A094159 -id:A094159 - cp's OEIS frontend

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

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)

A048790 Array read by antidiagonals: T(n,k) = number of rooted n-dimensional polycubes with k cells, with no symmetries removed (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 18, 4, 1, 8, 45, 76, 5, 1, 10, 84, 344, 315, 6, 1, 12, 135, 936, 2670, 1296, 7, 1, 14, 198, 1980, 10810, 20886, 5320, 8, 1, 16, 273, 3604, 30475, 127632, 164514, 21800, 9, 1, 18, 360, 5936, 69405, 483702, 1531180, 1303304, 89190, 10, 1

Offset: 1

Dan Hoey

			Array begins:
n\k 1..2...3.....4......5.......6........7........8........9.....10......11......12.......13
1 | 1..2...3.....4......5.......6........7........8........9.....10......11......12.......13
2 | 1..4..18....76....315....1296.....5320....21800....89190.364460.1487948.6070332.24750570
3 | 1..6..45...344...2670...20886...164514..1303304.10375830
4 | 1..8..84...936..10810..127632..1531180.18589840
5 | 1.10.135..1980..30475..483702..7847525
6 | 1.12.198..3604..69405.1386048.28403620
7 | 1.14.273..5936.137340.3307878
8 | 1.16.360..9104.246020
9 | 1.18.459.13236.409185

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

R. C. Schroeppel, A few mathematical experiments

Rows give A048663-A048668, A094101. Columns give A094159-A094161. Cf. A094100.

More terms from Joshua Zucker, Aug 14 2006

Edited by N. J. A. Sloane, Sep 15 2008 at the suggestion of R. J. Mathar

A195319 Three times second hexagonal numbers: 3n(2*n+1).

Original entry on oeis.org

0, 9, 30, 63, 108, 165, 234, 315, 408, 513, 630, 759, 900, 1053, 1218, 1395, 1584, 1785, 1998, 2223, 2460, 2709, 2970, 3243, 3528, 3825, 4134, 4455, 4788, 5133, 5490, 5859, 6240, 6633, 7038, 7455, 7884, 8325, 8778, 9243, 9720, 10209, 10710, 11223

Offset: 0

Omar E. Pol, Sep 17 2011

Sequence found by reading the line from 0, in the direction 0, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Semi-axis opposite to A094159 in the same spiral.
Sum of the numbers from 2*n to 4*n. - Wesley Ivan Hurt, Nov 27 2015
From Peter M. Chema, Jan 21 2017: (Start)
Also 0 together with the partial sums of A017629.
Digit root is 0 together with period 3: repeat [9,3,9].
Final digits cycle a length period 10: repeat [0,9,0,3,8,5,4,5,8,3]. (End)
Sequence found by reading the line from 0, in the direction 0, 9, ..., in the triangle spiral. - Hans G. Oberlack, Dec 08 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Hans G. Oberlack, Triangle spiral.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A045943.

Cf. A000217, A001318, A014105, A094159, A017629.

List([0..30], n -> 3*n*(2*n+1)); # G. C. Greubel, Dec 07 2018

[3*n*(2*n+1): n in [0..50]]; // Vincenzo Librandi, Sep 20 2011

A195319:=n->6*n^2 + 3*n: seq(A195319(n), n=0..50); # Wesley Ivan Hurt, Nov 27 2015

Table[6n^2+3n,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,9,30},50] (* Harvey P. Dale, Oct 13 2013 *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^3, {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 27 2015 *)

a(n)=3*n*(2*n+1) \\ Charles R Greathouse IV, Oct 16 2015

[3*n*(2*n+1) for n in range(50)] # G. C. Greubel, Dec 07 2018

a(n) = 6*n^2 + 3*n = 3*A014105(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Harvey P. Dale, Oct 13 2013
G.f.: 3*x*(3+x) / (1-x)^3. - Wesley Ivan Hurt, Nov 27 2015
a(n) = A000217(3*n) + 3*A000217(n). - Bruno Berselli, Aug 31 2017
E.g.f.: 3*x*(2*x+3)*exp(x). - G. C. Greubel, Dec 07 2018
From Amiram Eldar, Feb 27 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(1 - log(2))/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/2 + log(2) - 2)/3. (End)

A225144 a(n) = Sum_{i=n..2n} i^2(-1)^i.

Original entry on oeis.org

0, 3, 11, 18, 42, 45, 93, 84, 164, 135, 255, 198, 366, 273, 497, 360, 648, 459, 819, 570, 1010, 693, 1221, 828, 1452, 975, 1703, 1134, 1974, 1305, 2265, 1488, 2576, 1683, 2907, 1890, 3258, 2109, 3629, 2340, 4020, 2583, 4431, 2838, 4862, 3105, 5313, 3384

Offset: 0

Bruno Berselli, Jun 06 2013

3 and 11 are the only primes in the sequence.

			a(6) = 6^2-7^2+8^2-9^2+10^2-11^2+12^2 = 93.
a(7) = -7^2+8^2-9^2+10^2-11^2+12^2-13^2+14^2 = 84.

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

Cf. A050409: sum(i^2, i=n..2n); A064455: sum(i*(-1)^i, i=n..2n); A065679: A000217(n)+(-1)^n*A000217(n-1); A089594: sum(i^2*(-1)^i, i=1..n).

[&+[i^2*(-1)^i: i in [n..2*n]]: n in [0..50]];

Table[Sum[i^2 (-1)^i, {i, n, 2 n}], {n, 0, 50}]

G.f.: x*(3+11*x+9*x^2+9*x^3)/(1-x^2)^3.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6).
a(n) = n*(4*n+(n-1)*(-1)^n+2)/2.
a(n) = A000217(2n) +(-1)^n*A000217(n-1) with A000217(-1)=0.
a(2n-1) = A094159(n) for n>0; a(2n) = A055437(n) for A055437(0)=0.

A152994 Nine times hexagonal numbers: a(n) = 9n(2*n-1).

Original entry on oeis.org

0, 9, 54, 135, 252, 405, 594, 819, 1080, 1377, 1710, 2079, 2484, 2925, 3402, 3915, 4464, 5049, 5670, 6327, 7020, 7749, 8514, 9315, 10152, 11025, 11934, 12879, 13860, 14877, 15930, 17019, 18144, 19305, 20502, 21735, 23004, 24309, 25650, 27027, 28440, 29889, 31374

Offset: 0

Omar E. Pol, Dec 22 2008

Sequence found by reading the line from 0, in the direction 0, 9,..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Sep 18 2011

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

Cf. A000384, A094159.

Similar sequences are listed in A316466.

List([0..40], n-> 9*n*(2*n-1)); # G. C. Greubel, Sep 01 2019

[9*n*(2*n-1): n in [0..40]]; // G. C. Greubel, Sep 01 2019

seq(9*n*(2*n-1), n=0..40); # G. C. Greubel, Sep 01 2019

Table[9*n*(2*n-1), {n,0,40}] (* G. C. Greubel, Sep 01 2019 *)
9*PolygonalNumber[6,Range[0,50]] (* Harvey P. Dale, Jul 24 2022 *)

a(n)=9*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

[9*n*(2*n-1) for n in (0..40)] # G. C. Greubel, Sep 01 2019

a(n) = 18*n^2 - 9*n = A000384(n)*9 = A094159(n)*3.
a(n) = a(n-1) + 36*n - 27 for n>0, a(0)=0. - Vincenzo Librandi, Dec 15 2010
a(n) = Sum_{i = 2..10} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - Bruno Berselli, Jul 04 2018
From G. C. Greubel, Sep 01 2019: (Start)
G.f.: 9*x*(1+3*x)/(1-x)^3.
E.g.f.: 9*x*(1+2*x)*exp(x). (End)
From Amiram Eldar, Feb 27 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*log(2)/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi - 2*log(2))/18. (End)

A143698 12 times hexagonal numbers: 12n(2*n-1).

Original entry on oeis.org

0, 12, 72, 180, 336, 540, 792, 1092, 1440, 1836, 2280, 2772, 3312, 3900, 4536, 5220, 5952, 6732, 7560, 8436, 9360, 10332, 11352, 12420, 13536, 14700, 15912, 17172, 18480, 19836, 21240, 22692, 24192, 25740, 27336, 28980, 30672, 32412

Offset: 0

Omar E. Pol, Jan 23 2009

Sequence found by reading the line from 0, in the direction 0, 12,..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. - Omar E. Pol, Oct 02 2011

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

Cf. A000384, A002939, A085250, A094159, A152746, A154617.

seq(12*n*(2*n-1), n=0..40); # G. C. Greubel, May 30 2021

Table[24n^2-12n,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,12,72},40] (* Harvey P. Dale, Sep 24 2015 *)

a(n)=24*n^2-12*n \\ Charles R Greathouse IV, Jun 17 2017

[12*n*(2*n-1) for n in (0..40)] # G. C. Greubel, May 30 2021

a(n) = 24*n^2 - 12*n = 12*A000384(n) = 6*A002939(n) = 4*A094159(n) = 3*A085250(n) = 2*A152746(n).
a(n) = a(n-1) + 48*n - 36, with a(0)=0. - Vincenzo Librandi, Dec 14 2010
From G. C. Greubel, May 30 2021: (Start)
G.f.: 12*x*(1 + 3*x)/(1-x)^3.
E.g.f.: 12*x*(1 + 2*x)*exp(x). (End)

A154617 Eleven times hexagonal numbers: a(n) = 11n(2*n-1).

Original entry on oeis.org

0, 11, 66, 165, 308, 495, 726, 1001, 1320, 1683, 2090, 2541, 3036, 3575, 4158, 4785, 5456, 6171, 6930, 7733, 8580, 9471, 10406, 11385, 12408, 13475, 14586, 15741, 16940, 18183, 19470, 20801, 22176, 23595, 25058, 26565, 28116, 29711, 31350, 33033, 34760, 36531, 38346

Offset: 0

Omar E. Pol, Jan 13 2009

Sequence found by reading the line from 0, in the direction 0, 11, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. - Omar E. Pol, Sep 18 2011

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

Cf. A000384, A002939, A094159, A152745, A153781, A195313.

a(n)=11*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 22*n^2 - 11*n = 11*A000384(n).
a(n) = a(n-1) + 44*n - 33 (with a(0)=0). - Vincenzo Librandi, Dec 15 2010
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 11*x*(1 + 3*x)/(1 - x)^3.
E.g.f.: 11*x*(1 + 2*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A221912 Partial sums of floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008730.

			..0....0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
...

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242, A218530.

Accumulate[Floor[Range[0,70]/12]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2},70] (* Harvey P. Dale, Mar 23 2015 *)

a(12n) = A051866(n).
a(12n+1) = A139267(n).
a(12n+2) = A094159(n).
a(12n+3) = A033579(n).
a(12n+4) = A049452(n).
a(12n+5) = A033581(n).
a(12n+6) = A049453(n).
a(12n+7) = A033580(n).
a(12n+8) = A195319(n).
a(12n+9) = A202804(n).
a(12n+10) = A211014(n).
a(12n+11) = A049598(n).
G.f.: x^12/((1-x)^2*(1-x^12)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=1, a(13)=2, a(n)=2*a(n-1)- a(n-2)+ a(n-12)- 2*a(n-13)+ a(n-14). - Harvey P. Dale, Mar 23 2015

A227719 Floor(1/s(n)), where s(n) = n*log(1+1/n) - (2n-1)/(2n).

Original entry on oeis.org

5, 16, 33, 56, 86, 121, 162, 209, 263, 322, 387, 458, 536, 619, 708, 803, 905, 1012, 1125, 1244, 1370, 1501, 1638, 1781, 1931, 2086, 2247, 2414, 2588, 2767, 2952, 3143, 3341, 3544, 3753, 3968, 4190, 4417, 4650, 4889, 5135, 5386, 5643, 5906, 6176, 6451, 6732

Offset: 1

Clark Kimberling, Jul 22 2013

That s(n) > 0 for n >=1 follows from the chain 1 < log 2 < 3/4 < 2 log 3/2 < 5/6 < 3 log 4/3 < 7/8 < 4 log 5/4 < ... ; i.e., n*log((n+1)/n) - (2n-1)/(2n) > 0 and (2n+1)/(2n+2) - n* log((n+1)/n) > 0. For the first, closeness to 0 is indicated by A227719 and A227720, and for the second, by A227721 and a sequence which possibly equals A094159. Conjecture: the four sequences are linearly recurrent.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227720, A227721, A094159.

s[n_] := n*Log[1 + 1/n] - (2 n - 1)/(2 n);
Table[Floor[1/s[n]], {n, 1, 100}]  (* A227719 *)
Table[Round[1/s[n]], {n, 1, 100}]  (* A227720 *)

a(n) = -2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) (conjectured).

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

cp's OEIS Frontend

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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A048790 Array read by antidiagonals: T(n,k) = number of rooted n-dimensional polycubes with k cells, with no symmetries removed (n >= 1, k >= 1).

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A195319 Three times second hexagonal numbers: 3*n*(2*n+1).

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A225144 a(n) = Sum_{i=n..2*n} i^2*(-1)^i.

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A152994 Nine times hexagonal numbers: a(n) = 9*n*(2*n-1).

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A143698 12 times hexagonal numbers: 12*n*(2*n-1).

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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.

A195319 Three times second hexagonal numbers: 3n(2*n+1).

A225144 a(n) = Sum_{i=n..2n} i^2(-1)^i.

A152994 Nine times hexagonal numbers: a(n) = 9n(2*n-1).

A143698 12 times hexagonal numbers: 12n(2*n-1).

A154617 Eleven times hexagonal numbers: a(n) = 11n(2*n-1).