A195160 -id:A195160 - cp's OEIS frontend

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

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

A027468 9 times the triangular numbers A000217.

Original entry on oeis.org

0, 9, 27, 54, 90, 135, 189, 252, 324, 405, 495, 594, 702, 819, 945, 1080, 1224, 1377, 1539, 1710, 1890, 2079, 2277, 2484, 2700, 2925, 3159, 3402, 3654, 3915, 4185, 4464, 4752, 5049, 5355, 5670, 5994, 6327, 6669, 7020, 7380, 7749, 8127, 8514, 8910, 9315

Offset: 0

Olivier Gérard

Staggered diagonal of triangular spiral in A051682, between (0,1,11) spoke and (0,8,25) spoke. - Paul Barry, Mar 15 2003
Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-2 fixed points. - Zerinvary Lajos, Oct 15 2006
Number of n permutations (n>=2) of 4 objects u, v, z, x with repetition allowed, containing n-2=0 u's. Example: if n=2 then n-2 =zero (0) u, a(1)=9 because we have vv, zz, xx, vx, xv, zx, xz, vz, zv. A027465 formatted as a triangular array: diagonal: 9, 27, 54, 90, 135, 189, 252, 324, ... . - Zerinvary Lajos, Aug 06 2008
a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is a multiple of 3. - Augustine O. Munagi, Dec 18 2008
Also sequence found by reading the line from 0, in the direction 0, 9, ..., and the same line from 0, in the direction 0, 27, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Axis perpendicular to A195147 in the same spiral. - Omar E. Pol, Sep 18 2011
Sum of the numbers from 4*n to 5*n. - Wesley Ivan Hurt, Nov 01 2014

			The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4 are 3+3+3, 6+6+6+3+3+3, 9+9+9+6+6+6+3+3+3, 12+12+12+9+9+9+6+6+6+3+3+3. - _Augustine O. Munagi_, Dec 18 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Augustine O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., Vol. 308, No. 12 (2008), pp. 2492-2501.
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Leo Tavares, Illustration: Centroid Triangles.
D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, Vol. 7, No. 1 (2007), pp. 135-162.
D. Zvonkine, Home Page.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for two-way infinite sequences.

Third diagonal of A027465.
Cf. A000459, A002378, A008585, A024966, A028895, A028896, A038764, A033996, A045943, A046092, A049598, A059073, A080855, A134171, A283394.
Cf. A060544.

[9*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, Dec 29 2012

[seq(9*binomial(n+1,2), n=0..50)]; # Zerinvary Lajos, Nov 24 2006

Table[(9/2)*n*(n+1), {n,0,50}] (* G. C. Greubel, Aug 22 2017 *)

PARI
```
a(n)=9*n*(n+1)/2
```

[9*binomial(n+1, 2) for n in (0..50)] # G. C. Greubel, May 20 2021

Numerators of sequence a[n, n-2] in (a[i, j])^2 where a[i, j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.
a(n) = (9/2)*n*(n+1).
a(n) = 9*C(n, 1) + 9*C(n, 2) (binomial transform of (0, 9, 9, 0, 0, ...)). - Paul Barry, Mar 15 2003
G.f.: 9*x/(1-x)^3.
a(-1-n) = a(n).
a(n) = 9*C(n+1,2), n>=0. - Zerinvary Lajos, Aug 06 2008
a(n) = a(n-1) + 9*n (with a(0)=0). - Vincenzo Librandi, Nov 19 2010
a(n) = A060544(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A218470(9*n+8). - Philippe Deléham, Mar 27 2013
E.g.f.: (9/2)*x*(x+2)*exp(x). - G. C. Greubel, Aug 22 2017
a(n) = A060544(n+1) - 1. See Centroid Triangles illustration. - Leo Tavares, Dec 27 2021
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 2/9. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(9/(2*Pi))*cos(sqrt(17)*Pi/6).
Product_{n>=1} (1 + 1/a(n)) = 9*sqrt(3)/(4*Pi). (End)

More terms from Patrick De Geest, Oct 15 1999

A195159 Multiples of 7 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 7, 3, 14, 5, 21, 7, 28, 9, 35, 11, 42, 13, 49, 15, 56, 17, 63, 19, 70, 21, 77, 23, 84, 25, 91, 27, 98, 29, 105, 31, 112, 33, 119, 35, 126, 37, 133, 39, 140, 41, 147, 43, 154, 45, 161, 47, 168, 49, 175, 51, 182, 53, 189, 55, 196, 57, 203, 59, 210, 61

Offset: 0

Omar E. Pol, Sep 10 2011

This is 7*n if n is even, n if n is odd, if n>=0.
Partial sums give the generalized 11-gonal (or hendecagonal) numbers A195160.
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 11-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).

Cf. A008589 and A005408 interleaved.
Column k=7 of A195151.
Cf. 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, this sequence, A195161.
Cf. A056020, A195160.

&cat[[7*n, 2*n+1]: n in [0..40]]; // Vincenzo Librandi, Sep 27 2011

Table[If[EvenQ[n], 7(n/2), n], {n, 0, 61}] (* Alonso del Arte, Sep 14 2011 *)
With[{nn=40},Riffle[7*Range[0,nn],Range[1,2nn,2]]] (* Harvey P. Dale, Aug 01 2019 *)

a(n)=(5*(-1)^n+9)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

a(2n) = 7n, a(2n+1) = 2n+1. [corrected by Omar E. Pol, Jul 26 2018]
From Bruno Berselli, Sep 14 2011: (Start)
G.f.: x*(1+7*x+x^2)/((1-x)^2*(1+x)^2).
a(n) = (5*(-1)^n+9)*n/4.
a(n) + a(n-1) = A056020(n). (End)
Multiplicative with a(2^e) = 7*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 5/2^s). - Amiram Eldar, Oct 25 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

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, ...