A277082 -id:A277082 - cp's OEIS frontend

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

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

A316672 Numbers k for which 120*k + 169 is a square.

Original entry on oeis.org

-1, 0, 1, 3, 10, 14, 17, 22, 36, 43, 48, 56, 77, 87, 94, 105, 133, 146, 155, 169, 204, 220, 231, 248, 290, 309, 322, 342, 391, 413, 428, 451, 507, 532, 549, 575, 638, 666, 685, 714, 784, 815, 836, 868, 945, 979, 1002, 1037, 1121, 1158, 1183, 1221, 1312, 1352, 1379, 1420

Offset: 1

Bruno Berselli, Jul 10 2018

All terms of A303305 belong to this sequence.

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

Cf. A051869, A303305.
Subsequence of A047283.
Cf. Numbers k for which 8*(2*h+1)*k + (2*h-1)^2 is a square: A000217 (h=0), A001318 (h=1), A085787 (h=2), A118277 (h=3), A195160 (h=4), A195313 (h=5), A277082 (h=6), this sequence (h=7), A303813 (h=8), A303298 (h=9); A303815 (h=13).

[k: k in [0..1500] | IsSquare(120*k+169)];

select(k->issqr(120*k+169),[$-1..1500]); # Muniru A Asiru, Jul 10 2018

LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {-1, 0, 1, 3, 10, 14, 17, 22, 36}, 60]

isok(n) = issquare(120*n+169); \\ Michel Marcus, Jul 11 2018

Vec(x*(-1 + x + x^2 + 2*x^3 + 9*x^4 + 2*x^5 + x^6 + x^7 - x^8)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2) + O(x^40)) \\ Colin Barker, Jul 18 2018

print([k for k in (0..1500) if is_square(120*k+169)])

O.g.f.: x*(-1 + x + x^2 + 2*x^3 + 9*x^4 + 2*x^5 + x^6 + x^7 - x^8)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).
a(n) = a(1-n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9).
a(n) = (30*n^2 - 2*(15 + 3*(-1)^n + 10*i^(n*(n+1)))*n + 2*(5 + (-1)^n)*i^(n*(n+1)) + 3*(-1)^n - 79)/64, with i = sqrt(-1). Therefore:
a(4*k+1) = (3*k + 2)*(5*k - 1)/2;
a(4*k+2) = k*(15*k + 13)/2, first bisection of A303305;
a(4*k+3) = (k + 1)*(15*k + 2)/2, second bisection of A303305 (see A051869);
a(4*k+4) = (3*k + 1)*(5*k + 6)/2.

A317311 Multiples of 11 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 11, 3, 22, 5, 33, 7, 44, 9, 55, 11, 66, 13, 77, 15, 88, 17, 99, 19, 110, 21, 121, 23, 132, 25, 143, 27, 154, 29, 165, 31, 176, 33, 187, 35, 198, 37, 209, 39, 220, 41, 231, 43, 242, 45, 253, 47, 264, 49, 275, 51, 286, 53, 297, 55, 308, 57, 319, 59, 330, 61, 341, 63, 352, 65, 363, 67, 374, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 15-gonal numbers (A277082).

a(n) is also the length of the n-th line segment of the rectangular spiral wh0se vertices are the generalized 15-gonal numbers.

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

Cf. A008593 and A005408 interleaved.
Column 11 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A277082.

{0}~Join~Riffle[2 Range@ # - 1, 11 Range@ #] &@ 35 (* or *)
CoefficientList[Series[x (1 + 11 x + x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 69}], x] (* Michael De Vlieger, Jul 26 2018 *)
LinearRecurrence[{0,2,0,-1},{0,1,11,3},90] (* Harvey P. Dale, Aug 28 2022 *)

concat(0, Vec(x*(1 + 11*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jul 26 2018

a(2n) = 11*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 26 2018: (Start)
G.f.: x*(1 + 11*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 11*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 9/2^s). - Amiram Eldar, Oct 25 2023
a(n) = (13 + 9*(-1)^n)*n/4. - Aaron J Grech, Aug 20 2024

A244633 a(n) = 26*n^2.

Original entry on oeis.org

0, 26, 104, 234, 416, 650, 936, 1274, 1664, 2106, 2600, 3146, 3744, 4394, 5096, 5850, 6656, 7514, 8424, 9386, 10400, 11466, 12584, 13754, 14976, 16250, 17576, 18954, 20384, 21866, 23400, 24986, 26624, 28314, 30056, 31850, 33696, 35594, 37544, 39546, 41600, 43706

Offset: 0

Vincenzo Librandi, Jul 03 2014

Sequence found by reading the line from 0, in the direction 0, 26, ..., in the square spiral whose vertices are the generalized 15-gonal numbers. - Omar E. Pol, Jul 03 2014

Norms of purely imaginary numbers in Z[sqrt(-26)]. - Alonso del Arte, Dec 25 2014

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

Cf. similar sequences listed in A244630.

Cf. A000290, A001105, A005843, A008595, A152742, A252994, A277082.

Magma
```
[26*n^2: n in [0..40]];
```

A244633:=n->26*n^2: seq(A244633(n), n=0..50); # Wesley Ivan Hurt, Jul 04 2014

Table[26 n^2, {n, 0, 40}]
26 Range[0, 50]^2 (* Wesley Ivan Hurt, Jul 04 2014 *)

a(n)=26*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 26*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 26*A000290(n) = 13*A001105(n) = 2*A152742(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 26*x*(1 + x)*exp(x).
a(n) = n*A252994(n) = A005843(n)*A008595(n). (End)

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

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Jun 08 2018

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

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

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

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

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

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

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

Original entry on oeis.org

0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35, -24, -48, -35, -63, -48, -80, -63, -99, -80, -120, -99, -143, -120, -168, -143, -195, -168, -224, -195, -255, -224, -288, -255, -323, -288, -360, -323, -399, -360, -440, -399, -483, -440, -528, -483, -575, -528, -624, -575, -675, -624, -728, -675, -783

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 1 we have A317301.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

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).

Row 0 of A303301.
Cf. A001057, A008795, A008794, A174474, A317301.
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), A316729 (k=30).

concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 01 2018

a(n) = -A174474(n+1).
From Colin Barker, Aug 01 2018: (Start)
G.f.: x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = -n*(n + 4) / 4 for n even.
a(n) = -(n - 3)*(n + 1) / 4 for n odd.
(End)

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

Original entry on oeis.org

0, 1, -2, 1, -5, 0, -9, -2, -14, -5, -20, -9, -27, -14, -35, -20, -44, -27, -54, -35, -65, -44, -77, -54, -90, -65, -104, -77, -119, -90, -135, -104, -152, -119, -170, -135, -189, -152, -209, -170, -230, -189, -252, -209, -275, -230, -299, -252, -324, -275, -350, -299, -377, -324, -405, -350, -434

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 0 we have A317300.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

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).

Row 1 of A303301.
Cf. A000096, A001057, A008794, A008795, A317300.
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), A316729 (k=30).

/* By definition: */ k:=1; [0] cat [m*i*((k-2)*m*i-k+4)/2: i in [1,-1], m in [1..30]]; // Bruno Berselli, Jul 30 2018

Table[(-2 n (n + 1) - 5 (2 n + 1) (-1)^n + 5)/16, {n, 0, 60}] (* Bruno Berselli, Jul 30 2018 *)

concat(0, Vec(x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^50))) \\ Colin Barker, Aug 01 2018

From Bruno Berselli, Jul 30 2018: (Start)
O.g.f.: x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (-5*(1 + 2*x) + (5 - 2*x^2)*exp(2*x))*exp(-x)/16.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (-2*n*(n + 1) - 5*(2*n + 1)*(-1)^n + 5)/16. Therefore:
a(n) = -n*(n + 6)/8 for even n;
a(n) = -(n - 5)*(n + 1)/8 for odd n. Also:
a(n) = a(n-5) for odd n > 3.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) + n*(n^2 - 3) = 0. (End)

cp's OEIS Frontend

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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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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A316672 Numbers k for which 120*k + 169 is a square.

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A317311 Multiples of 11 and odd numbers interleaved.

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A244633 a(n) = 26*n^2.

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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, ...

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

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.