A001082 -id:A001082 - cp's OEIS frontend

A316724 Generalized 26-gonal (or icosihexagonal) numbers: m(12m - 11) with m = 0, +1, -1, +2, -2, +3, -3, ...

0, 1, 23, 26, 70, 75, 141, 148, 236, 245, 355, 366, 498, 511, 665, 680, 856, 873, 1071, 1090, 1310, 1331, 1573, 1596, 1860, 1885, 2171, 2198, 2506, 2535, 2865, 2896, 3248, 3281, 3655, 3690, 4086, 4123, 4541, 4580, 5020, 5061, 5523, 5566, 6050, 6095, 6601, 6648, 7176, 7225, 7775

Offset: 0

Omar E. Pol, Jul 11 2018

48*a(n) + 121 is a square. - Bruno Berselli, Jul 11 2018

Partial sums of A317322. - Omar E. Pol, Jul 28 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A255185, A317322.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), this sequence (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

[(12*n*(n+1) + 5*(-1)^n*(2*n+1) -5)/4: n in [0..60]]; // G. C. Greubel, Sep 24 2024

Table[(12 n (n + 1) + 5 (2 n + 1) (-1)^n - 5)/4, {n, 0, 60}] (* Bruno Berselli, Jul 11 2018 *)
CoefficientList[ Series[-x (x^2 + 22x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 60}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 23, 26, 70}, 60] (* Robert G. Wilson v, Jul 28 2018 *)
nn=30; Sort[Table[n (12 n - 11), {n, -nn, nn}]] (* Vincenzo Librandi, Jul 29 2018 *)

concat(0, Vec(x*(1 + 22*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^60))) \\ Colin Barker, Jul 12 2018

[(12*n*(n+1) + 5*(-1)^n*(2*n+1) -5)//4 for n in range(61)] # G. C. Greubel, Sep 24 2024

From Bruno Berselli, Jul 11 2018: (Start)
O.g.f.: x*(1 + 22*x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-1-n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (12*n*(n + 1) + 5*(2*n + 1)*(-1)^n - 5)/4. Therefore:
a(n) = n*(6*n + 11)/2 for n even; otherwise, a(n) = (n + 1)*(6*n - 5)/2.
(2*n - 1)*a(n) + (2*n + 1)*a(n-1) - n*(12*n^2 - 11) = 0. (End)
From Amiram Eldar, Mar 01 2022: (Start)
Sum_{n>=1} 1/a(n) = 12/121 + (sqrt(3)+2)*Pi/11.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(3)*log(sqrt(3)+2) + 6*log(2) + 3*log(3))/11 - 12/121. (End)
E.g.f.: (1/4)*(5*(1 - 2*x)*exp(-x) + (-5 + 24*x + 12*x^2)*exp(x)). - G. C. Greubel, Sep 24 2024

A316725 Generalized 27-gonal (or icosiheptagonal) numbers: m(25m - 23)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 24, 27, 73, 78, 147, 154, 246, 255, 370, 381, 519, 532, 693, 708, 892, 909, 1116, 1135, 1365, 1386, 1639, 1662, 1938, 1963, 2262, 2289, 2611, 2640, 2985, 3016, 3384, 3417, 3808, 3843, 4257, 4294, 4731, 4770, 5230, 5271, 5754, 5797, 6303, 6348, 6877, 6924, 7476, 7525, 8100, 8151, 8749, 8802

Offset: 0

Omar E. Pol, Jul 11 2018

Note that in the sequences of generalized k-gonal numbers always a(3) = k. In this case k = 27.
Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, with k >= 5.
A general formula for the generalized k-gonal numbers is given by m*((k-2)*m-k+4)/2, with m = 0, +1, -1, +2, -2, +3, -3, ..., k >= 5.
Partial sums of A317323. - Omar E. Pol, Jul 28 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A255186; A317323.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), this sequence (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

a:=[0,1,24,27,73];;  for n in [6..60] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Jul 16 2018

a:= n-> (m-> m*(25*m-23)/2)(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 11 2018

CoefficientList[Series[-x (x^2 + 23x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 53}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 24, 27, 73, 78, 147}, 53] (* Robert G. Wilson v, Jul 28 2018; corrected by Georg Fischer, Apr 03 2019 *)
nn=30; Sort[Table[n (25 n - 23) / 2, {n, -nn, nn}]] (* Vincenzo Librandi, Jul 29 2018 *)

concat(0, Vec(x*(1 + 23*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jul 11 2018

From Colin Barker, Jul 11 2018: (Start)
G.f.: x*(1 + 23*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = n*(25*n + 46)/8 for n even.
a(n) = (25*n - 21)*(n + 1)/8 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)
Sum_{n>=1} 1/a(n) =  2*(25 + 23*Pi*cot(2*Pi/25))/529. - Amiram Eldar, Mar 01 2022

A316729 Generalized 30-gonal (or triacontagonal) numbers: m(14m - 13) with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 27, 30, 82, 87, 165, 172, 276, 285, 415, 426, 582, 595, 777, 792, 1000, 1017, 1251, 1270, 1530, 1551, 1837, 1860, 2172, 2197, 2535, 2562, 2926, 2955, 3345, 3376, 3792, 3825, 4267, 4302, 4770, 4807, 5301, 5340, 5860, 5901, 6447, 6490, 7062, 7107, 7705, 7752, 8376, 8425, 9075, 9126, 9802, 9855

Offset: 0

Omar E. Pol, Jul 11 2018

Note that in the sequences of generalized k-gonal numbers always a(3) = k. In this case k = 30.
Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, with k >= 5.
A general formula for the generalized k-gonal numbers is given by m*((k-2)*m-k+4)/2, with m = 0, +1, -1, +2, -2, +3, -3, ..., k >= 5.
Every sequence of generalized k-gonal numbers can be represented as vertices of a rectangular spiral constructed with line segments on the square grid, with k >= 5.
56*a(n) + 169 is a square. - Vincenzo Librandi, Jul 12 2018
Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, with k >= 5. - Omar E. Pol, Jul 27 2018
Also partial sums of A317326. - Omar E. Pol, Jul 28 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A254474, A317326.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), this sequence (k=30).

CoefficientList[Series[x (1 + 26 x + x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 55}], x] (* Vincenzo Librandi, Jul 12 2018 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 27, 30, 82}, 47] (* Robert G. Wilson v, Jul 28 2018 *)

concat(0, Vec(x*(1 + 26*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jul 16 2018

G.f.: x*(1 + 26*x + x^2)/((1 + x)^2*(1 - x)^3). - Vincenzo Librandi, Jul 12 2018
From Amiram Eldar, Mar 01 2022: (Start)
a(n) = (28*n*(n + 1) + 12*(2*n + 1)*(-1)^n - 12)/8.
a(n) = n*(7*n + 13)/2, if n is even, or (n + 1)*(7*n - 6)/2 otherwise.
Sum_{n>=1} 1/a(n) = 14/169 + Pi*cot(Pi/14)/13. (End)

Duplicated term (1551) deleted by Colin Barker, Jul 16 2018

A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1. Moreover, any integer n > 1 can be written as x*(3*x+2) + y*(3*y+2) + 3^z + 3^w, where x is an integer and y,z,w are nonnegative integers.
a(n) > 0 for all n = 2..3*10^8. Those x*(3*x-2) with x integral are called generalized octagonal numbers (A001082). 76683391 is the least integer n > 1 not representable as the sum of two generalized octagonal numbers and two powers of 2.
See also A303389, A303401 and A303432 for similar conjectures.

			a(2) = 1 with 2 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^0.
a(3) = 1 with 3 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^0.
a(4) = 2 with 4 = 1*(3*1-2) + 1*(3*1-2) + 3^0 + 3^0 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^1.
a(5) = 1 with 5 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^1.
a(9) = 1 with 9 = (-1)*(3*(-1)-2) + 0*(3*0-2) + 3^0 + 3^1.
a(4360) = 4 with 4360 = (-35)*(3*(-35)-2) + (-13)*(3*(-13)-2) + 3^0 + 3^4 = (-37)*(3*(-37)-2) + (-7)*(3*(-7)-2) + 3^2 + 3^2 = (-27)*(3*(-27)-2) + (-23)*(3*(-23)-2) + 3^5 + 3^5 = (-25)*(3*(-25)-2) + (-1)*(3*(-1)-2) + 3^5 + 3^7.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000244, A000567, A001082, A045944, A280472, A303233, A303338, A303363, A303389, A303393, A303399, A303401, A303432.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[3(n-3^j-3^k)+2],Do[If[SQ[3(n-3^j-3^k-x(3x-2))+1],r=r+1],{x,-Floor[(Sqrt[3(n-3^j-3^k)/2+1]-1)/3],(Sqrt[3(n-3^j-3^k)/2+1]+1)/3}]],
{j,0,Log[3,n/2]},{k,j,Log[3,n-3^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A089801 a(n) = 0 unless n = 3j^2 + 2j or 3j^2 + 4j + 1 for some j >= 0, in which case a(n) = 1.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 12 2003

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also characteristic function of generalized octagonal numbers A001082. - Omar E. Pol, Jul 13 2012
Number 12 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 + x + x^5 + x^8 + x^16 + x^21 + x^33 + x^40 + x^56 + x^65 + x^85 + ...
G.f. = q + q^4 + q^16 + q^25 + q^49 + q^64 + q^100 + q^121 + q^169 + q^196 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Th. 2.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See X(q).
James A. Sellers, Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice, arXiv:2409.12321 [math.NT], 2024. See p. 4.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.
Index entries for characteristic functions.

Cf. A001082, A033684, A080995, A089802, A089807, A280739.

Basis( ModularForms( Gamma0(36), 1/2), 87) [2]; /* Michael Somos, Jul 02 2014 */

A089801 := proc(n)
        A033684(3*n+1) ;
end proc: # R. J. Mathar, Oct 07 2011
M:=33;
S:=f->series(f,q,500);
L:=f->seriestolist(f);
X:=add(q^(3*n^2+2*n),n=-M..M);
S(%);
L(%); # N. J. A. Sloane, Jan 31 2012
eps:=Array(0..120,0);
for j from 0 to 120 do
if 3*j^2+2*j <= 120 then eps[3*j^2+2*j] := 1; fi;
if 3*j^2+4*j+1 <= 120 then eps[3*j^2+4*j+1] := 1; fi;
end do;  # N. J. A. Sloane, Aug 12 2017

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/3) (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3]), {x, 0, n}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Jun 29 2012 *)

{a(n) = issquare(3*n + 1)}; /* Michael Somos, Apr 12 2005 */

from sympy.ntheory.primetest import is_square
def A089801(n): return int(is_square(3*n+1)) # Chai Wah Wu, May 17 2023

G.f.: Sum_{n=-oo..oo} q^(3n^2+2n).
Expansion of Jacobi theta function (theta_3(q^(1/3)) - theta_3(q^3))/(2 q^(1/3)) in powers of q.
Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, -1, ...]. - Michael Somos, Apr 12 2005
a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p != 3. - Michael Somos, Jun 06 2005; b=A033684. - R. J. Mathar, Oct 07 2011
Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q. - Michael Somos, Apr 12 2005
Expansion of chi(x) * psi(-x^3) in powers of x where chi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 19 2007
Expansion of f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 29 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089807.
a(8*n + 4) = a(4*n + 2) = a(4*n + 3) = 0, a(4*n + 1) = a(n), a(8*n) = A080995(n). - Michael Somos, Jan 28 2011
a(n) = (-1)^n * A089802(n).
For n > 0, a(n) = b(n)-b(n-1) + c(n)-c(n-1), where b(n) = floor(sqrt(n/3+1/9)+2/3) and c(n) = floor(sqrt(n/3+1/9)+4/3). - Mikael Aaltonen, Jan 22 2015
a(n) = A033684(3*n + 1). - Michael Somos, Jan 10 2017

Edited with simpler definition by N. J. A. Sloane, Jan 31 2012

Further edited by N. J. A. Sloane, Aug 12 2017

A062717 Numbers m such that 6*m+1 is a perfect square.

Original entry on oeis.org

0, 4, 8, 20, 28, 48, 60, 88, 104, 140, 160, 204, 228, 280, 308, 368, 400, 468, 504, 580, 620, 704, 748, 840, 888, 988, 1040, 1148, 1204, 1320, 1380, 1504, 1568, 1700, 1768, 1908, 1980, 2128, 2204, 2360, 2440, 2604, 2688, 2860, 2948, 3128, 3220, 3408, 3504

Offset: 1

Jason Earls, Jul 14 2001

X values of solutions to the equation 6*X^3 + X^2 = Y^2. - Mohamed Bouhamida, Nov 06 2007
Arithmetic averages of the k triangular numbers 0, 1, 3, 6, ..., (k-1)*k/2 that take integer values. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009 [edited by Jon E. Schoenfield, Jan 10 2015]
Even terms in A186423; union of A033579 and A033580, A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011
a(n) are integers produced by Sum_{i = 1..k-1} i*(k-i)/k, for some k > 0. Values for k are given by A007310 = sqrt(6*a(n)+1), the square roots of those perfect squares. - Richard R. Forberg, Feb 16 2015
Equivalently, numbers of the form 2*h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the sixth comment of A152749). - Bruno Berselli, Feb 02 2017

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Equals 4 * A001318.
Cf. A005563, A046092, A001082, A002378, A036666.
Cf. A160757, A000217. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009
Cf. A007310.
Diagonal of array A323674. - Sally Myers Moite, Feb 03 2019

[(6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4: n in [1..70]]; // Wesley Ivan Hurt, Apr 21 2021

seq(n^2+n+2*ceil(n/2)^2,n=0..48); # Gary Detlefs, Feb 23 2010

Select[Range[0, 3999], IntegerQ[Sqrt[6# + 1]] &] (* Harvey P. Dale, Mar 10 2013 *)

je=[]; for(n=0,7000, if(issquare(6*n+1),je=concat(je,n))); je

{ n=0; for (m=0, 10^9, if (issquare(6*m + 1), write("b062717.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 09 2009

def A062717(n): return (n*(3*n + 4) + 1 if n&1 else n*(3*n + 2))>>1 # Chai Wah Wu, Jan 31 2023

G.f.: 4*x^2*(1 + x + x^2) / ( (1+x)^2*(1-x)^3 ).
a(2*k) = k*(6*k+2), a(2*k+1) = 6*k^2 + 10*k + 4. - Mohamed Bouhamida, Nov 06 2007
a(n) = n^2 - n + 2*ceiling((n-1)/2)^2. - Gary Detlefs, Feb 23 2010
From Bruno Berselli, Nov 28 2010: (Start)
a(n) = (6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4.
6*a(n) + 1 = A007310(n)^2. (End)
E.g.f.: (3*x^2*exp(x) - x*exp(-x) + sinh(x))/2. - Ilya Gutkovskiy, Jun 18 2016
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 21 2021
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=2} 1/a(n) = (9-sqrt(3)*Pi)/6.
Sum_{n>=2} (-1)^n/a(n) = 3*(log(3)-1)/2. (End)

A069190 Centered 24-gonal numbers.

Original entry on oeis.org

1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, 11161, 11905, 12673, 13465, 14281, 15121, 15985, 16873, 17785, 18721, 19681, 20665, 21673

Offset: 1

Terrel Trotter, Jr., Apr 10 2002

Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - Omar E. Pol, Sep 16 2011

			a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Odd Ordered Star Perimeters.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.

FoldList[#1 + #2 &, 1, 24 Range@ 45] (* Robert G. Wilson v *)
Table[12n^2-12n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,25,73},50] (* Harvey P. Dale, Jul 17 2011 *)

a(n)=12*n^2-12*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 12*n^2 - 12*n + 1.
a(n) = 24*n + a(n-1) - 24 with a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - Harvey P. Dale, Jul 17 2011
G.f.: x*(1+22*x+x^2)/(1-x)^3. - Harvey P. Dale, Jul 17 2011
Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).
Sum_{n>=1} a(n)/n! = 13*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)
E.g.f.: exp(x)*(1 + 12*x^2) - 1. - Stefano Spezia, May 31 2022

More terms from Harvey P. Dale, Jul 17 2011

A195152 Square array read by antidiagonals with T(n,k) = n((k+2)n-k)/2, n=0, +- 1, +- 2,..., k>=0.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 4, 5, 3, 1, 0, 9, 7, 6, 4, 1, 0, 9, 12, 10, 7, 5, 1, 0, 16, 15, 15, 13, 8, 6, 1, 0, 16, 22, 21, 18, 16, 9, 7, 1, 0, 25, 26, 28, 27, 21, 19, 10, 8, 1, 0, 25, 35, 36, 34, 33, 24, 22, 11, 9, 1, 0, 36, 40, 45, 46, 40, 39, 27, 25, 12, 10, 1, 0

Offset: 0

Omar E. Pol, Sep 14 2011

Also, column k lists the partial sums of the column k of A195151. The first differences in row n are always the n-th term of the triangular numbers repeated 0,0,1,1,3,3,6,6,... ([0,0] together with A008805).

Also, for k >= 1, this is a table of generalized polygonal numbers since column k lists the generalized m-gonal numbers, where m = k+4, for example: if k = 1 then m = 5, so the column 1 lists the generalized pentagonal numbers A001318 (see example).

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,...
.  4,   5,   6,   7,   8,   9,  10,  11,  12,  13,...
.  4,   7,  10,  13,  16,  19,  22,  25,  28,  31,...
.  9,  12,  15,  18,  21,  24,  27,  30,  33,  36,...
.  9,  15,  21,  27,  33,  39,  45,  51,  57,  63,...
. 16,  22,  28,  34,  40,  46,  52,  58,  64,  70,...
. 16,  26,  36,  46,  56,  66,  76,  86,  96, 106,...
. 25,  35,  45,  55,  65,  75,  85,  95, 105, 115,...
. 25,  40,  55,  70,  85, 100, 115, 130, 145, 160,...
...

Rows: A000004, A000012, A000027.
Column 0 gives A008794, except its first term.
Columns >= 1: A001318, (A000217), A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818.
Cf. A008805, A152151.

T(n,k) = (k+2)*n*(n+1)/8+(k-2)*((2*n+1)*(-1)^n-1)/16, n >= 0 and k >= 0. - Omar E. Pol, Oct 01 2011

A195848 Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 7, 10, 12, 13, 14, 16, 21, 27, 32, 35, 38, 44, 54, 67, 78, 86, 94, 107, 128, 153, 176, 194, 213, 241, 282, 331, 376, 415, 456, 512, 590, 680, 767, 845, 928, 1037, 1180, 1345, 1506, 1657, 1818, 2020, 2278, 2570, 2862, 3142, 3442

Offset: 0

Omar E. Pol, Sep 24 2011

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also column 4 of A195825, therefore this sequence contains two plateaus: [1, 1, 1, 1, 1], [4, 4, 4]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 26 2012
The number of partitions of n into parts congruent to 0, 1 or 5 ( mod 6 ). - Peter Bala, Dec 09 2020

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ...
G.f. = 1/q + q^2 + q^5 + q^8 + q^11 + 2*q^14 + 3*q^17 + 4*q^20 + 4*q^23 + 4*q^26 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Bringmann, J. Lovejoy, and K. Mahlburg, A partition identity and the universal mock theta function g_2(x;q), Mathematical Research Letters, 23 (2016), 67-80.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Column 1 of triangle A195838. Also 1 together with the row sums of triangle A195838. Column 4 of array A195825.
Cf. A000041, A001082, A006950, A036820, A057077, A195825, A195828, A195849, A195850, A195851, A195852, A196933, A210843, A210964, A211971.
Cf. A089802.

A001082 := proc(n)
        if type(n,'even') then
                n*(3*n-4)/4 ;
        else
                (n-1)*(3*n+1)/4 ;
        end if;
end proc:
A195838 := proc(n,k)
        option remember;
        local ks,a,j ;
        if A001082(k+1) > n then
                0 ;
        elif n <= 5 then
                return 1;
        elif k = 1 then
                a := 0 ;
                for j from 1 do
                        if A001082(j+1) <= n-1 then
                                a := a+procname(n-1,j) ;
                        else
                                break;
                        end if;
                end do;
                return a;
        else
                ks := A001082(k+1) ;
                (-1)^floor((k-1)/2)*procname(n-ks+1,1) ;
        end if;
end proc:
A195848 := proc(n)
        A195838(n+1,1) ;
end proc:
seq(A195848(n),n=0..60) ; # R. J. Mathar, Oct 07 2011

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]^2), {x, 0, n}]; (* Michael Somos, Oct 18 2014 *)
a[ n_] := SeriesCoefficient[ 2 q^(3/8) / (QPochhammer[ q, q^2] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Oct 18 2014 *)
nmax = 60; CoefficientList[Series[Product[(1+x^k) / ((1+x^(3*k)) * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 07 2012 */
From Omar E. Pol, Jun 10 2012: (Start)
(GW-BASIC)' A program with two A-numbers:
10 Dim A001082(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 58: For j = 1 to n
30 If A001082(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A001082(j))
40 Next j: Print a(n-1);: Next n (End)

Expansion of 1 / (psi(x^3) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Jun 07 2012
Expansion of q^(1/3) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of q. - Michael Somos, Jun 07 2012
Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, 1, ...]. - Michael Somos, Oct 18 2014
Convolution inverse of A089802. - Michael Somos, Oct 18 2014
a(n) ~ exp(Pi*sqrt(n/3))/(4*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = (1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
From Peter Bala, Dec 09 2020: (Start)
O.g.f.: 1/( Product_{n >= 1} (1 - x^(6*n-5))*(1 - x^(6*n-1))*(1 - x^(6*n)) ).
a(n) = a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - - ... (with the convention a(n) = 0 for negative n), where 1, 5, 8, 16, ... is the sequence of generalized octagonal numbers A001082. (End)

New sequence name from Michael Somos, Oct 18 2014

A219257 Numbers k such that 11*k+1 is a square.

Original entry on oeis.org

0, 9, 13, 40, 48, 93, 105, 168, 184, 265, 285, 384, 408, 525, 553, 688, 720, 873, 909, 1080, 1120, 1309, 1353, 1560, 1608, 1833, 1885, 2128, 2184, 2445, 2505, 2784, 2848, 3145, 3213, 3528, 3600, 3933, 4009, 4360, 4440, 4809, 4893, 5280, 5368, 5773, 5865

Offset: 1

Bruno Berselli, Nov 16 2012

Equivalently, numbers of the form m*(11*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/11.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. numbers k such that h*k+1 is a square: A005563 (h=1), A046092 (h=2), A001082 (h=3), A002378 (h=4), A036666 (h=5), A062717 (h=6), A132354 (h=7), A000217 (h=8), A132355 (h=9), A132356 (h=10), A152749 (h=12), A219389 (h=13), A219390 (h=14), A204221 (h=15), A074378 (h=16), A219394 (h=17), A219395 (h=18), A219396 (h=19), A219190 (h=20), A219391 (h=21), A219392 (h=22), A219393 (h=23), A001318 (h=24), A219259 (h=25), A217441 (h=26), A219258 (h=27), A219191 (h=28).

Cf. A175885 (square roots of 11*a(n)+1).

[n: n in [0..7000] | IsSquare(11*n+1)];

I:=[0,9,13,40,48]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Select[Range[0, 7000], IntegerQ[Sqrt[11 # + 1]] &]
CoefficientList[Series[x (9 + 4 x + 9 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x^2*(9+4*x+9*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (22*n*(n-1)+7*(-1)^n*(2*n-1)-1)/8 + 1 = (1/176)*(22*n+7*(-1)^n-15)*(22*n+7*(-1)^n-7).
Sum_{n>=2} 1/a(n) = 11/4 - cot(2*Pi/11)*Pi/2. - Amiram Eldar, Mar 15 2022

cp's OEIS Frontend

A316724 Generalized 26-gonal (or icosihexagonal) numbers: m*(12*m - 11) with m = 0, +1, -1, +2, -2, +3, -3, ...

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A316725 Generalized 27-gonal (or icosiheptagonal) numbers: m*(25*m - 23)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

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A316729 Generalized 30-gonal (or triacontagonal) numbers: m*(14*m - 13) with m = 0, +1, -1, +2, -2, +3, -3, ...

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A303428 Number of ways to write n as x*(3*x-2) + y*(3*y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

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A089801 a(n) = 0 unless n = 3j^2 + 2j or 3j^2 + 4j + 1 for some j >= 0, in which case a(n) = 1.

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A062717 Numbers m such that 6*m+1 is a perfect square.

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A316724 Generalized 26-gonal (or icosihexagonal) numbers: m(12m - 11) with m = 0, +1, -1, +2, -2, +3, -3, ...

A316725 Generalized 27-gonal (or icosiheptagonal) numbers: m(25m - 23)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

A316729 Generalized 30-gonal (or triacontagonal) numbers: m(14m - 13) with m = 0, +1, -1, +2, -2, +3, -3, ...

A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

A195152 Square array read by antidiagonals with T(n,k) = n((k+2)n-k)/2, n=0, +- 1, +- 2,..., k>=0.