A152751 -id:A152751 - cp's OEIS frontend

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3n(4*n-3).

0, 3, 30, 81, 156, 255, 378, 525, 696, 891, 1110, 1353, 1620, 1911, 2226, 2565, 2928, 3315, 3726, 4161, 4620, 5103, 5610, 6141, 6696, 7275, 7878, 8505, 9156, 9831, 10530, 11253, 12000, 12771, 13566, 14385, 15228, 16095, 16986, 17901, 18840, 19803, 20790, 21801

Offset: 0

Omar E. Pol, Dec 15 2008

3*A172078(n) = n*a(n) - Sum_{k=0..n-1} a(k). - Bruno Berselli, Dec 12 2010

			For n=8, a(8) = (1*3 + 5*7 + 9*11 +..+ 29*31) - (2*4 + 6*8 + 10*12 +..+ 26*28) = 696 (see Problem 1052 in References). - _Bruno Berselli_, Dec 12 2010

"Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno), Jan. 1910 p. 47 (Problem 1052).

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. A001107, A001539, A002939, A139271, A153794, A172078.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=24: see Comments lines of A226492.

Table[3n(4n-3),{n,0,40}] (* Harvey P. Dale, May 26 2012 *)

a(n)=3*n*(4*n-3) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 12*n^2 - 9*n = 3*A001107(n).
a(n) = a(n-1) + 24*n - 21, n > 0. - Vincenzo Librandi, Nov 26 2010
a(n) = Sum_{k=0..n-1} A001539(k) - Sum_{k=0..n-1} 4*A002939(k) if n > 0 (see References, Problem 1052). - Bruno Berselli, Dec 08 2010 - Jan 21 2011
G.f.: -3*x*(1+7*x)/(x-1)^3.
a(0)=0, a(1)=3, a(2)=30, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 26 2012
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 4*x).
a(n) = A153794(n) - n. (End)

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

Original entry on oeis.org

0, 3, 36, 99, 192, 315, 468, 651, 864, 1107, 1380, 1683, 2016, 2379, 2772, 3195, 3648, 4131, 4644, 5187, 5760, 6363, 6996, 7659, 8352, 9075, 9828, 10611, 11424, 12267, 13140, 14043, 14976, 15939, 16932, 17955, 19008, 20091, 21204

Offset: 0

Omar E. Pol, Dec 26 2008

This sequence is related to A172117 by 3*A172117(n) = n*a(n) - Sum_{i=0..n-1} a(i) and this is the case d=10 in the identity n*(3*n*(d*n - d + 2)/2) - Sum_{k=0..n-1} 3*k*(d*k - d + 2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/2. - Bruno Berselli, Aug 26 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051624, A152965.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153875, A172117.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=30: see Comments lines of A226492.

Table[3n(5n-4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,3,36},40] (* Harvey P. Dale, Jun 18 2014 *)
3*PolygonalNumber[12,Range[0,60]] (* Harvey P. Dale, May 13 2022 *)

a(n)=3*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 15*n^2 - 12*n = A051624(n)*3.
a(n) = 30*n + a(n-1) - 27 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 9*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=36. - Harvey P. Dale, Jun 18 2014
E.g.f.: 3*x*(1 + 5*x)*exp(x). - G. C. Greubel, Aug 21 2016
a(n) = (4*n-2)^2 - (n-2)^2. In general, if P(k,n) is the k-th n-gonal number, then (2*k+1)*P(8*k+4,n) = ((3*k+1)*n-2*k)^2 - (k*n-2*k)^2. - Charlie Marion, Jul 29 2021

A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3n(9*n-7)/2.

Original entry on oeis.org

0, 3, 33, 90, 174, 285, 423, 588, 780, 999, 1245, 1518, 1818, 2145, 2499, 2880, 3288, 3723, 4185, 4674, 5190, 5733, 6303, 6900, 7524, 8175, 8853, 9558, 10290, 11049, 11835, 12648, 13488, 14355, 15249, 16170, 17118, 18093, 19095

Offset: 0

Omar E. Pol, Jan 02 2009

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

Cf. A051682, A152995.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=27: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,27}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Table[3*n*(9*n-7)/2, {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,3,33},25] (* G. C. Greubel, Aug 28 2016 *)

a(n)=3*n*(9*n-7)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (27*n^2 - 21*n)/2 = A051682(n)*3.
a(n) = 27*n + a(n-1) - 24, with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 8*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From G. C. Greubel, Aug 28 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*x*(2 + 9*x)*exp(x). (End)

A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3n(11*n - 9)/2.

Original entry on oeis.org

0, 3, 39, 108, 210, 345, 513, 714, 948, 1215, 1515, 1848, 2214, 2613, 3045, 3510, 4008, 4539, 5103, 5700, 6330, 6993, 7689, 8418, 9180, 9975, 10803, 11664, 12558, 13485, 14445, 15438, 16464, 17523, 18615, 19740, 20898, 22089

Offset: 0

Omar E. Pol, Jan 03 2009

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

Cf. A051865, A152997.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=33: see Comments lines of A226492.

Table[(33*n^2 - 27*n)/2, {n,0,25}] (* G. C. Greubel, Aug 31 2016 *)

a(n)=3*n*(11*n-9)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (33*n^2 - 27*n)/2 = A051865(n)*3.
a(n) = a(n-1) + 33*n - 30, with n>0, a(0)=0. - Vincenzo Librandi, Dec 14 2010
G.f.: 3*x*(1 + 10*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From G. C. Greubel, Aug 31 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*x*(2 + 11*x)*exp(x). (End)

A153794 4 times octagonal numbers: a(n) = 4n(3*n-2).

Original entry on oeis.org

0, 4, 32, 84, 160, 260, 384, 532, 704, 900, 1120, 1364, 1632, 1924, 2240, 2580, 2944, 3332, 3744, 4180, 4640, 5124, 5632, 6164, 6720, 7300, 7904, 8532, 9184, 9860, 10560, 11284, 12032, 12804, 13600, 14420, 15264, 16132, 17024

Offset: 0

Omar E. Pol, Jan 19 2009

Sequence found by reading the segment (0, 4) together with the line from 4, in the direction 4, 32, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

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

Cf. A000567, A139267, A152751, A153808.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,4,7!,24}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Table[4n(3n-2),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,4,32},41] (* Harvey P. Dale, Jul 14 2011 *)
4*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, Dec 04 2022 *)

a(n) = 12*n^2 - 8*n; \\ Altug Alkan, Aug 29 2016

a(n) = 12*n^2 - 8*n = 4*A000567(n) = 2*A139267(n).
a(n) = 24*n + a(n-1) - 20 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=4, a(2)=32, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 14 2011
G.f.: 4*(x + 5*x^2)/(1-x)^3. - Harvey P. Dale, Jul 14 2011
E.g.f.: 4*x*(1 + 3*x)*exp(x). - G. C. Greubel, Aug 29 2016

A153796 6 times octagonal numbers: a(n) = 6n(3*n-2).

Original entry on oeis.org

0, 6, 48, 126, 240, 390, 576, 798, 1056, 1350, 1680, 2046, 2448, 2886, 3360, 3870, 4416, 4998, 5616, 6270, 6960, 7686, 8448, 9246, 10080, 10950, 11856, 12798, 13776, 14790, 15840, 16926, 18048, 19206, 20400, 21630, 22896, 24198, 25536, 26910, 28320, 29766

Offset: 0

Omar E. Pol, Jan 19 2009

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

Cf. A000567, A139267, A152751.

[6*n*(3*n-2): n in [0..60]]; // Wesley Ivan Hurt, Aug 29 2016

A153796:=n->6*n*(3*n-2): seq(A153796(n), n=0..60); # Wesley Ivan Hurt, Aug 29 2016

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,6,8!,36}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 03 2009 *)
Table[6*n*(3*n-2), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1}, {0,6,48}, 25] (* G. C. Greubel, Aug 29 2016 *)
6*PolygonalNumber[8,Range[0,50]] (* Harvey P. Dale, Dec 17 2023 *)

a(n)=6*n*(3*n-2) \\ Charles R Greathouse IV, Aug 29 2016

a(n) = 18*n^2 - 12*n = 6*A000567(n) = 3*A139267(n) = 2*A152751(n).
a(n) = a(n-1) + 36*n - 30 (with a(0)=0). - Vincenzo Librandi, Dec 15 2010
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
G.f.: 6*x*(1 + 5*x)/(1 - x)^3.
E.g.f.: 6*x*(1 + 3*x)*exp(x). (End)

A224273 Decimal expansion of Baxter's four-coloring constant.

Original entry on oeis.org

1, 4, 6, 0, 9, 9, 8, 4, 8, 6, 2, 0, 6, 3, 1, 8, 3, 5, 8, 1, 5, 8, 8, 7, 3, 1, 1, 7, 8, 4, 6, 0, 5, 9, 6, 9, 7, 0, 3, 8, 9, 3, 1, 3, 5, 5, 8, 0, 7, 4, 6, 1, 7, 8, 8, 2, 0, 5, 7, 7, 5, 4, 3, 4, 4, 4, 1, 5, 2, 1, 3, 5, 5, 8, 8, 5, 7, 3, 1, 4, 4, 0, 7, 7, 6, 5, 3

Offset: 1

Bruno Berselli, Apr 02 2013

The constant is named after Australian physicist Rodney James Baxter. - Amiram Eldar, Aug 13 2020

			1.46099848620631835815887311784605969703893135580746178820577543...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See p. 413.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
R. J. Baxter, Colorings of a hexagonal lattice, Journal of Mathematical Physics, Vol. 11, No. 3 (1970), pp. 784-789.
R. J. Baxter, q colourings of the triangular lattice, Journal of Physics A: Mathematical and General, Vol. 19, No. 14 (1986), pp. 2821-2839.
Eric Weisstein's World of Mathematics, Baxter's Four-Coloring Constant.

Cf. A004117, A081760, A152751.

RealDigits[3 Gamma[1/3]^3/(4 Pi^2), 10, 90][[1]]

3*gamma(1/3)^3/(4*Pi^2) \\ Michel Marcus, Mar 23 2020

Equals 1/Product_{n>=1} (1-1/(3n-1)^2) = 3*Gamma(1/3)^3/(4*Pi^2).
Equals 1/(2^(1/3)*A081760). - Kritsada Moomuang, Mar 15 2020
Equals 2*Pi/(sqrt(3)*Gamma(2/3)^3). - Vaclav Kotesovec, Mar 23 2020
Equals Product_{k>=1} (1 + 1/A152751(k)). - Amiram Eldar, Aug 13 2020
Equals Sum_{k>=0} binomial(-1/3, k)^2. - Gerry Martens, Jul 24 2023

A064201 9 times octagonal numbers: a(n) = 9n(3*n-2).

Original entry on oeis.org

0, 9, 72, 189, 360, 585, 864, 1197, 1584, 2025, 2520, 3069, 3672, 4329, 5040, 5805, 6624, 7497, 8424, 9405, 10440, 11529, 12672, 13869, 15120, 16425, 17784, 19197, 20664, 22185, 23760, 25389, 27072, 28809, 30600, 32445, 34344, 36297, 38304, 40365, 42480, 44649

Offset: 0

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Sep 22 2001

L. Berzolari, Allgemeine Theorie der Höheren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B. G. Teubner, 1906. p. 341.

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

Cf. A000567, A152751.

9*PolygonalNumber[8,Range[0,40]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3,-3,1},{0,9,72},40] (* Harvey P. Dale, Aug 01 2020 *)

a(n)=9*n*(3*n-2) \\ Charles R Greathouse IV, Jun 16 2017

a(n) = 9*(n-2)*(3*n-8), with offset 2.
a(n) = 9*A000567(n). - Omar E. Pol, Dec 11 2008
a(n) = a(n-1) + 54*n - 45, with n > 0, a(0)=0. - Vincenzo Librandi, Dec 15 2010
G.f.: 9*x*(1+5*x)/(1-x)^3. - Colin Barker, Feb 29 2012
From Elmo R. Oliveira, Dec 25 2024: (Start)
E.g.f.: 9*exp(x)*x*(1 + 3*x).
a(n) = 3*A152751(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Better definition, corrected offset and edited from Omar E. Pol, Dec 11 2008

A153808 8 times octagonal numbers: 8n(3*n-2).

Original entry on oeis.org

0, 8, 64, 168, 320, 520, 768, 1064, 1408, 1800, 2240, 2728, 3264, 3848, 4480, 5160, 5888, 6664, 7488, 8360, 9280, 10248, 11264, 12328, 13440, 14600, 15808, 17064, 18368, 19720, 21120, 22568, 24064, 25608, 27200, 28840, 30528, 32264

Offset: 0

Omar E. Pol, Jan 19 2009

G. C. Greubel, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567 (octagonal numbers), A064201 (9 times octagonal numbers), A139267 (twice octagonal numbers), A152751 (3 times octagonal numbers), A153794 (4 times octagonal numbers).

Magma
```
[ 8*n*(3*n-2): n in [0..40] ];
```

Table[8*n*(3*n-2), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,8,64}, 25] (* G. C. Greubel, Aug 29 2016 *)
8*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, Nov 22 2023 *)

a(n)=24*n^2-16*n \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 24*n^2 - 16*n = 8*A000567(n) = 4*A139267(n) = 2*A153794(n).
a(n) = a(n-1) + 48*n - 40 (with a(0)=0). - Vincenzo Librandi, Nov 27 2010
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 8*x*(1 + 5*x)/(1 - x)^3.
E.g.f.: 8*x*(1 + 3*x)*exp(x). (End)

A194273 Concentric triangular numbers (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55, 63, 72, 81, 90, 99, 109, 120, 132, 144, 156, 168, 181, 195, 210, 225, 240, 255, 271, 288, 306, 324, 342, 360, 379, 399, 420, 441, 462, 483, 505, 528, 552, 576, 600, 624, 649, 675, 702, 729, 756, 783, 811

Offset: 0

Omar E. Pol, Aug 20 2011

This can be interpreted as a cellular automaton on the infinite hexagonal net. The sequence gives the number of cells "ON" in the structure after n-th stage. A194272 gives the first differences. For a definition without words see the illustration of initial terms in the example section. For other concentric polygonal numbers see A194274, A194275 and A032528.

Also, row sums of an infinite square array T(n,k) in which column k lists 6*k-1 zeros followed by the numbers A008486 (see example).

			Using the numbers A008486 we can write:
0, 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,...
0, 0, 0, 0, 0,  0,  0,  1,  3,  6,  9, 12, 15, 18,...
0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  1,...
And so on.
=========================================================
The sums of the columns give this sequence:
0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55,...
...
Illustration of initial terms:
.                                              o
.                                 o           o o
.                      o         o o         o   o
.             o       o o       o   o       o     o
.      o     o o     o   o     o     o     o       o
. o   o o   o o o   o o o o   o o o o o   o o o o o o
.
. 1    3      6        9          12           15
.
.                                           o
.                        o                 o o
.       o               o o               o   o
.      o o             o   o             o     o
.     o   o           o     o           o   o   o
.    o     o         o   o   o         o   o o   o
.   o   o   o       o   o o   o       o   o o o   o
.  o         o     o           o     o             o
. o o o o o o o   o o o o o o o o   o o o o o o o o o
.
.       19               24                 30

Index entries for sequences related to cellular automata

Cf. A000217, A008486, A032528, A069131, A152751, A194272, A194274, A194275.

G.f.: x/(1-3*x+3*x^2-3*x^4+3*x^5-x^6) = x/((1-x)^3*(1+x)*(1-x+x^2)).

cp's OEIS Frontend

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3*n*(4*n-3).

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A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3*n*(5*n-4).

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A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3*n*(9*n-7)/2.

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A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3*n*(11*n - 9)/2.

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A153794 4 times octagonal numbers: a(n) = 4*n*(3*n-2).

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A153796 6 times octagonal numbers: a(n) = 6*n*(3*n-2).

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A224273 Decimal expansion of Baxter's four-coloring constant.

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A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3n(4*n-3).

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3n(9*n-7)/2.

A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3n(11*n - 9)/2.

A153794 4 times octagonal numbers: a(n) = 4n(3*n-2).

A153796 6 times octagonal numbers: a(n) = 6n(3*n-2).

A064201 9 times octagonal numbers: a(n) = 9n(3*n-2).

A153808 8 times octagonal numbers: 8n(3*n-2).