A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.
1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761
Offset: 0
Keywords
A108576 Number of 3 X 3 magic squares (with distinct positive entries) having all entries < n.
0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 40, 64, 96, 128, 184, 240, 320, 400, 504, 608, 744, 880, 1056, 1232, 1440, 1648, 1904, 2160, 2464, 2768, 3120, 3472, 3880, 4288, 4760, 5232, 5760, 6288, 6888, 7488, 8160, 8832, 9576, 10320, 11144, 11968, 12880, 13792, 14784, 15776
Offset: 1
Comments
From Thomas Zaslavsky, Mar 12 2010: (Start)
A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.
a(n) is given by a quasipolynomial of period 12. (End)
Examples
a(10) = 8 because there are 8 3 X 3 magic squares with distinct entries < 10 (they are the standard magic squares).
Links
- T. Zaslavsky, Table of n, a(n) for n = 1..10000.
- M. Beck and T. Zaslavsky, An enumerative geometry for magic and magilatin labellings, arXiv:math/0506315 [math.CO], 2005; Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071. - _Thomas Zaslavsky_, Jan 29 2010
- M. Beck and T. Zaslavsky, Six little squares and how their numbers grow, submitted. - _Thomas Zaslavsky_, Jan 29 2010
- Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
- Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,2,-2,1,0,-1,2,-1).
Programs
-
Mathematica
LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 1, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 40}, 60] (* Jean-François Alcover, Nov 12 2018 *) CoefficientList[Series[(8 x^10 (2 x^2 + 1)) / ((1 - x^6) (1 - x^4) (1 - x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Nov 12 2018 *) -
PARI
a(n)=1/6*(n^3-16*n^2+(76-3*(n%2))*n -[96,58,96,102,112,90,96,70,96,90,112,102][(n%12)+1])
Formula
G.f.: (8*x^10*(2*x^2+1)) / ((1-x^6)*(1-x^4)*(1-x)^2).
a(n) is given by a quasipolynomial of period 12.
Extensions
Edited by N. J. A. Sloane, Feb 05 2010
A108577 Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having all entries < n.
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 8, 12, 16, 23, 30, 40, 50, 63, 76, 93, 110, 132, 154, 180, 206, 238, 270, 308, 346, 390, 434, 485, 536, 595, 654, 720, 786, 861, 936, 1020, 1104, 1197, 1290, 1393, 1496, 1610, 1724, 1848, 1972, 2108, 2244, 2392, 2540, 2700, 2860
Offset: 1
Comments
A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal. The symmetries are those of the square. - Thomas Zaslavsky, Mar 12 2010
Examples
a(10) = 1 because there is only one symmetry type of 3 X 3 magic square with entries 1,...,9.
Links
- Thomas Zaslavsky, Table of n, a(n) for n = 1..10000.
- Matthias Beck and Thomas Zaslavsky, Auxiliary files for "Six little squares".
- Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
- Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,2,-2,1,0,-1,2,-1).
Programs
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Mathematica
LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 1, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5}, 58] (* Mike Sheppard, Feb 04 2025 *)
Formula
G.f.: (x^10*(2*x^2+1)) / ((1-x^6)*(1-x^4)*(1-x)^2).
a(n) is given by a quasipolynomial of period 12.
A108578 Number of 3 X 3 magic squares with magic sum 3n.
0, 0, 0, 0, 8, 24, 32, 56, 80, 104, 136, 176, 208, 256, 304, 352, 408, 472, 528, 600, 672, 744, 824, 912, 992, 1088, 1184, 1280, 1384, 1496, 1600, 1720, 1840, 1960, 2088, 2224, 2352, 2496, 2640, 2784, 2936, 3096, 3248, 3416, 3584, 3752, 3928, 4112, 4288
Offset: 1
Keywords
Comments
Contribution from Thomas Zaslavsky, Mar 12 2010: (Start)
A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.
a(n) is given by a quasipolynomial of period 6. (End)
Examples
a(5) = 8 because there are 8 3 X 3 magic squares with entries 1,...,9 and magic sum 15.
Links
- T. Zaslavsky, Table of n, a(n) for n = 1..10000.
- M. Beck and T. Zaslavsky, An enumerative geometry for magic and magilatin labellings, Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071. - _Thomas Zaslavsky_, Jan 29 2010
- Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
- Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1, -1,1).
Programs
-
Magma
I:=[0,0,0,0,8,24]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..60]]; // Vincenzo Librandi, Sep 01 2018
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Mathematica
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 8, 24}, 50] (* Jean-François Alcover, Sep 01 2018 *) CoefficientList[Series[8 x^4 (1 + 2 x) / ((1 - x) (1 - x^2) (1 - x^3)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 01 2018 *) -
PARI
a(n)=(1/9)*(2*n^2-32*n+[144,78,120,126,96,102][(n%18)/3+1])
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PARI
x='x+O('x^99); concat(vector(4), Vec(8*x^5*(1+2*x)/((1-x)*(1-x^2)*(1-x^3)))) \\ Altug Alkan, Sep 01 2018
Formula
G.f.: [8*x^5*(1+2*x)] / [(1-x)*(1-x^2)*(1-x^3)].
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - Vincenzo Librandi, Sep 01 2018
Extensions
Edited by N. J. A. Sloane, Feb 05 2010
Corrected g.f. to account for previous change in parameter n from magic sum s to s/3; by Thomas Zaslavsky, Mar 12 2010
A174257 Number of symmetry classes of 3 X 3 reduced magic squares with distinct values and maximum value 2n; also, with magic sum 3n.
0, 0, 0, 1, 2, 1, 3, 3, 3, 4, 5, 4, 6, 6, 6, 7, 8, 7, 9, 9, 9, 10, 11, 10, 12, 12, 12, 13, 14, 13, 15, 15, 15, 16, 17, 16, 18, 18, 18, 19, 20, 19, 21, 21, 21, 22, 23, 22, 24, 24, 24, 25, 26, 25, 27, 27, 27, 28, 29, 28, 30, 30, 30, 31, 32, 31, 33, 33, 33, 34, 35, 34, 36, 36, 36, 37
Offset: 1
Comments
In a reduced magic square the row, column, and two diagonal sums must all be equal (the "magic sum") and the minimum entry is 0. The maximum entry is necessarily even and = (2/3)*(magic sum). The symmetries are those of the square.
a(n) is a quasipolynomial with period 6.
The second differences of A108577 are a(n/2) for even n and 0 for odd n. The first differences of A108579 are a(n/3).
For n>=3 equals a(n) the number of partitions of n-3 using parts 1 and 2 only, with distinct multiplicities. Example: a(7) = 3 because [2,2], [2,1,1], [1,1,1,1] are such partitions of 7-3=4. - T. Amdeberhan, May 13 2012
a(n) is equal to the number of partitions of n of length 3 with exactly two equal entries (see below example). - John M. Campbell, Jan 29 2016
a(k) + 2 =:t(k), k >= 1, based on sequence A300069, is used to obtain for 2^t(k)*O_{-k} integer coordinates in the quadratic number field Q(sqrt(3)), where O_{-k} is the center of a k-family of regular hexagons H_{-k} forming part of a discrete spiral. See the linked W. Lang paper, Lemma 5, and Table 7. - Wolfdieter Lang, Mar 30 2018
a(n) is equal to the number of incongruent isosceles triangles (excluding equilateral triangles) formed from the vertices of a regular n-gon. - Frank M Jackson, Oct 30 2022
Examples
From _John M. Campbell_, Jan 29 2016: (Start) For example, there are a(16)=7 partitions of 16 of length 3 with exactly two equal entries: (14,1,1) |- 16 (12,2,2) |- 16 (10,3,3) |- 16 (8,4,4) |- 16 (7,7,2) |- 16 (6,6,4) |- 16 (6,5,5) |- 16 (End)
Links
- Thomas Zaslavsky, Table of n, a(n) for n = 1..10000.
- Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
- Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow, Journal of Integer Sequences, Vol. 13 (2010), Article 10.6.2.
- Wolfdieter Lang, On a Conformal Mapping of Regular Hexagons and the Spiral of its Centers.
- Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).
Programs
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Maple
seq(floor((n-1)/2)+floor((n-1)/3)-floor(n/3),n=1..100) # Mircea Merca, May 14 2013
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Mathematica
Rest@ CoefficientList[Series[x^4 (1 + 2 x)/((1 + x) (1 + x + x^2) (x - 1)^2), {x, 0, 76}], x] (* Michael De Vlieger, Jan 29 2016 *) Table[Length@Select[Length/@Union/@IntegerPartitions[n, {3}], # == 2 &], {n, 1, 100}] (* Frank M Jackson, Oct 30 2022 *) -
PARI
concat(vector(3), Vec(x^4*(1+2*x) / ( (1+x)*(1+x+x^2)*(x-1)^2 ) + O(x^90))) \\ Michel Marcus, Jan 29 2016
Formula
G.f.: x^4*(1+2*x) / ( (1+x)*(1+x+x^2)*(x-1)^2 ).
a(n) = (1/8)*A174256(n).
a(n) = floor((n-1)/2) + floor((n-1)/3) - floor(n/3). - Mircea Merca, May 14 2013
a(n) = A300069(n-1) + 3*floor((n-1)/6), n >= 1. Proof via g.f.. - Wolfdieter Lang, Feb 24 2018
a(n) = (6*n - 13 - 8*cos(2*n*Pi/3) - 3*cos(n*Pi))/12. - Wesley Ivan Hurt, Oct 04 2018
Extensions
Information added to name and comments by Thomas Zaslavsky, Apr 24 2010
A173546 Number of 3 X 3 semimagic squares with distinct positive values < n. In a semimagic squares the row and column sums must all be equal (the "magic sum").
72, 288, 936, 2592, 5760, 11520, 20952, 35712, 57168, 88272, 131112, 189504, 265752, 365760, 492480, 653040, 851472, 1096416, 1392768, 1751904, 2178864, 2687184, 3283632, 3983760, 4794984, 5736528, 6816456, 8056224, 9466128
Offset: 10
Keywords
Comments
a(n) is given by a quasipolynomial of degree 5 and period 60.
References
- Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Annals of Combinatorics, 10 (2006), no. 4, pages 395-413. MR 2007m:05010. Zbl 1116.05071.
Links
- Thomas Zaslavsky, Table of n, a(n) for n = 10..10000.
- Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
- Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
- Index entries for linear recurrences with constant coefficients, signature (0, 2, 2, 0, -3, -3, -2, 1, 4, 4, 1, -2, -3, -3, 0, 2, 2, 0, -1).
Crossrefs
Formula
G.f.: 72 * x^2/(1-x)^2 * { x^5/[(1-x)^3*(1-x^2)] - 2x^5/[(1-x)*(1-x^2)^2] - x^5/[(1-x)^2*(1-x^3)] - 2x^6/[(1-x)*(1-x^2)*(1-x^3)] - x^6/(1-x^2)^3 - x^7/[(1-x^2)^2*(1-x^3)] + x^5/[(1-x)*(1-x^4)] + 2x^5/[(1-x^2)*(1-x^3)] + 2x^6/[(1-x^2)*(1-x^4)] + x^6/(1-x^3)^2 + x^7/[(1-x^2)*(1-x^5)] + x^7/[(1-x^3)*(1-x^4)] + x^8/[(1-x^3)*(1-x^5)] - x^5/(1-x^5) }. - Thomas Zaslavsky, Mar 03 2010
A173547 Number of 3 X 3 semimagic squares with distinct positive values and magic sum n.
72, 144, 288, 576, 864, 1440, 2088, 3024, 3888, 5904, 6984, 9432, 12168, 14904, 17928, 23832, 26784, 33048, 39672, 46584, 53640, 65592, 72504, 85248, 98928, 111816, 125208, 147528, 160632, 182808, 206424, 229176, 252648, 287928, 310752
Offset: 15
Keywords
Comments
In a semimagic squares the row and column sums must all be equal to the magic sum. a(n) is given by a quasipolynomial of degree 4 and period 840.
a(15) is the first term because the values 1,...,9 make magic sum 15. [From Thomas Zaslavsky, Mar 03 2010]
References
- Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Annals of Combinatorics, 10 (2006), no. 4, pages 395-413. MR 2007m:05010. Zbl 1116.05071.
Links
- Thomas Zaslavsky, Table of n, a(n) for n=15..10000.
- Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
- Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
- Index entries for linear recurrences with constant coefficients, signature (-2, -3, -2, 0, 3, 6, 8, 9, 7, 3, -4, -10, -15, -16, -14, -8, 0, 8, 14, 16, 15, 10, 4, -3, -7, -9, -8, -6, -3, 0, 2, 3, 2, 1).
Crossrefs
Formula
G.f.: 72 * x^3/(1-x)^3 * { x^7/[(x-1)*(x^2-1)^3] + 2x^7/[(x-1)*(x^2-1)*(x^4-1)] + x^7/[(x-1)*(x^6-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^3-1)*(x^4-1)] + x^7/(x^7-1) + x^9/[(x-1)*(x^4-1)^2] + 2*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 2*x^9/[(x^3-1)*(x^6-1)] + x^9/[(x^4-1)*(x^5-1)] + x^11/[(x^3-1)*(x^4-1)^2] + x^11/[(x^3-1)*(x^8-1)] + x^11/[(x^5-1)*(x^6-1)] + x^13/[(x^5-1)*(x^8-1)] }
A304507 a(n) = 5*(n+1)*(9*n+4).
20, 130, 330, 620, 1000, 1470, 2030, 2680, 3420, 4250, 5170, 6180, 7280, 8470, 9750, 11120, 12580, 14130, 15770, 17500, 19320, 21230, 23230, 25320, 27500, 29770, 32130, 34580, 37120, 39750, 42470, 45280, 48180, 51170, 54250, 57420, 60680, 64030, 67470, 71000, 74620
Offset: 0
Comments
The first Zagreb index of the single-defect 5-gonal nanocone CNC(5,n) (see definition in the Doslic et al. reference, p. 27).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of CNC(5,n) is M(CNC(5,n); x,y) = 5*x^2*y^2 + 10*n*x^2*y^3 + 5*n*(3*n+1)*x^3*y^3/2.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
- T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, Journal of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
- A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
-
GAP
List([0..50], n -> 5*(n+1)*(9*n+4)); # Muniru A Asiru, May 15 2018
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Maple
seq((5*(n+1))*(9*n+4), n = 0 .. 40);
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Mathematica
Array[5 (# + 1) (9 # + 4) &, 41, 0] (* or *) LinearRecurrence[{3, -3, 1}, {20, 130, 330}, 41] (* or *) CoefficientList[Series[10 (2 + 7 x)/(1 - x)^3, {x, 0, 40}], x] (* Michael De Vlieger, May 14 2018 *) -
PARI
a(n) = 5*(n+1)*(9*n+4); \\ Altug Alkan, May 14 2018
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PARI
Vec(10*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018
Formula
From Colin Barker, May 14 2018: (Start)
G.f.: 10*(2 + 7*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
a(n) = 10*A062708(n+1) for n >= 0. - Robert G. Wilson v, May 14 2018
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 5*exp(x)*(4 + 22*x + 9*x^2).
A363839 Numbers in the witch's multiplication table (German: "Hexeneinmaleins") in Goethe's Faust.
1, 10, 2, 3, 4, 5, 6, 7, 8, 9, 1, 10, 0, 1
Offset: 1
Comments
In German, the witch declaims the following magic spell to Faust:
Du mußt verstehn!
Aus Eins[1] mach' Zehn[10],
Und Zwey[2] laß gehn,
Und Drey[3] mach' gleich,
So bist du reich.
Verlier' die Vier[4]!
Aus Fünf[5] und Sechs[6],
So sagt die Hex',
Mach' Sieben[7] und Acht[8],
So ist's vollbracht:
Und Neun[9] ist Eins[1],
Und Zehn[10] ist keins[0].
Das ist das Hexen-Einmal-Eins[1]!
(The numbers in "[...]" do not apppear in the original.)
Literal translation into English:
You must understand!
From one make ten,
And two let go,
And three make equal,
So you are rich.
Lose the four!
From five and six,
So says the witch,
Make seven and eight,
So it is finished:
And nine is one,
And ten is none.
This is the Witch's one-times-one!
.
From Peter Luschny, Jun 23 2023: (Start)
Sometimes the witch's one-times-one is interpreted as a construction guide for a magic square, which describes a transformation like this:
1 2 3 4 9 2
4 5 6 -> 3 5 7
7 8 9 8 1 6
(End)
Links
- Johann Wolfgang von Goethe: Faust - Der Tragödie erster Teil. Page 161 and page 162. Tübingen: Cotta, 1808. Digital full-text edition at Wikisource, version of Aug 18 2016.
- Poetry in Translation, Faust Scene VI, translated by A. S. Kline, 2003.
- Wikipedia (German), Hexeneinmaleins.
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_162.jpg&oldid=-){kind=link}
Comments
Examples
Links
Crossrefs
Programs
Mathematica
t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]] c[n_] := Count[t[n], 0] t = Table[c[n], {n, 0, 70}] (* A211422 *) (t - 1)/8 (* A120486 *)