A045944 -id:A045944 - cp's OEIS frontend

A212195 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the staggered hexagonal square grid graph SH_(k,k).

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 48, 5, 0, 0, 6, 1056, 180, 6, 0, 0, 6, 45696, 32940, 480, 7, 0, 0, 6, 4038432, 30847500, 393600, 1050, 8, 0, 0, 6, 743601024, 148046704020, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 03 2012

The staggered hexagonal square grid graph SH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges; see A212194 for example. The chromatic polynomial of SH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212163 first at (n,k) = (4,5): A(4,5) = 4038432, A212163(4,5) = 4034304.

			Square array A(n,k) begins:
  1,    0,       0,            0,                 0, ...
  2,    0,       0,            0,                 0, ...
  3,    6,       6,            6,                 6, ...
  4,   48,    1056,        45696,           4038432, ...
  5,  180,   32940,     30847500,      148046704020, ...
  6,  480,  393600,   3312560640,   286170443437440, ...
  7, 1050, 2735250, 123791435250, 97337320223288250, ...

Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068248, 6*A068249.
Rows n=1-10, 16-18 give: A000007, A000038, A040006, 4*A068283, 5*A068284, 6*A068285, 7*A068286, 8*A068287, 9*A068288, 10*A068289, 16*A068290, 17*A068291, 18*A068292.
Cf. A212163, A212194.

A140676 a(n) = n(3n + 4).

Original entry on oeis.org

0, 7, 20, 39, 64, 95, 132, 175, 224, 279, 340, 407, 480, 559, 644, 735, 832, 935, 1044, 1159, 1280, 1407, 1540, 1679, 1824, 1975, 2132, 2295, 2464, 2639, 2820, 3007, 3200, 3399, 3604, 3815, 4032, 4255, 4484, 4719, 4960, 5207, 5460, 5719, 5984, 6255, 6532, 6815

Offset: 0

Omar E. Pol, May 22 2008

The number of peers of a cell of an n^2 X n^2 sudoku is a(n-1). - Neven Sajko, Apr 20 2016

First differences are in A016921. - Wesley Ivan Hurt, Apr 21 2016

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A016921, A033428, A045944, A067707, A067725, A140677, A140678, A140679, A140680, A140681, A140689.

[n*(3*n+4) : n in [0..80]]; // Wesley Ivan Hurt, Apr 21 2016

A140676:=n->n*(3*n+4): seq(A140676(n), n=0..100); # Wesley Ivan Hurt, Apr 21 2016

Table[n (3 n + 4), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 7, 20}, 50] (* Harvey P. Dale, May 04 2013 *)

a(n)=n*(3*n+4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 4*n.
a(n) = 6*n + a(n-1) + 1 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
O.g.f.: x*(7 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Harvey P. Dale, May 04 2013
E.g.f.: x*(7 + 3*x)*exp(x). - Ilya Gutkovskiy, Apr 20 2016
a(n) = A000567(n+1) - 1. - Neven Sajko, Apr 20 2016
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 15/16 - Pi/(8*sqrt(3)) - 3*log(3)/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 9/16 - Pi/(4*sqrt(3)). (End)

A140681 a(n) = 3n(n+6).

Original entry on oeis.org

0, 21, 48, 81, 120, 165, 216, 273, 336, 405, 480, 561, 648, 741, 840, 945, 1056, 1173, 1296, 1425, 1560, 1701, 1848, 2001, 2160, 2325, 2496, 2673, 2856, 3045, 3240, 3441, 3648, 3861, 4080, 4305, 4536, 4773, 5016, 5265, 5520, 5781

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028560, A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140680, A140689, A000217, A005563, A067728, A140091, A212331.

A140681:=n->3*n*(n+6); seq(A140681(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[3n(n+6), {n,0,100}] (* Wesley Ivan Hurt, Dec 10 2013 *)
LinearRecurrence[{3,-3,1},{0,21,48},50] (* Harvey P. Dale, May 03 2023 *)

a(n)=3*n*(n+6) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A028560(n)*3 = 3*n^2 + 18*n = n*(3*n+18).
a(n) = 6*n + a(n-1) + 15 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
from G. C. Greubel, Jul 20 2017: (Start)
G.f.: 3*x*(7 - 5*x)/(1-x)^3.
E.g.f.: 3*x*(x+7)*exp(x). (End)
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 49/360.
Sum_{n>=1} (-1)^(n+1)/a(n) = 37/1080. (End)

A061045 Numerator of 1/36 - 1/n^2.

Original entry on oeis.org

-35, -2, -1, -5, -11, 0, 13, 7, 5, 4, 85, 1, 133, 10, 7, 55, 253, 2, 325, 91, 5, 28, 493, 5, 589, 40, 77, 187, 805, 2, 925, 247, 13, 70, 1189, 35, 1333, 88, 55, 391, 1645, 4, 1813, 475, 221, 130, 2173, 7, 2365, 154, 95, 667, 2773, 20, 2989, 775, 119, 208, 3445, 11, 3685, 238, 437, 1015, 4189, 10

Offset: 1

N. J. A. Sloane, May 26 2001

Sixth case of Rydberg's formula. From Humphrey's spectrum of hydrogen. See A045944, A000567, A061043, A061046, A061047. - Paul Curtz, Dec 08 2008

			The fractions are -35/36, -2/9, -1/12, -5/144, -11/900, 0, 13/1764, 7/576, 5/324, 4/225, 85/4356, 1/48, 133/6084, 10/441, 7/300, 55/2304, 253/10404, 2/81, 325/12996, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Curtis J. Humphreys, The Sixth Series in the Hydrogen Spectrum, Journal of the Optical Society of America, 1952, 42, p. 432.

A061046 gives denominators.

Cf. A000567, A045944, A061035 - A061050.

import Data.Ratio ((%), numerator)
a061045 = numerator . (1 % 36 -) . recip . (^ 2) . fromIntegral
-- Reinhard Zumkeller, Jan 06 2014

[Numerator(1/6^2 -1/n^2): n in [1..80]]; // G. C. Greubel, Feb 24 2023

Numerator[(1/36-1/Range[100]^2)] (* Harvey P. Dale, Mar 17 2013 *)

def A061045(n): return ((n^2-36)/(6*n)^2).numerator()
[A061045(n) for n in range(1,81)] # G. C. Greubel, Feb 24 2023

A160378 a(n) = n^3 - n*(n+1)/2.

Original entry on oeis.org

0, 0, 5, 21, 54, 110, 195, 315, 476, 684, 945, 1265, 1650, 2106, 2639, 3255, 3960, 4760, 5661, 6669, 7790, 9030, 10395, 11891, 13524, 15300, 17225, 19305, 21546, 23954, 26535, 29295, 32240, 35376, 38709, 42245, 45990, 49950, 54131, 58539

Offset: 0

Gil Broussard, May 11 2009

n-th cube (A000578(n)) minus n-th triangular number (A000217(n)).
Partial sums of A045944. - Vladimir Joseph Stephan Orlovsky, Jun 25 2009
The sum of the n-1 numbers between n^2 and n*(n+1) = a(n). - J. M. Bergot, Apr 15 2013
Use the terms in A061885 to form the antidiagonals for an array. The antidiagonals begin: 0;2,3;6,7,8;12,13,14,15;20,21,22,23,24,25.  The sum of the terms in these antidiagonals = a(n)for n > 0. - J. M. Bergot, Jul 08 2013
a(n) is the sum of the n numbers strictly between n^2-n-1 and n^2. - Charlie Marion, Feb 21 2020

			a(4) = 4^3 - 4*5/2 = 64 - 10 = 54.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic and B. Petkovic, A counting function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A000578, A045944.

Magma
```
[ n^3-n*(n+1)/2: n in [0..50] ];
```

Table[n^3 - n*(n+1)/2, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jun 25 2009 *)

a(n)=n^3-n*(n+1)/2 \\ Charles R Greathouse IV, Oct 18 2022

[n^3 -binomial(n+1,2) for n in range(41)] # G. C. Greubel, Oct 14 2023

a(n) = (2*n^3 - n^2 - n)/2. - Vincenzo Librandi, Dec 12 2010; edited by Klaus Brockhaus, Dec 12 2010
From Chai Wah Wu, Aug 03 2022: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
G.f.: x^2*(5 + x)/(1 - x)^4. (End)
E.g.f.: (x^2/2)*(5 + 2*x)*exp(x). - G. C. Greubel, Oct 14 2023

Definition clarified and offset changed from 1 to 0 by Klaus Brockhaus, Dec 12 2010

A165806 a(n) = 15n^2 + 3n + 1.

Original entry on oeis.org

19, 67, 145, 253, 391, 559, 757, 985, 1243, 1531, 1849, 2197, 2575, 2983, 3421, 3889, 4387, 4915, 5473, 6061, 6679, 7327, 8005, 8713, 9451, 10219, 11017, 11845, 12703, 13591, 14509, 15457, 16435, 17443, 18481, 19549, 20647, 21775, 22933, 24121, 25339

Offset: 1

A.K. Devaraj, Sep 28 2009

Polynomials f(x) have the following property: f(x + n*f(x)) is congruent to f(x); here n is an integer.
This can be proved by Taylor's theorem.
After rationalization of the denominator, the quotient q(n,x) = f(x + n*f(x))/f(x) consists of two parts:
a) a rational integer and b) an irrational part.
The present sequence is the integer part for f(x) = x^2 + 3x + 13 and x = sqrt(2), i.e., q(n,x) = a(n) + sqrt(2)*A045944(n).

			When we substitute sqrt(2) for x in the quadratic expression x^2 + 3x + 13 we get 15 + 3*sqrt(2).
sqrt(2) + (15 + 3*sqrt(2)) = (15 + 4*sqrt(2)). When this is substituted in f(x) we get 270 + 132*sqrt(2).
(270 + 132*sqrt(2))/(15+3*sqrt(2)) = 19 + 5*sqrt(2).

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

Cf. A005563, A045944.

[15*n^2 + 3*n + 1: n in [1..50]]; // Vincenzo Librandi, Sep 29 2011

LinearRecurrence[{3,-3,1},{19,67,145}, 100] (* G. C. Greubel, Apr 08 2016 *)
Table[15n^2+3n+1,{n,50}] (* Harvey P. Dale, Mar 14 2020 *)

a(n)=15*n^2+3*n+1 \\ Charles R Greathouse IV, Sep 28 2011

G.f.: x*(19 + 10*x + x^2)/(1-x)^3. - R. J. Mathar, Sep 29 2009
From G. C. Greubel, Apr 08 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (15*x^2 + 18*x + 1)*exp(x). (End)

Definition simplified, sequence extended by R. J. Mathar, Sep 29 2009

A140677 a(n) = n(3n + 8).

Original entry on oeis.org

0, 11, 28, 51, 80, 115, 156, 203, 256, 315, 380, 451, 528, 611, 700, 795, 896, 1003, 1116, 1235, 1360, 1491, 1628, 1771, 1920, 2075, 2236, 2403, 2576, 2755, 2940, 3131, 3328, 3531, 3740, 3955, 4176, 4403, 4636, 4875, 5120, 5371, 5628

Offset: 0

Omar E. Pol, May 22 2008

			a(1) = 6*1 + 0 + 5 = 11; a(2) = 6*2 + 11 + 5 = 28; a(3) = 6*3 + 28 + 5 = 51. - _Vincenzo Librandi_, Aug 03 2010

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033428, A045944, A140676, A067725, A140678, A067707, A140679, A140680, A140681, A140689.

Table[n(3n+8),{n,0,45}]  (* Harvey P. Dale, Feb 20 2011 *)

a(n)=n*(3*n+8) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 8*n.
a(n) = 6*n + a(n-1) + 5, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(11 - 5*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (3*x^2 + 11*x)*exp(x). - G. C. Greubel, Jul 20 2017

A168244 a(n) = 1 + 3n - 2n^2.

Original entry on oeis.org

1, 2, -1, -8, -19, -34, -53, -76, -103, -134, -169, -208, -251, -298, -349, -404, -463, -526, -593, -664, -739, -818, -901, -988, -1079, -1174, -1273, -1376, -1483, -1594, -1709, -1828, -1951, -2078, -2209, -2344, -2483, -2626, -2773, -2924, -3079, -3238, -3401, -3568, -3739, -3914, -4093, -4276, -4463, -4654, -4849

Offset: 0

A.K. Devaraj, Nov 21 2009

Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients f(x + n*f(x))/f(x), as in A168235 and A168240. a(n) is the real part of the quotient at x = 1+sqrt(-5).
The imaginary part of the quotient is sqrt(5)*A045944(n).
As stated in short description of A168244 the quotient is in two parts: rational integers (cf. A168244) and rational integer multiples of sqrt(-5). It so happens that the sequence of rational integer coefficients of sqrt(-5) is A045944. - A.K. Devaraj, Nov 22 2009
This sequence contains half of all integers m such that -8*m +17 is an odd square. The other half are found in A091823 multiplied by -1. The squares resulting from A168244 are (4*n - 3)^2, those from A091823 are (4*n + 3)^2. - Klaus Purath, Jul 11 2021

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

Cf. A168235, A168240, A165806, A165808, A165809, A045944, A091823.

[1+3*n-2*n^2: n in [1..50]]; // Vincenzo Librandi, Jul 10 2012

Table[-(n + (n + 1)^2 - 3)/2, {n, -1, 200, 2}]  (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
LinearRecurrence[{3,-3,1},{2,-1,-8},60] (* Harvey P. Dale, Jun 06 2015 *)

a(n) = 1 + 3*n - 2*n^2; \\ Altug Alkan, Apr 09 2016

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 1 + x*(2-7*x+x^2)/(1-x)^3.
a(-n) = -A091823(n), a(0) = 1. - Michael Somos, May 11 2014
E.g.f.: (1 + x - 2*x^2)*exp(x). - G. C. Greubel, Apr 09 2016
a(n) = a(n-2) + (-2)*sqrt((-8)*a(n-1) + 17), n > 1. - Klaus Purath, Jul 08 2021

Edited, definition simplified, sequence extended beyond a(5) by R. J. Mathar, Nov 23 2009

a(0)=1 added by N. J. A. Sloane, Apr 09 2016

A187507 T(n,k)=Number of n-step S, E, and NW-moving king's tours on a kXk board summed over all starting positions.

Original entry on oeis.org

1, 4, 0, 9, 5, 0, 16, 16, 6, 0, 25, 33, 31, 2, 0, 36, 56, 74, 36, 0, 0, 49, 85, 135, 115, 40, 0, 0, 64, 120, 214, 236, 184, 36, 0, 0, 81, 161, 311, 399, 435, 272, 20, 0, 0, 100, 208, 426, 604, 788, 772, 330, 12, 0, 0, 121, 261, 559, 851, 1243, 1525, 1224, 390, 6, 0, 0, 144, 320, 710

Offset: 1

R. H. Hardin Mar 10 2011

Table starts
.1.4..9..16...25....36....49....64....81....100....121....144....169....196
.0.5.16..33...56....85...120...161...208....261....320....385....456....533
.0.6.31..74..135...214...311...426...559....710....879...1066...1271...1494
.0.2.36.115..236...399...604...851..1140...1471...1844...2259...2716...3215
.0.0.40.184..435...788..1243..1800..2459...3220...4083...5048...6115...7284
.0.0.36.272..772..1525..2524..3769..5260...6997...8980..11209..13684..16405
.0.0.20.330.1224..2726..4807..7458.10679..14470..18831..23762..29263..35334
.0.0.12.390.1910..4880..9250.14969.22026..30421..40154..51225..63634..77381
.0.0..6.450.2872..8522.17564.29834.45255..63814..85511.110346.138319.169430
.0.0..0.398.3868.13796.31548.56952.89684.129637.176796.231161.292732.361509

			Some n=4 solutions for 4X4
..0..0..0..0....0..0..0..0....1..0..0..0....0..0..0..0....0..0..0..0
..0..0..0..0....0..0..0..0....2..0..0..0....0..0..0..0....2..3..4..0
..0..4..2..0....0..3..4..0....3..0..0..0....3..1..0..0....0..1..0..0
..0..0..3..1....0..1..2..0....4..0..0..0....4..2..0..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..290

Row 2 is A045944(n-1)

Empirical: T(1,k) = k^2
Empirical: T(2,k) = 3*k^2 - 4*k + 1
Empirical: T(3,k) = 9*k^2 - 20*k + 10 for k>1
Empirical: T(4,k) = 21*k^2 - 68*k + 51 for k>2
Empirical: T(5,k) = 51*k^2 - 208*k + 200 for k>3
Empirical: T(6,k) = 123*k^2 - 600*k + 697 for k>4
Empirical: T(7,k) = 285*k^2 - 1624*k + 2210 for k>5
Empirical: T(8,k) = 669*k^2 - 4316*k + 6681 for k>6
Empirical: T(9,k) = 1569*k^2 - 11252*k + 19434 for k>7
Empirical: T(10,k) = 3603*k^2 - 28504*k + 54377 for k>8

A140678 a(n) = n(3n + 10).

Original entry on oeis.org

0, 13, 32, 57, 88, 125, 168, 217, 272, 333, 400, 473, 552, 637, 728, 825, 928, 1037, 1152, 1273, 1400, 1533, 1672, 1817, 1968, 2125, 2288, 2457, 2632, 2813, 3000, 3193, 3392, 3597, 3808, 4025, 4248, 4477, 4712, 4953, 5200, 5453, 5712

Offset: 0

Omar E. Pol, May 22 2008

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

Cf. A033428, A045944, A140676, A067725, A140677, A067707, A140679, A140680, A140681, A140689.

Table[n (3 n + 10), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 13, 32}, 50] (* Harvey P. Dale, Jun 05 2012 *)

a(n)=n*(3*n+10) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 10*n.
a(n) = 6*n + a(n-1) + 7, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(13 - 7*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=13, a(2)=32. - Harvey P. Dale, Jun 05 2012
E.g.f.: (3*x^2 + 13*x)*exp(x). - G. C. Greubel, Jul 20 2017

cp's OEIS Frontend

A212195 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the staggered hexagonal square grid graph SH_(k,k).

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A140676 a(n) = n*(3*n + 4).

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A140681 a(n) = 3*n*(n+6).

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A061045 Numerator of 1/36 - 1/n^2.

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A160378 a(n) = n^3 - n*(n+1)/2.

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Magma

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A165806 a(n) = 15n^2 + 3n + 1.

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A140677 a(n) = n*(3*n + 8).

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A140676 a(n) = n(3n + 4).

A140681 a(n) = 3n(n+6).

A140677 a(n) = n(3n + 8).

A168244 a(n) = 1 + 3n - 2n^2.

A140678 a(n) = n(3n + 10).