A155955 -id:A155955 - cp's OEIS frontend

A016742 Even squares: a(n) = (2*n)^2.

0, 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 6084, 6400, 6724, 7056, 7396, 7744, 8100, 8464

Offset: 0

N. J. A. Sloane

4 times the squares.
Number of edges in the complete bipartite graph of order 5n, K_{n,4n} - Roberto E. Martinez II, Jan 07 2002
It is conjectured (I think) that a regular Hadamard matrix of order n exists iff n is an even square (cf. Seberry and Yamada, Th. 10.11). A Hadamard matrix is regular if the sum of the entries in each row is the same. - N. J. A. Sloane, Nov 13 2008
Sequence arises from reading the line from 0, in the direction 0, 16, ... and the line from 4, in the direction 4, 36, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008
The entries from a(1) on can be interpreted as pair sums of (2, 2), (8, 8), (18, 18), (32, 32) etc. that arise from a re-arrangement of the subshell orbitals in the periodic table of elements. 8 becomes the maximum number of electrons in the (2s,2p) or (3s,3p) orbitals, 18 the maximum number of electrons in (4s,3d,4p) or (5s,3d,5p) shells, for example. - Julio Antonio Gutiérrez Samanez, Jul 20 2008
The first two terms of the sequence (n=1, 2) give the numbers of chemical elements using only n types of atomic orbitals, i.e., there are a(1)=4 elements (H,He,Li,Be) where electrons reside only on s-orbitals, there are a(2)=16 elements (B,C,N,O,F,Ne,Na,Mg,Al,Si,P,S,Cl,Ar,K,Ca) where electrons reside only on s- and p-orbitals. However, after that, there is 37 (which is one more than a(3)=36) elements (from Sc, Scandium, atomic number 21 to La, Lanthanum, atomic number 57) where electrons reside only on s-, p- and d-orbitals. This is because Lanthanum (with the electron configuration [Xe]5d^1 6s^2) is an exception to the Aufbau principle, which would predict that its electron configuration is [Xe]4f^1 6s^2. - Antti Karttunen, Aug 14 2008.
Number of cycles of length 3 in the king's graph associated with an (n+1) X (n+1) chessboard. - Anton Voropaev (anton.n.voropaev(AT)gmail.com), Feb 01 2009
a(n+1) is the molecular topological index of the n-star graph S_n. - Eric W. Weisstein, Jul 11 2011
a(n) is the sum of two consecutives odd numbers 2*n^2-1 and 2*n^2+1 and the difference of two squares (n^2+1)^2 - (n^2-1)^2. - Pierre CAMI, Jan 02 2012
For n > 3, a(n) is the area of the irregular quadrilateral created by the points ((n-4)*(n-3)/2,(n-3)*(n-2)/2), ((n-2)*(n-1)/2,(n-1)*n/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+3)*(n+4)/2,(n+2)*(n+3)/2). - J. M. Bergot, May 27 2014
Number of terms less than 10^k: 1, 2, 5, 16, 50, 159, 500, 1582, 5000, 15812, 50000, 158114, 500000, ... - Muniru A Asiru, Jan 28 2018
Right-hand side of the binomial coefficient identity Sum_{k = 0..2*n} (-1)^(k+1)* binomial(2*n,k)*binomial(2*n + k,k)*(2*n - k) = a(n). - Peter Bala, Jan 12 2022

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
Seberry, Jennifer and Yamada, Mieko; Hadamard matrices, sequences and block designs, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431-560, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
W. D. Wallis, Anne Penfold Street and Jennifer Seberry Wallis, Combinatorics: Room squares, sum-free sets, Hadamard matrices, Lecture Notes in Mathematics, Vol. 292, Springer-Verlag, Berlin-New York, 1972. iv+508 pp.

Vincenzo Librandi, Table of n, a(n) for n = 0..900
R. P. Boas and N. J. A. Sloane, Correspondence, 1974.
Leo Tavares, Illustration: X Squares
Various, Electron Configuration (Discussion in Physics Forums).
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Molecular Topological Index.
Wikipedia, Aufbau principle.
Index entries for sequences related to Hadamard matrices
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A001105, A001539, A016754, A016802, A016814, A016826, A016838, A007742, A033991, A245058.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. sequences listed in A254963.
Other n X n king graph cycle counts: A288918 (4-cycles), A288919 (5-cycles), A288920 (6-cycles).
Cf. A000384, A014105.
Cf. A016813.

List([0..100], n -> (2*n)^2); # Muniru A Asiru, Jan 28 2018

a016742 = (* 4) . (^ 2)
a016742_list = 0 : map (subtract 4) (zipWith (+) a016742_list [8, 16 ..])
-- Reinhard Zumkeller, Jun 28 2015, Apr 20 2015

[(2*n)^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

seq((2*n)^2, n=0..100); # Muniru A Asiru, Jan 28 2018

Table[(2n)^2, {n, 0, 46}] (* Alonso del Arte, Apr 26 2011 *)

makelist((2*n)^2,n,0,20); /* Martin Ettl, Jan 22 2013 */

a(n)=4*n^2 \\ Charles R Greathouse IV, Jul 28 2015

O.g.f.: 4*x*(1+x)/(1-x)^3. - R. J. Mathar, Jul 28 2008
a(n) = A000290(n)*4 = A001105(n)*2. - Omar E. Pol, May 21 2008
a(n) = A155955(n,2) for n > 1. - Reinhard Zumkeller, Jan 31 2009
Sum_{n>=1} 1/a(n) = (1/4)*Pi^2/6 = Pi^2/24. - Ant King, Nov 04 2009
a(n) = a(n-1) + 8*n - 4 (with a(0)=0). - Vincenzo Librandi, Nov 19 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 4, a(2) = 16. - Philippe Deléham, Mar 26 2013
a(n) = A118729(8n+3). - Philippe Deléham, Mar 26 2013
Pi = 2*Product_{n>=1} (1 + 1/(a(n)-1)). - Adriano Caroli, Aug 04 2013
Pi = Sum_{n>=0} 8/(a(2n+1)-1). - Adriano Caroli, Aug 06 2013
E.g.f.: exp(x)*(4x^2 + 4x). - Geoffrey Critzer, Oct 07 2013
a(n) = A000384(n) + A014105(n). - Bruce J. Nicholson, Nov 11 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 (A245058). - Amiram Eldar, Oct 10 2020
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/2)/(Pi/2) (A308716).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/2)/(Pi/2) = 2/Pi (A060294). (End)
a(n) = A016754(n) - A016813(n). - Leo Tavares, Feb 24 2022

More terms from Sabir Abdus-Samee (sabdulsamee(AT)prepaidlegal.com), Mar 13 2006

A062206 a(n) = n^(2n).

Original entry on oeis.org

1, 1, 16, 729, 65536, 9765625, 2176782336, 678223072849, 281474976710656, 150094635296999121, 100000000000000000000, 81402749386839761113321, 79496847203390844133441536, 91733330193268616658399616009, 123476695691247935826229781856256

Offset: 0

Jason Earls, Jun 13 2001

a(n) is also the number of sequences of length 2n on n symbols. - Washington Bomfim, Oct 06 2009

a(n) is the number of endofunctions on [n] that map each even number to an even number and each odd number to an odd number. - Enrique Navarrete, Sep 30 2022

Winston de Greef, Table of n, a(n) for n = 0..213 (first 101 terms from Harry J. Smith)

Cf. A062207, A086648, A155957.
Cf. A000290, A000312, A155955.
Cf. A085741, A212333.
Column k=0 of A245910 and A245980.

f[n_]:=n^(2*n); Join[{1},f[Range[20]]] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n) = n^(2*n); \\ Harry J. Smith, Aug 02 2009

def A062206(n): return n**(n<<1) # Chai Wah Wu, Nov 18 2022

a(n) = A000312(n)^2 = A000290(n)^n.
(-1)^n*determinant of the 2n X 2n matrix M_(i, j) = i+j if (i + j) is a multiple of n, M_(i, j) = 1 otherwise. - Benoit Cloitre, Aug 06 2003
a(n) = A155955(n,n) = A000290(A000312(n)). - Reinhard Zumkeller, Jan 31 2009
a(n) = n! * [x^n] 1/(1 + LambertW(-n*x)). - Ilya Gutkovskiy, Oct 03 2017
Sum_{n>=1} 1/a(n) = A086648. - Amiram Eldar, Nov 16 2020

Initial term corrected by Reinhard Zumkeller, Jan 30 2009

A016767 a(n) = (3*n)^3.

Original entry on oeis.org

0, 27, 216, 729, 1728, 3375, 5832, 9261, 13824, 19683, 27000, 35937, 46656, 59319, 74088, 91125, 110592, 132651, 157464, 185193, 216000, 250047, 287496, 328509, 373248, 421875, 474552, 531441, 592704

Offset: 0

N. J. A. Sloane

			G.f. = 27*x + 216*x^2 + 729*x^3 * 1728*x^4 + 3375*x^5 + 5832*x^6 + ... - _Michael Somos_, Sep 16 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[(3*n)^3: n in [0..40]]; // Vincenzo Librandi, Jul 29 2011

Table[(3*n)^3, {n,0,30}] (* G. C. Greubel, Sep 15 2018 *)

a(n)=27*n^3 \\ Charles R Greathouse IV, Jul 29 2011

From Reinhard Zumkeller, Jan 29 2009: (Start)
a(n) = 27*A000578(n).
A020639(a(n)) = A010693(n), for n>0. (End)
a(n) = A155955(n,3) for n>2. - Reinhard Zumkeller, Jan 31 2009
G.f.: 27*x*(1+4*x+x^2)/(x-1)^4 . - R. J. Mathar, Jul 07 2017
E.g.f.: 27*x*(1 + 3*x + x^2)*exp(x). - G. C. Greubel, Sep 15 2018
a(n) = -a(-n) for all n in Z. - Michael Somos, Sep 16 2018

A155956 a(n) = Sum_{k=0..n} (k*n)^k.

Original entry on oeis.org

1, 2, 19, 769, 67333, 9929106, 2201420095, 683765250589, 283214405613321, 150820803395086078, 100389106493001087411, 81663040762574177095289, 79709457342800206602843277, 91941570967455757927336110570, 123717598784707453088183544702311

Offset: 0

Reinhard Zumkeller, Jan 31 2009

nonn

Sums of rows of the triangle in A155955.

G. C. Greubel, Table of n, a(n) for n = 0..210

Cf. A000312.

[&+[(k*n)^k: k in [0..n]]: n in [0..20]]; // Bruno Berselli, Jul 04 2016

Table[Sum[(k*n)^k,{k,n}],{n,0,20}]+1 (* Harvey P. Dale, Apr 04 2017 *)

vector(30, n, n--; sum(k=0,n, (k*n)^k)) \\ G. C. Greubel, Sep 14 2018

a(n) = h(n,n) with h(n,k) = if k<0 then 0 otherwise n*h(n,k-1)+(n-k)^(n-k).

A016804 a(n) = (4*n)^4.

Original entry on oeis.org

0, 256, 4096, 20736, 65536, 160000, 331776, 614656, 1048576, 1679616, 2560000, 3748096, 5308416, 7311616, 9834496, 12960000, 16777216, 21381376, 26873856, 33362176, 40960000, 49787136, 59969536

Offset: 0

N. J. A. Sloane

a(n) = A155955(n,4) for n > 3. - Reinhard Zumkeller, Jan 31 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[(4*n)^4: n in [0..40]]; // Vincenzo Librandi, Jul 29 2011

a(n)=n^4<<8 \\ Charles R Greathouse IV, Jul 29 2011

A016853 a(n) = (5*n)^5.

Original entry on oeis.org

0, 3125, 100000, 759375, 3200000, 9765625, 24300000, 52521875, 102400000, 184528125, 312500000, 503284375, 777600000, 1160290625, 1680700000, 2373046875, 3276800000, 4437053125, 5904900000, 7737809375, 10000000000, 12762815625, 16105100000, 20113571875, 24883200000

Offset: 0

N. J. A. Sloane

a(n) = A155955(n,5) for n > 4. - Reinhard Zumkeller, Jan 31 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A000584.

(5 Range[0, 20])^5 (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 3125, 100000, 759375, 3200000, 9765625}, 20] (* Harvey P. Dale, Jul 11 2015 *)
CoefficientList[Series[3125 x (1 + 26 x + 66 x^2 + 26 x^3 + x^4) / (1-x)^6, {x, 0, 33}], x] (* Vincenzo Librandi, Oct 24 2017 *)

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), with a(0) = 0, a(1) = 3125, a(2) = 100000, a(3) = 759375, a(4) = 3200000, a(5) = 9765625. - Harvey P. Dale, Jul 11 2015
a(n) = 3125*A000584(n). - Michel Marcus, Oct 24 2017
G.f.: 3125*x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^6. - Vincenzo Librandi, Oct 24 2017

A155957 a(n) = (2*n^2)^n.

Original entry on oeis.org

1, 2, 64, 5832, 1048576, 312500000, 139314069504, 86812553324672, 72057594037927936, 76848453272063549952, 102400000000000000000000, 166712830744247830760081408, 325619086145088897570576531456

Offset: 0

Reinhard Zumkeller, Jan 31 2009

nonn

Central terms of the triangle in A155955;

G. C. Greubel, Table of n, a(n) for n = 0..200

Cf. A000079, A062206, A155955.

[(2*n^2)^n: n in [0..30]]; // G. C. Greubel, Sep 14 2018

Table[If[n==0,1,(2*n^2)^n], {n, 0, 30}] (* G. C. Greubel, Sep 14 2018 *)

vector(30, n, n--; (2*n^2)^n) \\ G. C. Greubel, Sep 14 2018

a(n) = A062206(n)*A000079(n).

a(n) = n! * [x^n] 1/(1 + LambertW(-2*n*x)). - Ilya Gutkovskiy, Oct 03 2017

A016914 a(n) = (6*n)^6.

Original entry on oeis.org

0, 46656, 2985984, 34012224, 191102976, 729000000, 2176782336, 5489031744, 12230590464, 24794911296, 46656000000, 82653950016, 139314069504, 225199600704, 351298031616, 531441000000, 782757789696

Offset: 0

N. J. A. Sloane

a(n) = A155955(n,6) for n > 5. - Reinhard Zumkeller, Jan 31 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

[(6*n)^6: n in [0..35]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,20])^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,46656,2985984,34012224,191102976,729000000,2176782336},20] (* Harvey P. Dale, Jan 28 2024 *)

A016987 a(n) = (7*n)^7.

Original entry on oeis.org

0, 823543, 105413504, 1801088541, 13492928512, 64339296875, 230539333248, 678223072849, 1727094849536, 3938980639167, 8235430000000, 16048523266853, 29509034655744, 51676101935731

Offset: 0

N. J. A. Sloane

a(n) = A155955(n,7) for n > 6. - Reinhard Zumkeller, Jan 31 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

[(7*n)^7: n in [0..25]]; // Vincenzo Librandi, Jun 18 2011

(7*Range[0,20])^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,823543,105413504,1801088541,13492928512,64339296875,230539333248,678223072849},20] (* Harvey P. Dale, Apr 01 2013 *)

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8); a(0)=0, a(1)=823543, a(2)=105413504, a(3)=1801088541, a(4)=13492928512, a(5)=64339296875, a(6)=230539333248, a(7)=678223072849. - Harvey P. Dale, Apr 01 2013

A017072 a(n) = (8*n)^8.

Original entry on oeis.org

0, 16777216, 4294967296, 110075314176, 1099511627776, 6553600000000, 28179280429056, 96717311574016, 281474976710656, 722204136308736, 1677721600000000, 3596345248055296, 7213895789838336, 13685690504052736

Offset: 0

N. J. A. Sloane

a(n) = A155955(n,8) for n > 7. - Reinhard Zumkeller, Jan 31 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

[(8*n)^8: n in [0..35]]; // Vincenzo Librandi, Jul 11 2011

(8*Range[0,20])^8 (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,16777216,4294967296,110075314176,1099511627776,6553600000000,28179280429056,96717311574016,281474976710656},20] (* Harvey P. Dale, May 25 2019 *)

cp's OEIS Frontend

A016742 Even squares: a(n) = (2*n)^2.

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A062206 a(n) = n^(2n).

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A016767 a(n) = (3*n)^3.

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A155956 a(n) = Sum_{k=0..n} (k*n)^k.

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A016804 a(n) = (4*n)^4.

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A016853 a(n) = (5*n)^5.

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A155957 a(n) = (2*n^2)^n.

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