A002492 -id:A002492 - cp's OEIS frontend

A188777 T(n,k) = Number of n-turn bishop's tours on a k X k board summed over all starting positions.

1, 4, 0, 9, 4, 0, 16, 20, 0, 0, 25, 56, 28, 0, 0, 36, 120, 152, 24, 0, 0, 49, 220, 488, 328, 8, 0, 0, 64, 364, 1192, 1720, 584, 0, 0, 0, 81, 560, 2468, 5816, 5464, 840, 0, 0, 0, 100, 816, 4560, 15424, 26360, 15824, 784, 0, 0, 0, 121, 1140, 7760, 34736, 91120, 112680, 40496

Offset: 1

R. H. Hardin, Apr 10 2011

Table starts
.1.4..9..16.....25......36.......49.......64.......81......100.....121....144
.0.4.20..56....120.....220......364......560......816.....1140....1540...2024
.0.0.28.152....488....1192.....2468.....4560.....7760....12400...18860..27560
.0.0.24.328...1720....5816....15424....34736....69776...128528..221448.361528
.0.0..8.584...5464...26360....91120...252720...603696..1288592.2525400
.0.0..0.840..15824..112680...516160..1778608..5082912.12622640
.0.0..0.784..40496..451104..2803552.12139552.41792672
.0.0..0.384..88264.1665344.14497784.80088992
.0.0..0...0.159704.5607456
.0.0..0...0.229296

			Some n=4 solutions for 4 X 4
..0..0..0..0....0..4..0..1....0..0..0..4....3..0..0..0....0..1..0..0
..0..3..0..0....0..0..3..0....0..0..0..0....0..2..0..0....4..0..2..0
..2..0..4..0....0..2..0..0....0..3..0..1....0..0..4..0....0..3..0..0
..0..1..0..0....0..0..0..0....0..0..2..0....0..0..0..1....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..113

Row 2 is A002492(n-1).

Empirical: T(1,k) = k^2.
Empirical: T(2,k) = (4/3)*k^3 - 2*k^2 + (2/3)*k.
Empirical: T(3,k) = 4*T(3,k-1)-5*T(3,k-2)+5*T(3,k-4)-4*T(3,k-5)+T(3,k-6).
Empirical: T(4,k) = 4*T(4,k-1)-4*T(4,k-2)-4*T(4,k-3)+10*T(4,k-4)-4*T(4,k-5)-4*T(4,k-6)+4*T(4,k-7)-T(4,k-8).
Empirical: T(5,k) = 4*T(5,k-1)-3*T(5,k-2)-8*T(5,k-3)+14*T(5,k-4)-14*T(5,k-6)+8*T(5,k-7)+3*T(5,k-8)-4*T(5,k-9)+T(5,k-10).
Empirical: T(6,k) = 4*T(6,k-1)-2*T(6,k-2)-12*T(6,k-3)+17*T(6,k-4)+8*T(6,k-5)-28*T(6,k-6)+8*T(6,k-7)+17*T(6,k-8)-12*T(6,k-9)-2*T(6,k-10)+4*T(6,k-11)-T(6,k-12).

A300192 Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2x + 1)^n + (x^2 - 1)(x + 1)^n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 2, 6, 6, 2, 0, 3, 13, 22, 18, 7, 1, 0, 4, 23, 56, 75, 60, 29, 8, 1, 0, 5, 36, 115, 215, 261, 215, 121, 45, 10, 1, 0, 6, 52, 206, 495, 806, 938, 798, 496, 220, 66, 12, 1, 0, 7, 71, 336, 987, 2016, 3031, 3452, 3010, 2003, 1001, 364, 91

Offset: 0

Franck Maminirina Ramaharo, Feb 28 2018

			The triangle T(n, k) begins:
n\k  0  1   2    3    4     5     6     7     8     9    10   11  12  13 14
0:   0  0   1
1:   0  1   2    1
2:   0  2   6    6    2
3:   0  3  13   22   18     7     1
4:   0  4  23   56   75    60    29     8     1
5:   0  5  36  115  215   261   215   121    45    10     1
6:   0  6  52  206  495   806   938   798   496   220    66   12   1
7:   0  7  71  336  987  2016  3031  3452  3010  2003  1001  364  91  14  1

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996.

Paul Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806, 10 pages.
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
A. M. Mathai and P. N. Rathie, Enumeration of almost cubic maps, Journal of Combinatorial Theory, Series B, Vol 13 (1972), 83-90.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Row sums: A000302 (powers of 4).

Cf. A034870, A034871, A032443.

T := (n, k) -> binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k);
for n from 0 to 10 do seq(T(n, k), k = 0 .. max(2*n, n + 2)) od;

T(n, k) := binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k)$
a : []$
for n:0 thru 10 do
  a : append(a, makelist(T(n, k), k, 0, max(2*n, n + 2)))$
a;

row(n) = Vecrev((x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n); \\ Michel Marcus, Nov 12 2022

T(n,k) = binomial(2*n,k) + binomial(n,k-2) - binomial(n,k).
T(n,k) = T(n-1,k-1)+ T(n-1,k) + A034871(n-1,k-1), with T(n,0) = T(0,1) = 0 and T(0,2) = 1
T(n,1) = A001477(n).
T(n,2) = A143689(n).
T(n,3) = n + A002492(n-1) - A000292(n-2).
T(n,n) = A247493(n+1,n).
T(n,n+1) = n + A001791(n).
T(n,n+2) = 1 + A002694(n), n >= 2.
T(n,n+k) = binomial(2*n, n-k) = A094527(n,k), for k >= 3 and n>=k.
G.f.: 1/(1 - y*(x^2 + 2*x + 1)) + (x^2 - 1)/(1 - y*(x + 1)).

A051895 Partial sums of second pentagonal numbers with even index (A049453).

Original entry on oeis.org

0, 7, 33, 90, 190, 345, 567, 868, 1260, 1755, 2365, 3102, 3978, 5005, 6195, 7560, 9112, 10863, 12825, 15010, 17430, 20097, 23023, 26220, 29700, 33475, 37557, 41958, 46690, 51765, 57195, 62992, 69168, 75735, 82705, 90090, 97902, 106153, 114855, 124020, 133660

Offset: 0

Barry E. Williams, Dec 17 1999

For A049453(n+1), the corresponding formula would be a(n)=(n+1)*(6*n+7) and its partial sums would be given by a(n)=(n+1)*(n+2)*(4*n+7)/2.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002492, A016061, A017605, A049453.

I:=[0, 7, 33, 90]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Apr 27 2012

Table[(n(4n-1)(n-1))/2,{n,40}]  (* Harvey P. Dale, Mar 11 2011 *)
CoefficientList[Series[x*(7+5*x)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Apr 27 2012 *)

a(n) = n*(n+1)*(4*n+3)/2; \\ Altug Alkan, Apr 20 2018

a(n) = n*(n+1)*(4*n+3)/2.
G.f.: x*(7+5*x)/(1-x)^4. - Colin Barker, Jan 12 2012
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Vincenzo Librandi, Apr 27 2012
a(n) = A002492(n) + A016061(n). - J. M. Bergot, Apr 20 2018

A116955 a(n+1) = a(n) + (if a(n) is odd then (next odd square) else (next even square)), a(0) = 1.

Original entry on oeis.org

1, 10, 14, 30, 66, 130, 230, 374, 570, 826, 1150, 1550, 2034, 2610, 3286, 4070, 4970, 5994, 7150, 8446, 9890, 11490, 13254, 15190, 17306, 19610, 22110, 24814, 27730, 30866, 34230, 37830, 41674, 45770, 50126, 54750, 59650, 64834, 70310, 76086, 82170, 88570

Offset: 0

Reinhard Zumkeller, Mar 15 2006

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002492.

A116955:=n->2*(15+n-3*n^2+2*n^3)/3: 1,seq(A116955(n), n=1..60); # Wesley Ivan Hurt, Feb 12 2017

LinearRecurrence[{4,-6,4,-1},{1,10,14,30,66},50] (* Harvey P. Dale, Aug 21 2016 *)

a(n+1) = A002492(n) + 10.

a(n) = 2*(15+n-3*n^2+2*n^3)/3 for n>0. G.f.: -(9*x^4-30*x^3+20*x^2-6*x-1) / (x-1)^4. - Colin Barker, Jul 18 2013

More terms from Colin Barker, Jul 18 2013

A168574 a(n) = (4n + 3)(1 + 2*n^2)/3.

Original entry on oeis.org

1, 7, 33, 95, 209, 391, 657, 1023, 1505, 2119, 2881, 3807, 4913, 6215, 7729, 9471, 11457, 13703, 16225, 19039, 22161, 25607, 29393, 33535, 38049, 42951, 48257, 53983, 60145, 66759, 73841, 81407, 89473, 98055, 107169, 116831, 127057, 137863, 149265, 161279

Offset: 0

Paul Curtz, Nov 30 2009

Binomial transform of quasi-finite sequence 1, 6, 20, 16, 0, 0, ... (0 continued).

a(n+1) is the sum of the first and last number at the bottom (2nd row) of each block in A172002, 3+4, 13+20, 39+56, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A168547, A168234.

[(4*n+3)*(1+2*n^2)/3 : n in [0..40]]; // Vincenzo Librandi, Aug 06 2011

Table[ (4*n+3)*(1+2*n^2)/3 , {n,0,25}] (* G. C. Greubel, Jul 26 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,7,33,95},40] (* Harvey P. Dale, May 16 2019 *)

a(n)=(4*n+3)*(1+2*n^2)/3 \\ Charles R Greathouse IV, Jul 26 2016

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 16.
a(n) = A168582(2*n+1) .
a(n+1) = A166911(n) + A002492(n+1).
G.f.: (1 + 3*x + 11*x^2 + x^3)/(1 - x)^4.
E.g.f.: (1/3)*(3 + 18*x + 30*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Jul 26 2016

Edited and extended by R. J. Mathar, Mar 25 2010

A249120 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A210843 multiplied by A000330(k), and the first element of column k is in row A000217(k).

Original entry on oeis.org

1, 4, 13, 5, 35, 20, 86, 65, 194, 175, 14, 415, 430, 56, 844, 970, 182, 1654, 2075, 490, 3133, 4220, 1204, 30, 5773, 8270, 2716, 120, 10372, 15665, 5810, 390, 18240, 28865, 11816, 1050, 31449, 51860, 23156, 2580, 53292, 91200, 43862, 5820, 55, 88873, 157245, 80822, 12450, 220, 146095, 266460, 145208, 25320, 715

Offset: 1

Omar E. Pol, Dec 14 2014

Conjecture: gives an identity for the sum of all divisors of all positive integers <= n. Alternating sum of row n equals A024916(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A024916(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Column 1 is A210843.
Column k lists the partial sums of the k-th column of triangle A252117 which gives an identity for sigma.
The first element of column k is A000330(k).
The second element of column k is A002492(k).

			Triangle begins:
       1;
       4;
      13,       5;
      35,      20;
      86,      65;
     194,     175,      14;
     415,     430,      56;
     844,     970,     182;
    1654,    2075,     490;
    3133,    4220,    1204,     30;
    5773,    8270,    2716,    120;
   10372,   15665,    5810,    390;
   18240,   28865,   11816,   1050;
   31449,   51860,   23156,   2580;
   53292,   91200,   43862,   5820,    55;
   88873,  157245,   80822,  12450,   220;
  146095,  266460,  145208,  25320,   715;
  236977,  444365,  255360,  49620,  1925;
  379746,  730475,  440286,  93990,  4730;
  601656, 1184885,  746088, 173190, 10670;
  943305, 1898730, 1244222, 311160, 22825,   91;
  ...
For n = 6 the sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 1 + 3 + 4 + 7 + 6 + 12 = 33. On the other hand the 6th row of triangle is 194, 175, 14, so the alternating row sum is 194 - 175 + 14 = 33, equaling the sum of all divisors of all positive integers <= 6.

Cf. A000203, A000217, A000330, A002492, A003056, A024916, A195825, A196020, A210843, A211970, A236104, A252117.

A294630 Partial sums of A294629.

Original entry on oeis.org

4, 20, 48, 104, 172, 292, 424, 616, 844, 1140, 1448, 1888, 2340, 2876, 3488, 4224, 4972, 5892, 6824, 7936, 9140, 10460, 11792, 13416, 15092, 16900, 18816, 20960, 23116, 25612, 28120, 30880, 33764, 36812, 39968, 43568, 47180, 50972, 54904, 59240, 63588, 68372, 73168, 78288, 83676, 89276, 94888, 101112

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

a(n) is also the volume of a stepped pyramid with n levels which is another version of the stepped pyramid described in A244050. Both pyramids have the same top view and the same front view, that is to say externally both pyramids are equal, but this pyramid with n levels contains a central chamber whose volume is 4*A072481(n). For more information about the central chamber see the diagrams in A294629.

a(n) is the number of unit cubes of the pyramid with n levels.

			Illustration of the top view of the pyramid with 16 levels and 4224 unit cubes:
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the stepped pyramid. For more information about the hidden pattern see A237593 and A245092.

Muniru A Asiru, Table of n, a(n) for n = 1..2000

For other related diagrams see A294016 and A294629.

Cf. A000203, A002492, A004125, A072481, A237593, A239050, A243980, A244050, A245092, A294015, A294017, A294628.

List([1..50],n->Sum([1..n],m->Sum([1..m],k->8*(Sigma(k)-k+(1/2))))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sum(sum(8*(sigma(j)-j+(1/2)),j=1..k),k=1..n),n=1..50); # Muniru A Asiru, Mar 04 2018

f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@ Accumulate@ Array[f, 48] (* Robert G. Wilson v, Dec 12 2017 *)

from math import isqrt
def A294630(n): return ((((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1, s+1))<<2)-(n*(n+1)*((n<<1)+1)<<1))//3 # Chai Wah Wu, Nov 01 2023

a(n) = 4*A294017(n).
a(n) = A002492(n) - 8*A072481(n).
a(n) = A244050(n) - 4*A072481(n).

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

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

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A300758 a(n) = 2n(n+1)(2n+1).

Original entry on oeis.org

0, 12, 60, 168, 360, 660, 1092, 1680, 2448, 3420, 4620, 6072, 7800, 9828, 12180, 14880, 17952, 21420, 25308, 29640, 34440, 39732, 45540, 51888, 58800, 66300, 74412, 83160, 92568, 102660, 113460, 124992, 137280, 150348, 164220, 178920, 194472, 210900, 228228

Offset: 0

Christopher Purcell, Mar 12 2018

The altitude h(n) = a(n)/A001844(n) of the (A005408(n), A046092(n) and A001844(n)) rectangular triangle is an irreducible fraction. - Ralf Steiner, Feb 25 2020

In this case, area A = a(n)/2 = A055112(n). - Bernard Schott, Feb 27 2020

S. P. Borgatti and M. G. Everett, Notions of Position in Social Network Analysis, Sociological Methodology, 22 (1992), 1-35.
C. Purcell and P. Rombach, Role colouring graphs in hereditary classes, arXiv:1802.10180 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A002492, A006331, A055112, A059270, A005408, A046092, A001844, A000384, A016754, A060300.

Cf. A027480.

a(n) = 12*A000330(n).
G.f.: 12*x*(1+x)/(1-x)^4. - Colin Barker, Mar 12 2018
a(n) = 6*A006331(n) = 4*A059270(n) = 3*A002492(n) = 2*A055112(n). - Omar E. Pol, Apr 04 2018
From Ralf Steiner, Feb 27 2020: (Start)
a(n) = 2*n*A000384(n+1).
a(n) = sqrt(A016754(n)*A060300(n)).
(End)
a(n) = A005408(n) * A046092(n). - Bruce J. Nicholson, Apr 24 2020

Edited by N. J. A. Sloane, Aug 01 2019

A181773 Molecular topological indices of the cocktail party graphs.

Original entry on oeis.org

0, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, 13680, 18480, 24288, 31200, 39312, 48720, 59520, 71808, 85680, 101232, 118560, 137760, 158928, 182160, 207552, 235200, 265200, 297648, 332640, 370272, 410640

Offset: 1

Eric W. Weisstein, Jul 10 2011

a(n) is the number of 2 X 2 matrices (all four elements distinct) having entries in {-n,...,0,...,n} with determinant equal to the permanent. - Indranil Ghosh, Dec 25 2016

Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A280059 (2 X 2 matrices, elements can be repeated).

A181773:=n->8*(n-1)*n*(2*n-1): seq(A181773(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2017

a(n) = 8*(n-1)*n*(2n-1).
a(n) = 16*A059270(n-1).
G.f.: 48*x^2*(x+1)/(x-1)^4. - Colin Barker, Oct 17 2012
a(n) = 48*A000330(n-1). - R. J. Mathar, Jan 04 2017
From Omar E. Pol, Jan 05 2017: (Start)
a(n) = 24*A006331(n-1) = 12*A002492(n-1) = 8*A055112(n-1).
a(n) = 2*A069074(n-2), n >= 2. (End)

cp's OEIS Frontend

A188777 T(n,k) = Number of n-turn bishop's tours on a k X k board summed over all starting positions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A300192 Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Maxima

PARI

Formula

A051895 Partial sums of second pentagonal numbers with even index (A049453).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A116955 a(n+1) = a(n) + (if a(n) is odd then (next odd square) else (next even square)), a(0) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A168574 a(n) = (4*n + 3)*(1 + 2*n^2)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A249120 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A210843 multiplied by A000330(k), and the first element of column k is in row A000217(k).

Views

Author

Keywords

Comments

Examples

Crossrefs

A294630 Partial sums of A294629.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Python

A300192 Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2x + 1)^n + (x^2 - 1)(x + 1)^n.

A168574 a(n) = (4n + 3)(1 + 2*n^2)/3.

A294774 a(n) = 2n^2 + 2n + 5.

A300758 a(n) = 2n(n+1)(2n+1).