A118277 -id:A118277 - cp's OEIS frontend

A303305 Generalized 17-gonal (or heptadecagonal) numbers: m(15m - 13)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

0, 1, 14, 17, 43, 48, 87, 94, 146, 155, 220, 231, 309, 322, 413, 428, 532, 549, 666, 685, 815, 836, 979, 1002, 1158, 1183, 1352, 1379, 1561, 1590, 1785, 1816, 2024, 2057, 2278, 2313, 2547, 2584, 2831, 2870, 3130, 3171, 3444, 3487, 3773, 3818, 4117, 4164, 4476, 4525, 4850

Offset: 0

Omar E. Pol, Jun 06 2018

120*a(n) + 169 is a square. - Bruno Berselli, Jun 08 2018
Partial sums of A317313. - Omar E. Pol, Jul 28 2018
Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. They are also the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, k >= 5. In this case k = 17. - Omar E. Pol, Apr 25 2021

			From _Omar E. Pol_, Apr 24 2021: (Start)
Illustration of initial terms as vertices of a rectangular spiral:
        43_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _17
         |                                                   |
         |                         0                         |
         |                         |_ _ _ _ _ _ _ _ _ _ _ _ _|
         |                         1                         14
         |
        48
More generally, all generalized k-gonal numbers can be represented with this kind of spirals, k >= 5". (End)

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. A051869, A194715, A244636, A317313.

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), this sequence (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), A316729 (k=30).

With[{pp = 17, nn = 55}, {0}~Join~Riffle[Array[PolygonalNumber[pp, #] &, Ceiling[nn/2]], Array[PolygonalNumber[pp, -#] &, Ceiling[nn/2]]]] (* Michael De Vlieger, Jun 06 2018 *)
Table[(30 n (n + 1) + 11 (2 n + 1) (-1)^n - 11)/16, {n, 0, 60}] (* Bruno Berselli, Jun 08 2018 *)
CoefficientList[ Series[-x (x^2 + 13x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 14, 17, 43}, 51] (* Robert G. Wilson v, Jul 28 2018 *)

concat(0, Vec(x*(1 + 13*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jun 12 2018

From Bruno Berselli, Jun 08 2018: (Start)
G.f.: x*(1 + 13*x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (30*n*(n + 1) + 11*(2*n + 1)*(-1)^n - 11)/16. Therefore:
a(n) = n*(15*n + 26)/8, if n is even, or (n + 1)*(15*n - 11)/8 otherwise.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(15*n^2 - 13) = 0. (End)

A303299 Generalized 22-gonal (or icosidigonal) numbers: m(10m - 9) with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 19, 22, 58, 63, 117, 124, 196, 205, 295, 306, 414, 427, 553, 568, 712, 729, 891, 910, 1090, 1111, 1309, 1332, 1548, 1573, 1807, 1834, 2086, 2115, 2385, 2416, 2704, 2737, 3043, 3078, 3402, 3439, 3781, 3820, 4180, 4221, 4599, 4642, 5038, 5083, 5497, 5544, 5976, 6025, 6475, 6526, 6994, 7047, 7533, 7588

Offset: 0

Omar E. Pol, Jun 23 2018

Partial sums of A317318. - Omar E. Pol, Jul 28 2018

Exponents in expansion of Product_{n >= 1} (1 + x^(20*n-19))*(1 + x^(20*n-1))*(1 - x^(20*n)) = 1 + x + x^19 + x^22 + x^58 + .... - Peter Bala, Dec 10 2020

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. A051874, A317318.

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), this sequence (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

a:= n-> (m-> m*(10*m-9))(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 23 2018

CoefficientList[ Series[-x (x^2 + 18x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 19, 22, 58}, 51] (* Robert G. Wilson v, Jul 28 2018 *)
nn=30; Sort[Table[n (10 n - 9), {n, -nn, nn}]] (* Vincenzo Librandi, Jul 29 2018 *)

a(n) = n++; my(m = (-1) ^ n * (n >> 1)); m * (10 * m - 9) \\ David A. Corneth, Jun 23 2018

concat(0, Vec(x*(1 + 18*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jun 23 2018

From Colin Barker, Jun 23 2018: (Start)
G.f.: x*(1 + 18*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = (5*n^2 + 9*n)/2 for n even.
a(n) = (5*n^2 + n - 4)/2 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) = (10 + 9*sqrt(5+2*sqrt(5))*Pi)/81. - Amiram Eldar, Mar 01 2022

A303303 Generalized 23-gonal (or icositrigonal) numbers: m(21m - 19)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 20, 23, 61, 66, 123, 130, 206, 215, 310, 321, 435, 448, 581, 596, 748, 765, 936, 955, 1145, 1166, 1375, 1398, 1626, 1651, 1898, 1925, 2191, 2220, 2505, 2536, 2840, 2873, 3196, 3231, 3573, 3610, 3971, 4010, 4390, 4431, 4830, 4873, 5291, 5336, 5773, 5820, 6276, 6325, 6800, 6851, 7345, 7398, 7911, 7966

Offset: 0

Omar E. Pol, Jun 24 2018

168*a(n) + 361 is a square. - Bruno Berselli, Jul 10 2018

Partial sums of A317319. - 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. A051875, A317319.

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), this sequence (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

CoefficientList[ Series[-x (x^2 + 19x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 20, 23, 61}, 51] (* Robert G. Wilson v, Jul 28 2018 *)

concat(0, Vec(x*(1 + 19*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Jun 27 2018

From Colin Barker, Jun 27 2018: (Start)
G.f.: x*(1 + 19*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = n*(21*n + 38) / 8 for n even.
a(n) = (21*n - 17)*(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) = 42/361 + 2*Pi*cot(2*Pi/21)/19. - Amiram Eldar, Mar 01 2022

A303304 Generalized 25-gonal (or icosipentagonal) numbers: m(23m - 21)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 22, 25, 67, 72, 135, 142, 226, 235, 340, 351, 477, 490, 637, 652, 820, 837, 1026, 1045, 1255, 1276, 1507, 1530, 1782, 1807, 2080, 2107, 2401, 2430, 2745, 2776, 3112, 3145, 3502, 3537, 3915, 3952, 4351, 4390, 4810, 4851, 5292, 5335, 5797, 5842, 6325, 6372, 6876, 6925

Offset: 0

Omar E. Pol, Jul 10 2018

Numbers k for which 184*k + 441 is a square. - Bruno Berselli, Jul 10 2018

Partial sums of A317321. - 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. A255184, A317321.

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), this sequence (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

a:=[0,1,22,25,67];;  for n in [6..50] 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 10 2018

seq(coeff(series(x*(x^2+21*x+1)/((1-x)^3*(1+x)^2), x,n+1),x,n),n=0..50); # Muniru A Asiru, Jul 10 2018

CoefficientList[Series[x (1 + 21 x + x^2)/((1 - x)^3*(1 + x)^2), {x, 0, 49}], x] (* or *)
Array[PolygonalNumber[25, (1 - 2 Boole[EvenQ@ #]) Ceiling[#/2]] &, 50, 0] (* Michael De Vlieger, Jul 10 2018 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 22, 25, 67}, 50] (* Robert G. Wilson v, Jul 15 2018 *)

concat(0, Vec(x*(1 + 21*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jul 10 2018

From Colin Barker, Jul 10 2018: (Start)
G.f.: x*(1 + 21*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = n*(23*n + 42)/8 for n even.
a(n) = (23*n - 19)*(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) = 46/441 + 2*Pi*cot(2*Pi/23)/21. - Amiram Eldar, Mar 01 2022

A303812 Generalized 28-gonal (or icosioctagonal) numbers: m(13m - 12) with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 25, 28, 76, 81, 153, 160, 256, 265, 385, 396, 540, 553, 721, 736, 928, 945, 1161, 1180, 1420, 1441, 1705, 1728, 2016, 2041, 2353, 2380, 2716, 2745, 3105, 3136, 3520, 3553, 3961, 3996, 4428, 4465, 4921, 4960, 5440, 5481, 5985, 6028, 6556, 6601, 7153, 7200, 7776, 7825, 8425, 8476, 9100, 9153

Offset: 0

Omar E. Pol, Jun 12 2018

Partial sums of A317324. - Omar E. Pol, Jul 28 2018

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

Cf. A161935, A303814, A303815, A317324.

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), this sequence (k=28), A303815 (k=29), A316729 (k=30).

I:=[0,1,25,28,76]; [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..60]]; // Vincenzo Librandi, Jun 23 2018

With[{nn = 54, s = 28}, {0}~Join~Riffle[Array[PolygonalNumber[s, #] &, Ceiling[nn/2]], Array[PolygonalNumber[s, -#] &, Ceiling[nn/2]]]] (* Michael De Vlieger, Jun 14 2018 *)
CoefficientList[Series[x (1 + 24 x + x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Jun 23 2018 *)

G.f.: x*(1 + 24*x + x^2) / ((1 + x)^2*(1 - x)^3). - Vincenzo Librandi, Jun 23 2018
From Amiram Eldar, Mar 01 2022: (Start)
a(n) = (26*n*(n + 1) + 11*(2*n + 1)*(-1)^n - 11)/8.
a(n) = n*(13*n + 24)/4, if n is even, or (n + 1)*(13*n - 11)/4 otherwise.
Sum_{n>=1} 1/a(n) = 13/144 + Pi*cot(Pi/13)/12. (End)

A303814 Generalized 24-gonal (or icositetragonal) numbers: m(11m - 10) with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 21, 24, 64, 69, 129, 136, 216, 225, 325, 336, 456, 469, 609, 624, 784, 801, 981, 1000, 1200, 1221, 1441, 1464, 1704, 1729, 1989, 2016, 2296, 2325, 2625, 2656, 2976, 3009, 3349, 3384, 3744, 3781, 4161, 4200, 4600, 4641, 5061, 5104, 5544, 5589, 6049, 6096, 6576, 6625

Offset: 0

Omar E. Pol, Jun 06 2018

a(25) = 1729 is the Hardy-Ramanujan number.
Numbers k such that 11*k + 25 is a square. - Bruno Berselli, Jun 08 2018
Partial sums of A317320. - 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. A051876, A317320.

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), this sequence (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

With[{pp = 24, nn = 55}, {0}~Join~Riffle[Array[PolygonalNumber[pp, #] &, Ceiling[nn/2]], Array[PolygonalNumber[pp, -#] &, Ceiling[nn/2]]]] (* Michael De Vlieger, Jun 06 2018 *)
Table[(22 n (n + 1) + 9 (2 n + 1) (-1)^n - 9)/8, {n, 0, 50}] (* Bruno Berselli, Jun 08 2018 *)
CoefficientList[ Series[-x (x^2 + 20x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 21, 24, 64}, 50] (* Robert G. Wilson v, Jul 28 2018 *)

concat(0, Vec(x*(1 + 20*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jun 12 2018

From Bruno Berselli, Jun 08 2018: (Start)
G.f.: x*(1 + 20*x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (22*n*(n + 1) + 9*(2*n + 1)*(-1)^n - 9)/8. Therefore:
a(n) = n*(11*n + 20)/4, if n is even, or (n + 1)*(11*n - 9)/4 otherwise.
(2*n - 1)*a(n) + (2*n + 1)*a(n-1) - n*(11*n^2 - 10) = 0. (End)
Sum_{n>=1} 1/a(n) = (11 + 10*Pi*cot(Pi/11))/100. - Amiram Eldar, Mar 01 2022

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

Original entry on oeis.org

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

A144555 a(n) = 14*n^2.

Original entry on oeis.org

0, 14, 56, 126, 224, 350, 504, 686, 896, 1134, 1400, 1694, 2016, 2366, 2744, 3150, 3584, 4046, 4536, 5054, 5600, 6174, 6776, 7406, 8064, 8750, 9464, 10206, 10976, 11774, 12600, 13454, 14336, 15246, 16184, 17150, 18144, 19166, 20216, 21294, 22400, 23534, 24696

Offset: 0

N. J. A. Sloane, Jan 01 2009

Sequence found by reading the line from 0, in the direction 0, 14, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line and direction in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 10 2011

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

See also A033428, A033429, A033581, A033582, A033583, A033584, ... and A249327 for the whole table.
Cf. A000290, A001105, A064761, A118277, A152742, A195019, A195020.
Cf. A008596, A195145.

Table[14*n^2, {n, 0, 45}] (* Amiram Eldar, Feb 03 2021 *)

A144555(n)=14*n^2 \\ M. F. Hasler, Oct 31 2014

a(n) = 14*A000290(n) = 7*A001105(n) = 2*A033582(n). - Omar E. Pol, Jan 01 2009
a(n) = a(n-1) + 14*(2*n-1), with a(0) = 0. - Vincenzo Librandi, Nov 25 2010
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/84.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/168.
Product_{n>=1} (1 + 1/a(n)) = sqrt(14)*sinh(Pi/sqrt(14))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(14)*sin(Pi/sqrt(14))/Pi. (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 14*x*(1 + x)/(1-x)^3.
E.g.f.: 14*x*(1 + x)*exp(x).
a(n) = n*A008596(n) = A195145(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

cp's OEIS Frontend

A303305 Generalized 17-gonal (or heptadecagonal) numbers: m*(15*m - 13)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

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A303299 Generalized 22-gonal (or icosidigonal) numbers: m*(10*m - 9) with m = 0, +1, -1, +2, -2, +3, -3, ...

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A303303 Generalized 23-gonal (or icositrigonal) numbers: m*(21*m - 19)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

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A303304 Generalized 25-gonal (or icosipentagonal) numbers: m*(23*m - 21)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

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A303812 Generalized 28-gonal (or icosioctagonal) numbers: m*(13*m - 12) with m = 0, +1, -1, +2, -2, +3, -3, ...

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A303814 Generalized 24-gonal (or icositetragonal) numbers: m*(11*m - 10) with m = 0, +1, -1, +2, -2, +3, -3, ...

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

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A303305 Generalized 17-gonal (or heptadecagonal) numbers: m(15m - 13)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

A303299 Generalized 22-gonal (or icosidigonal) numbers: m(10m - 9) with m = 0, +1, -1, +2, -2, +3, -3, ...

A303303 Generalized 23-gonal (or icositrigonal) numbers: m(21m - 19)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

A303304 Generalized 25-gonal (or icosipentagonal) numbers: m(23m - 21)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

A303812 Generalized 28-gonal (or icosioctagonal) numbers: m(13m - 12) with m = 0, +1, -1, +2, -2, +3, -3, ...

A303814 Generalized 24-gonal (or icositetragonal) numbers: m(11m - 10) with m = 0, +1, -1, +2, -2, +3, -3, ...

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, ...