A051867 -id:A051867 - cp's OEIS frontend

A098230 75-gonal numbers: a(n) = n(73n-71)/2.

0, 1, 75, 222, 442, 735, 1101, 1540, 2052, 2637, 3295, 4026, 4830, 5707, 6657, 7680, 8776, 9945, 11187, 12502, 13890, 15351, 16885, 18492, 20172, 21925, 23751, 25650, 27622, 29667, 31785, 33976, 36240, 38577, 40987, 43470, 46026, 48655, 51357, 54132, 56980, 59901, 62895, 65962, 69102, 72315, 75601, 78960, 82392, 85897, 89475

Offset: 0

Parthasarathy Nambi, Oct 25 2004

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051867, A051873.

[ n*(73*n - 71)/2: n in [0..50] ]; // Vincenzo Librandi, Feb 04 2011

A098230 := proc(n) n*(73*n-71)/2 ; end proc:
seq(A098230(n),n=0..20) ; # R. J. Mathar, Feb 04 2011

a(n)=n*(73*n-71)/2 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: -x*(1+72*x) / (x-1)^3. - R. J. Mathar, Feb 05 2011
a(n) = n*(73*n - 71)/2.
E.g.f.: exp(x)*(x + 73*x^2/2). - Nikolaos Pantelidis, Feb 10 2023

A098924 45-gonal numbers: n(43n-41)/2.

Original entry on oeis.org

1, 45, 132, 262, 435, 651, 910, 1212, 1557, 1945, 2376, 2850, 3367, 3927, 4530, 5176, 5865, 6597, 7372, 8190, 9051, 9955, 10902, 11892, 12925, 14001, 15120, 16282, 17487, 18735, 20026, 21360, 22737, 24157, 25620, 27126, 28675, 30267

Offset: 1

Parthasarathy Nambi, Oct 18 2004

Similar to 21-gonal and 15-gonal numbers (A051873, A051867).

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

Cf. A051867, A051873.

[n*(43*n-41)/2: n in [1..50]]; // Vincenzo Librandi, Jul 08 2012

Table[n(43n - 41)/2, {n, 1, 40}] (* Stefan Steinerberger, Feb 15 2006 *)
CoefficientList[Series[(1+42*x)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
LinearRecurrence[{3,-3,1},{1,45,132},40] (* Harvey P. Dale, Jan 24 2015 *)

a(n)=n*(43*n-41)/2 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = n*(43*n-41)/2.
G.f.: x*(1+42*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 08 2012
E.g.f.: exp(x)*(x + 43*x^2/2). - Nikolaos Pantelidis, Feb 10 2023

More terms from Stefan Steinerberger, Feb 15 2006

A373633 Positive integers that cannot be written as a sum of a practical number and a 15-gonal number.

Original entry on oeis.org

10, 11, 14, 15, 22, 26, 34, 38, 52, 53, 59, 68, 76, 77, 92, 107, 116, 117, 125, 131, 134, 149, 152, 158, 164, 173, 179, 184, 185, 187, 188, 206, 212, 227, 230, 236, 245, 248, 251, 254, 259, 268, 269, 283, 293, 299, 317, 326, 332, 347, 356, 371, 389, 398, 403

Offset: 1

Duc Van Khanh Tran, Jun 11 2024

Somu and Tran (2024) conjectured that there are finitely many such integers. It was also conjectured that 1486748 is the largest such integer. This conjecture was checked up to 10^8.

Duc Van Khanh Tran, Table of n, a(n) for n = 1..767
Sai Teja Somu and Duc Van Khanh Tran, On sums of practical numbers and polygonal numbers, Journal of Integer Sequences, 27(5), 2024.

Cf. A005153, A051867.

Lim=403;Lim15=Ceiling[Sqrt[2Lim/13]];
PracticalQ[nn_] := Module[{f, p, e, prod=1, ok=True}, If[nn<1 || (nn>1 && OddQ[n]), False, If[nn==1, True, f=FactorInteger[nn]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];prac= Select[Range[Lim], PracticalQ] ;
seq={};Do[p15=i(13i-11)/2;p15i=prac+p15;AppendTo[seq,p15i],{i,0,Lim15}] (* sums of 15gonal and practical numbers *);
Complement[Range[Lim],Union[Flatten[seq]]] (* James C. McMahon, Jun 12 2024 *)

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0

Offset: 1

Robert G. Wilson v, Apr 27 2020

\c  0 1  2  3  4   5   6   7   8   9  10  11   12   13    14  15  ...
r\
_0  0 1  0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195  A067998
_1  0 1  1  0 -2  -5  -9 -14 -20 -27 -35 -44  -54  -65  -77  -90  A080956
_2  0 1  2  3  4   5   6   7   8   9  10  11   12   13   14   15  A001477
_3  0 1  3  6 10  15  21  28  36  45  55  66   78   91  105  120  A000217
_4  0 1  4  9 16  25  36  49  64  81 100 121  144  169  196  225  A000290
_5  0 1  5 12 22  35  51  70  92 117 145 176  210  247  287  330  A000326
_6  0 1  6 15 28  45  66  91 120 153 190 231  276  325  378  435  A000384
_7  0 1  7 18 34  55  81 112 148 189 235 286  342  403  469  540  A000566
_8  0 1  8 21 40  65  96 133 176 225 280 341  408  481  560  645  A000567
_9  0 1  9 24 46  75 111 154 204 261 325 396  474  559  651  750  A001106
10  0 1 10 27 52  85 126 175 232 297 370 451  540  637  742  855  A001107
11  0 1 11 30 58  95 141 196 260 333 415 506  606  715  833  960  A051682
12  0 1 12 33 64 105 156 217 288 369 460 561  672  793  924 1065  A051624
13  0 1 13 36 70 115 171 238 316 405 505 616  738  871 1015 1170  A051865
14  0 1 14 39 76 125 186 259 344 441 550 671  804  949 1106 1275  A051866
15  0 1 15 42 82 135 201 280 372 477 595 726  870 1027 1197 1380  A051867
...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Cf. A317302 (the same array) but read by ascending antidiagonals.
Sub-arrays: A089000, A139600, A206735;
Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;
Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
Number of times k>1 appears: A129654, First occurrence of k: A063778.

Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

P(r, c) = (r - 2)(c(c-1)/2) + c.

A360488 31-gonal numbers: a(n) = n(29n-27)/2.

Original entry on oeis.org

0, 1, 31, 90, 178, 295, 441, 616, 820, 1053, 1315, 1606, 1926, 2275, 2653, 3060, 3496, 3961, 4455, 4978, 5530, 6111, 6721, 7360, 8028, 8725, 9451, 10206, 10990, 11803, 12645, 13516, 14416, 15345, 16303, 17290, 18306, 19351, 20425, 21528, 22660, 23821, 25011, 26230

Offset: 0

Nikolaos Pantelidis, Feb 09 2023

nonn

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

Cf. A360436, A051867, A051869.

Cf. A126385, A126418, A126499.

Mathematica
```
Table[n (29 n - 27)/2, {n, 0, 30}]
```

G.f.: x*(1 + 28*x)/(1 - x)^3.

E.g.f.: exp(x)*(x + 29*x^2/2).

cp's OEIS Frontend

A098230 75-gonal numbers: a(n) = n*(73*n-71)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

PARI

Formula

A098924 45-gonal numbers: n*(43*n-41)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A373633 Positive integers that cannot be written as a sum of a practical number and a 15-gonal number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A360488 31-gonal numbers: a(n) = n*(29*n-27)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A098230 75-gonal numbers: a(n) = n(73n-71)/2.

A098924 45-gonal numbers: n(43n-41)/2.

A360488 31-gonal numbers: a(n) = n(29n-27)/2.