A195818 -id:A195818 - cp's OEIS frontend

A195817 Multiples of 10 and odd numbers interleaved.

0, 1, 10, 3, 20, 5, 30, 7, 40, 9, 50, 11, 60, 13, 70, 15, 80, 17, 90, 19, 100, 21, 110, 23, 120, 25, 130, 27, 140, 29, 150, 31, 160, 33, 170, 35, 180, 37, 190, 39, 200, 41, 210, 43, 220, 45, 230, 47, 240, 49, 250, 51, 260, 53, 270, 55, 280, 57, 290, 59, 300

Offset: 0

Omar E. Pol, Sep 29 2011

A008592 and A005408 interleaved.
Partial sums give the generalized 14-gonal (or tetradecagonal) numbers A195818.
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized 14-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 10 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, A195161, A195312, this sequence.
Cf. A091998, A195152.

[(2*(-1)^n+3)*n: n in [0..60]]; // Vincenzo Librandi, Sep 30 2011

With[{nn=30},Riffle[10*Range[0,nn],Range[1,2*nn+1,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,10,3},70] (* Harvey P. Dale, Nov 24 2013 *)

a(n) = (2*(-1)^n+3)*n; \\ Andrew Howroyd, Jul 23 2018

a(n) = (2*(-1)^n+3)*n. - Vincenzo Librandi, Sep 30 2011
From Bruno Berselli, Sep 30 2011: (Start)
G.f.: x*(1+10*x+x^2)/((1-x)^2*(1+x)^2).
a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).
a(n) * a(n+1) = a(n(n+1)).
a(n) + a(n+1) = A091998(n+1). (End)
a(0)=0, a(1)=1, a(2)=10, a(3)=3, a(n)=2*a(n-2)-a(n-4). - Harvey P. Dale, Nov 24 2013
Multiplicative with a(2^e) = 5*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 2^(3-s)). - Amiram Eldar, Oct 25 2023

A210964 Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 194, 195

Offset: 0

Omar E. Pol, Jun 16 2012

nonn

Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13, 13], [35, 35, 35, 35, 35], [86, 86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..3000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - The asymptotic ratio
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.

Cf. A247133.

nmax = 100; CoefficientList[Series[Product[1 / ((1 - x^(12*k)) * (1 - x^(12*k-1)) * (1 - x^(12*k-11))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

Expansion of 1 / f(-x, -x^11) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 10 2015
Partitions of n into parts of the form 12*k, 12*k+1, 12*k+11. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...]. - Michael Somos, Jan 10 2015
G.f.: Product_{k>0} 1 / ((1 - x^(12*k)) * (1 - x^(12*k - 1)) * (1 - x^(12*k - 11))).
Convolution inverse of A247133.
a(n) ~ sqrt(2)*(1+sqrt(3)) * exp(Pi*sqrt(n/6)) / (8*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = (1/n)*Sum_{k=1..n} A284372(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
a(n) = a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - - (with the convention a(n) = 0 for negative n), where 1, 11, 14, 34, ... is the sequence of generalized 14-gonal numbers A195818. - Peter Bala, Dec 10 2020

A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Jun 10 2012

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A211970
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.        (if k>=1)                         with a(0)=1,
.                                          if k >= 0]
========================================================
0    4    A008794        -         -         A211971
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
2,     2,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
4,     3,   2,   1,   1,   1,  1,  1,  1,  1,  1, ...
6,     5,   3,   2,   1,   1,  1,  1,  1,  1,  1, ...
10,    7,   4,   3,   2,   1,  1,  1,  1,  1,  1, ...
16,   11,   5,   4,   3,   2,  1,  1,  1,  1,  1, ...
24,   15,   7,   4,   4,   3,  2,  1,  1,  1,  1, ...
36,   22,  10,   5,   4,   4,  3,  2,  1,  1,  1, ...
54,   30,  13,   7,   4,   4,  4,  3,  2,  1,  1, ...
78,   42,  16,  10,   5,   4,  4,  4,  3,  2,  1, ...
112,  56,  21,  12,   7,   4,  4,  4,  4,  3,  2, ...
160,  77,  28,  14,  10,   5,  4,  4,  4,  4,  3, ...
224, 101,  35,  16,  12,   7,  4,  4,  4,  4,  4, ...
312, 135,  43,  21,  13,  10,  5,  4,  4,  4,  4, ...
432, 176,  55,  27,  14,  12,  7,  4,  4,  4,  4, ...
...

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A195825.
Cf. A057077, A195152.

T(n,k) = A211971(n), if k = 0.

T(n,k) = A195825(n,k), if k >= 1.

A195158 Concentric 24-gonal numbers.

Original entry on oeis.org

0, 1, 24, 49, 96, 145, 216, 289, 384, 481, 600, 721, 864, 1009, 1176, 1345, 1536, 1729, 1944, 2161, 2400, 2641, 2904, 3169, 3456, 3745, 4056, 4369, 4704, 5041, 5400, 5761, 6144, 6529, 6936, 7345, 7776, 8209, 8664, 9121, 9600, 10081, 10584, 11089

Offset: 0

Omar E. Pol, Sep 28 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., and the same line from 1, in the direction 1, 49, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. Main axis, perpendicular to A049598 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Column 24 of A195040.

Cf. A032527, A032528, A195143, A195149, A195058.

[(12*n^2+5*(-1)^n-5)/2: n in [0..50]]; // Vincenzo Librandi, Sep 30 2011

LinearRecurrence[{2,0,-2,1},{0,1,24,49},50] (* Harvey P. Dale, Jan 28 2021 *)

a(n)=6*n^2+5*((-1)^n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 6*n^2 + 5*((-1)^n-1)/2.
a(n) = -a(n-1) + A069190(n). - Vincenzo Librandi, Sep 30 2011
From Colin Barker, Sep 16 2012: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: x*(1+22*x+x^2)/((1-x)^3*(1+x)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/144 + tan(sqrt(5/6)*Pi/2)*Pi/(4*sqrt(30)). - Amiram Eldar, Jan 17 2023

A195824 a(n) = 24*n^2.

Original entry on oeis.org

0, 24, 96, 216, 384, 600, 864, 1176, 1536, 1944, 2400, 2904, 3456, 4056, 4704, 5400, 6144, 6936, 7776, 8664, 9600, 10584, 11616, 12696, 13824, 15000, 16224, 17496, 18816, 20184, 21600, 23064, 24576, 26136, 27744, 29400, 31104, 32856, 34656, 36504, 38400, 40344

Offset: 0

Omar E. Pol, Sep 28 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818.

Surface area of a cube with side 2n. - Wesley Ivan Hurt, Aug 05 2014

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

Bisection of A195158.

Cf. A000290, A001105, A008606, A016742, A033428, A033581, A135453, A139098, A195818.

[24*n^2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 05 2014

I:=[0,24,96]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 06 2014

A195824:=n->24*n^2: seq(A195824(n), n=0..50); # Wesley Ivan Hurt, Aug 05 2014

24 Range[0, 30]^2 (* or *) Table[24 n^2, {n, 0, 30}] (* or *) CoefficientList[Series[24 x (1 + x)/(1 - x)^3, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 05 2014 *)
LinearRecurrence[{3,-3,1},{0,24,96},40] (* Harvey P. Dale, Nov 11 2017 *)

a(n) = 24*n^2; \\ Michel Marcus, Aug 05 2014

a(n) = 24*A000290(n) = 12*A001105(n) = 8*A033428(n) = 6*A016742(n) = 4*A033581(n) = 3*A139098(n) = 2*A135453(n).
From Wesley Ivan Hurt, Aug 05 2014: (Start)
G.f.: 24*x*(1+x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 24*x*(1 + x)*exp(x).
a(n) = n*A008606(n) = A195158(2*n). (End)

A211014 Second 14-gonal numbers: n(6n+5).

Original entry on oeis.org

0, 11, 34, 69, 116, 175, 246, 329, 424, 531, 650, 781, 924, 1079, 1246, 1425, 1616, 1819, 2034, 2261, 2500, 2751, 3014, 3289, 3576, 3875, 4186, 4509, 4844, 5191, 5550, 5921, 6304, 6699, 7106, 7525, 7956, 8399, 8854, 9321, 9800, 10291, 10794, 11309, 11836, 12375

Offset: 0

Omar E. Pol, Aug 04 2012

Sequence found by reading the line from 0, in the direction 0, 34, ... and the line from 11 in the direction 11, 69, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016. See 3rd formula in Proposition 3 p. 36 giving a(n+1).
Leo Tavares, Star illustration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195818.
Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, A211013, this sequence.
Cf. A051866.
Cf. A003154.

List([0..50], n-> n*(6*n+5)); # G. C. Greubel, Jul 04 2019

[n*(6*n+5): n in [0..50]]; // G. C. Greubel, Jul 04 2019

Table[n*(6*n+5), {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)
LinearRecurrence[{3,-3,1},{0,11,34},50] (* Harvey P. Dale, Dec 12 2022 *)

a(n)=n*(6*n+5) \\ Charles R Greathouse IV, Jun 17 2017

[n*(6*n+5) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = -2*Sum_{k=0..n-1} binomial(6*n+5, 6*k+8)*Bernoulli(6*k+8). - Michel Marcus, Jan 11 2016
From G. C. Greubel, Jul 04 2019: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(11+x)/(1-x)^3.
E.g.f.: x*(11+6*x)*exp(x). (End)
From Amiram Eldar, Feb 28 2022: (Start)
Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/10 + 6/25 - 3*log(3)/10 - 2*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/5 + log(2)/5 - 6/25 - sqrt(3)*log(sqrt(3)+2)/5. (End)
a(n) = A003154(n+1) - n - 1. - Leo Tavares, Jan 29 2023

A284372 a(n) = Sum_{d|n, d = 0, 1, or 11 mod 12} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 13, 14, 1, 1, 1, 1, 1, 1, 1, 1, 12, 24, 37, 26, 14, 1, 1, 1, 1, 1, 1, 12, 1, 36, 49, 38, 1, 14, 1, 1, 1, 1, 12, 1, 24, 48, 85, 50, 26, 1, 14, 1, 1, 12, 1, 1, 1, 60, 73, 62, 1, 1, 1, 14, 12, 1, 1, 24, 36, 72, 145, 74, 38, 26, 1

Offset: 1

Seiichi Manyama, Mar 25 2017

			From _Peter Bala_, Dec 11 2020: (Start)
n = 24: n is not of the form m*(6*m +- 5), so e(n) = 0 and a(24) = a(23) + a(13) - a(10)  = 24 + 14 - 1  = 37;
n = 39: n = m*(6*m - 5) for m = 3, so e(n) = 39 and a(39) = 39 + a(38) + a(28) - a(25) - a(5) = 39 + 1 + 1 - 26 - 1 = 14;
n = 76: n = m*(6*m - 5) for m = 4, so e(n) = -76 and a(4) = -76 + a(75) + a(65) - a(62) - a(42) + a(37) + a(7) = -76 + 26 + 14  - 1 - 1 + 38 + 1 = 1. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A210964 (1/f(-x, -x^11)), A245058.

Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A113184(k=4), A284361 (k=5), A284362 (k=6), A284363 (k=7), this sequence (k=12).

Table[Sum[If[Mod[d, 12]<2 || Mod[d, 12]==11, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
sd12[n_]:=Total[Select[Divisors[n],MemberQ[{0,1,11},Mod[#,12]]&]]; Array[sd12,80] (* Harvey P. Dale, Aug 29 2024 *)

a(n) = sumdiv(n, d, ((d + 1) % 12 < 3) * d); \\ Amiram Eldar, Apr 12 2024

From Peter Bala, Dec 11 2020: (Start)
O.g.f.: Sum_{k >= 1, k == 0, 1 or 11 (mod 12)} k*x^k/(1 - x^k).
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - -, where [1, 11, 14, 34, ...] is the sequence of generalized 14-gonal numbers A195818, and e(n) = (-1)^(m+1)*n if n is a generalized 14-gonal number of the form m*(6*m+-5); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/48 = -A245058 = 0.205616... . - Amiram Eldar, Apr 12 2024

A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m((n - 2)m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, -3, 0, 1, -2, 0, 0, 1, -1, 1, -8, 0, 1, 0, 2, -5, -3, 0, 1, 1, 3, -2, 0, -15, 0, 1, 2, 4, 1, 3, -9, -8, 0, 1, 3, 5, 4, 6, -3, -2, -24, 0, 1, 4, 6, 7, 9, 3, 4, -14, -15, 0, 1, 5, 7, 10, 12, 9, 10, -4, -5, -35, 0, 1, 6, 8, 13, 15, 15, 16, 6, 5, -20, -24, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, -5, -9, -48

Offset: 0

Omar E. Pol, Jun 08 2018

Note that the formula mentioned in the definition gives several kinds of numbers, for example:
Row 0 and row 1 give A317300 and A317301 respectively.
Row 2 gives A001057 (canonical enumeration of integers).
Row 3 gives 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Row 4 gives A008794 (squares repeated) except the initial zero.
Finally, for n >= 5 row n gives the generalized k-gonal numbers (see Crossrefs section).

			Array begins:
------------------------------------------------------------------
n\m  Seq. No.    0   1  -1   2  -2   3   -3    4   -4    5   -5
------------------------------------------------------------------
0    A317300:    0,  1, -3,  0, -8, -3, -15,  -8, -24, -15, -35...
1    A317301:    0,  1, -2,  1, -5,  0,  -9,  -2, -14,  -5, -20...
2    A001057:    0,  1, -1,  2, -2,  3,  -3,   4,  -4,   5,  -5...
3   (A008795):   0,  1,  0,  3,  1,  6,   3,  10,   6,  15,  10...
4   (A008794):   0,  1,  1,  4,  4,  9,   9,  16,  16,  25,  25...
5    A001318:    0,  1,  2,  5,  7, 12,  15,  22,  26,  35,  40...
6    A000217:    0,  1,  3,  6, 10, 15,  21,  28,  36,  45,  55...
7    A085787:    0,  1,  4,  7, 13, 18,  27,  34,  46,  55,  70...
8    A001082:    0,  1,  5,  8, 16, 21,  33,  40,  56,  65,  85...
9    A118277:    0,  1,  6,  9, 19, 24,  39,  46,  66,  75, 100...
10   A074377:    0,  1,  7, 10, 22, 27,  45,  52,  76,  85, 115...
11   A195160:    0,  1,  8, 11, 25, 30,  51,  58,  86,  95, 130...
12   A195162:    0,  1,  9, 12, 28, 33,  57,  64,  96, 105, 145...
13   A195313:    0,  1, 10, 13, 31, 36,  63,  70, 106, 115, 160...
14   A195818:    0,  1, 11, 14, 34, 39,  69,  76, 116, 125, 175...
15   A277082:    0,  1, 12, 15, 37, 42,  75,  82, 126, 135, 190...
...

Alois P. Heinz, Antidiagonals n = 0..200 (first 45 antidiagonals from Robert G. Wilson v)

Columns 0..2 are A000004, A000012, A023445.
Column 3 gives A001477 which coincides with the row numbers.
Main diagonal gives A292551.
Row 0-2 gives A317300, A317301, A001057.
Row 3 gives 0 together with A008795.
Row 4 gives A008794.
For n >= 5, rows n gives the generalized n-gonal numbers: A001318 (n=5), A000217 (n=6), A085787 (n=7), A001082 (n=8), A118277 (n=9), A074377 (n=10), A195160 (n=11), A195162 (n=12), A195313 (n=13), A195818 (n=14), A277082 (n=15), A274978 (n=16), A303305 (n=17), A274979 (n=18), A303813 (n=19), A218864 (n=20), A303298 (n=21), A303299 (n=22), A303303 (n=23), A303814 (n=24), A303304 (n=25), A316724 (n=26), A316725 (n=27), A303812 (n=28), A303815 (n=29), A316729 (n=30).
Cf. A194801, A195152.
Cf. A317302 (a similar table but with polygonal numbers).

t[n_, r_] := PolygonalNumber[n, If[OddQ@ r, Floor[(r + 1)/2], -r/2]]; Table[ t[n - r, r], {n, 0, 11}, {r, 0, n}] // Flatten (* also *)
(* to view the square array *)  Table[ t[n, r], {n, 0, 15}, {r, 0, 10}] // TableForm (* Robert G. Wilson v, Aug 08 2018 *)

T(n,k) = A194801(n-3,k) if n >= 3.

A186742 Expansion of f(x, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jan 21 2012

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + x^11 + x^14 + x^34 + x^39 + x^69 + x^76 + x^116 + x^125 + ...
G.f. = q^25 + q^49 + q^289 + q^361 + q^841 + q^961 + q^1681 + q^1849 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A010815, A195818.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := With[{m = Sqrt[24 n + 25]}, If[ n >= 0 && IntegerQ @ m, Boole[ Mod[m, 12] == 5 || Mod[m, 12] == 7], 0]]; (* Michael Somos, Aug 28 2017 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^12] QPochhammer[ -x^1, x^12] QPochhammer[ -x^11, x^12], {x, 0, n}]; (* Michael Somos, Aug 28 2017 *)

{a(n) = my(m); n++; if( !issquare( 24*n + 1, &m), 0, m%12 == 5 || m%12 == 7)};

{a(n) = my(A); n = 3*n + 3; if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A) - eta(x^3 + A) - x * eta(x^18 + A)^2 / eta(x^9 + A)) / 2, n))}; /* Michael Somos, Aug 28 2017 */

Euler transform of period 24 sequence [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, ...].
a(n) is the characteristic function of A195818. a(n) = max( 0, -A010815(n + 1)).
G.f.: Sum_{k in Z} x^(6*k^2 - 5*k) = Product_{k>0} (1 + x^(12*k - 11)) * (1 + x^(12*k - 1)) * (1 - x^(12*k)).
Expansion of (f(x, x^2) - f(-x, -x^2)) / (2*x) in powers of x. - Michael Somos, Aug 28 2017
Sum_{k=1..n} a(k) ~ sqrt(2*n/3). - Amiram Eldar, Jan 13 2024

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)

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A195817 Multiples of 10 and odd numbers interleaved.

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A210964 Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954.

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A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

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A195158 Concentric 24-gonal numbers.

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A195824 a(n) = 24*n^2.

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A211014 Second 14-gonal numbers: n*(6*n+5).

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A284372 a(n) = Sum_{d|n, d = 0, 1, or 11 mod 12} d.

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A211014 Second 14-gonal numbers: n(6n+5).

A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m((n - 2)m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

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