E. Pol, Jan 23 2009

Sequence found by reading the line from 0, in the direction 0, 12,..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. - Omar E. Pol, Oct 02 2011

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A002939, A085250, A094159, A152746, A154617.

seq(12*n*(2*n-1), n=0..40); # G. C. Greubel, May 30 2021

Table[24n^2-12n,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,12,72},40] (* Harvey P. Dale, Sep 24 2015 *)

a(n)=24*n^2-12*n \\ Charles R Greathouse IV, Jun 17 2017

[12*n*(2*n-1) for n in (0..40)] # G. C. Greubel, May 30 2021

a(n) = 24*n^2 - 12*n = 12*A000384(n) = 6*A002939(n) = 4*A094159(n) = 3*A085250(n) = 2*A152746(n).
a(n) = a(n-1) + 48*n - 36, with a(0)=0. - Vincenzo Librandi, Dec 14 2010
From G. C. Greubel, May 30 2021: (Start)
G.f.: 12*x*(1 + 3*x)/(1-x)^3.
E.g.f.: 12*x*(1 + 2*x)*exp(x). (End)

cp's OEIS Frontend

A195320 7 times hexagonal numbers: a(n) = 7*n*(2*n-1).

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A207024 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

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A207169 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 1 0 1 horizontally and 0 0 1 and 1 0 1 vertically.

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A207453 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 1 0 1 vertically.

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A207599 T(n,k) = Number of n X k 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 1 0 1 vertically.

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A287318 Square array A(n,k) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

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A152745 5 times hexagonal numbers: 5*n*(2*n-1).

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A154105 a(n) = 12*n^2 + 18*n + 7.

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A195320 7 times hexagonal numbers: a(n) = 7n(2*n-1).

A287318 Square array A(n,k) = (2n)! [x^n] BesselI(0, 2sqrt(x))^k read by antidiagonals.

A152745 5 times hexagonal numbers: 5n(2*n-1).

A154105 a(n) = 12n^2 + 18n + 7.

A143698 12 times hexagonal numbers: 12n(2*n-1).