cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A303813 Generalized 19-gonal (or enneadecagonal) numbers: m*(17*m - 15)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 16, 19, 49, 54, 99, 106, 166, 175, 250, 261, 351, 364, 469, 484, 604, 621, 756, 775, 925, 946, 1111, 1134, 1314, 1339, 1534, 1561, 1771, 1800, 2025, 2056, 2296, 2329, 2584, 2619, 2889, 2926, 3211, 3250, 3550, 3591, 3906, 3949, 4279, 4324, 4669, 4716, 5076, 5125, 5500, 5551, 5941, 5994, 6399
Offset: 0

Views

Author

Omar E. Pol, Jun 06 2018

Keywords

Comments

Numbers k for which 136*k + 225 is a square. - Bruno Berselli, Jul 10 2018
Partial sums of A317315. - Omar E. Pol, Jul 28 2018

Links

Crossrefs

Cf. A051871, A316672, A317315.
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), this sequence (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).

Programs

  • GAP
    a:=[0,1,16,19,49];;  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 10 2018
  • Mathematica
    With[{nn = 54}, {0}~Join~Riffle[Array[PolygonalNumber[19, #] &, Ceiling[nn/2]], Array[PolygonalNumber[19, -#] &, Ceiling[nn/2]]]] (* Michael De Vlieger, Jun 06 2018 *)
CoefficientList[ Series[-x (x^2 + 15x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 16, 19, 49}, 51] (* Robert G. Wilson v, Jul 28 2018 *)
  • PARI
    concat(0, Vec(x*(1 + 15*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jun 08 2018

Formula

From Colin Barker, Jun 08 2018: (Start)
G.f.: x*(1 + 15*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = (34*n^2 + 60*n)/16 for n even.
a(n) = (34*n^2 + 8*n - 26)/16 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)

A303815 Generalized 29-gonal (or icosienneagonal) numbers: m*(27*m - 25)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 26, 29, 79, 84, 159, 166, 266, 275, 400, 411, 561, 574, 749, 764, 964, 981, 1206, 1225, 1475, 1496, 1771, 1794, 2094, 2119, 2444, 2471, 2821, 2850, 3225, 3256, 3656, 3689, 4114, 4149, 4599, 4636, 5111, 5150, 5650, 5691, 6216, 6259, 6809, 6854, 7429, 7476, 8076
Offset: 0

Views

Author

Omar E. Pol, Jun 06 2018

Keywords

Comments

Numbers k such that 216*k + 625 is a square. - Bruno Berselli, Jun 08 2018
Partial sums of A317325.

Links

Crossrefs

Cf. A255187, A277990 (see the third comment), A316672, A317325.
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), this sequence (k=29), A316729 (k=30).

Programs

  • Mathematica
    Table[(54 n (n + 1) + 23 (2 n + 1) (-1)^n - 23)/16, {n, 0, 50}] (* Bruno Berselli, Jun 07 2018 *)
CoefficientList[ Series[-x (x^2 + 25x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 26, 29, 79, 84}, 50] (* Robert G. Wilson v, Jul 28 2018 *)
With[{nn=25},Riffle[Table[1-(29x)/2+(27x^2)/2,{x,nn}],PolygonalNumber[ 29,Range[ nn]]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 26 2020 *)
  • PARI
    concat(0, Vec(x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jun 12 2018

Formula

From Bruno Berselli, Jun 07 2018: (Start)
G.f.: x*(1 + 25*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) = (54*n*(n + 1) + 23*(2*n + 1)*(-1)^n - 23)/16. Therefore:
a(n) = n*(27*n + 50)/8, if n is even, or (n + 1)*(27*n - 23)/8 otherwise.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(27*n^2 - 25) = 0. (End)
Sum_{n>=1} 1/a(n) = 2*(27 + 25*Pi*cot(2*Pi/27))/625. - Amiram Eldar, Mar 01 2022

A303298 Generalized 21-gonal (or icosihenagonal) numbers: m*(19*m - 17)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 18, 21, 55, 60, 111, 118, 186, 195, 280, 291, 393, 406, 525, 540, 676, 693, 846, 865, 1035, 1056, 1243, 1266, 1470, 1495, 1716, 1743, 1981, 2010, 2265, 2296, 2568, 2601, 2890, 2925, 3231, 3268, 3591, 3630, 3970, 4011, 4368, 4411, 4785, 4830, 5221, 5268, 5676, 5725, 6150, 6201, 6643, 6696, 7155, 7210
Offset: 0

Views

Author

Omar E. Pol, Jun 23 2018

Keywords

Comments

Numbers k for which 152*k + 289 is a square. - Bruno Berselli, Jul 10 2018
Partial sums of A317317. - Omar E. Pol, Jul 28 2018

Links

Crossrefs

Cf. A051873, A316672, A317317.
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), this sequence (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).

Programs

  • GAP
    a:=[0,1,18,21,55];;  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 10 2018
  • Maple
    a:= n-> (m-> m*(19*m-17)/2)(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 23 2018
  • Mathematica
    CoefficientList[Series[-(x^2 + 17 x + 1) x/((x + 1)^2*(x - 1)^3), {x, 0, 55}], x] (* or *)
Array[PolygonalNumber[21, (1 - 2 Boole[EvenQ@ #]) Ceiling[#/2]] &, 56, 0] (* Michael De Vlieger, Jul 10 2018 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 18, 21, 55}, 51] (* Robert G. Wilson v, Jul 28 2018 *)
  • PARI
    concat(0, Vec(x*(1 + 17*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jun 24 2018

Formula

G.f.: -(x^2+17*x+1)*x/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 23 2018
From Colin Barker, Jun 24 2018: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = (19*n^2 + 34*n) / 8 for n even.
a(n) = (19*n^2 + 4*n - 15) / 8 for n odd.
(End)
Sum_{n>=1} 1/a(n) = 38/289 + 2*Pi*cot(2*Pi/19)/17. - Amiram Eldar, Feb 28 2022

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

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jun 06 2018

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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).

Programs

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

Formula

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*(10*m - 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

Views

Author

Omar E. Pol, Jun 23 2018

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Maple
    a:= n-> (m-> m*(10*m-9))(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 23 2018
  • Mathematica
    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 *)
  • PARI
    a(n) = n++; my(m = (-1) ^ n * (n >> 1)); m * (10 * m - 9) \\ David A. Corneth, Jun 23 2018
  • PARI
    concat(0, Vec(x*(1 + 18*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jun 23 2018

Formula

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*(21*m - 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

Views

Author

Omar E. Pol, Jun 24 2018

Keywords

Comments

168*a(n) + 361 is a square. - Bruno Berselli, Jul 10 2018
Partial sums of A317319. - Omar E. Pol, Jul 28 2018

Links

Crossrefs

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).

Programs

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

Formula

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*(23*m - 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

Views

Author

Omar E. Pol, Jul 10 2018

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Formula

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*(13*m - 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

Views

Author

Omar E. Pol, Jun 12 2018

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Magma
    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
  • Mathematica
    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 *)

Formula

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*(11*m - 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

Views

Author

Omar E. Pol, Jun 06 2018

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Formula

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*(12*m - 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

Views

Author

Omar E. Pol, Jul 11 2018

Keywords

Comments

48*a(n) + 121 is a square. - Bruno Berselli, Jul 11 2018
Partial sums of A317322. - Omar E. Pol, Jul 28 2018

Links

Crossrefs

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).

Programs

  • Magma
    [(12*n*(n+1) + 5*(-1)^n*(2*n+1) -5)/4: n in [0..60]]; // G. C. Greubel, Sep 24 2024
  • Mathematica
    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 *)
  • PARI
    concat(0, Vec(x*(1 + 22*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^60))) \\ Colin Barker, Jul 12 2018
  • SageMath
    [(12*n*(n+1) + 5*(-1)^n*(2*n+1) -5)//4 for n in range(61)] # G. C. Greubel, Sep 24 2024

Formula

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
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