A051870 -id:A051870 - cp's OEIS frontend

A360417 a(n) = 8n^2 - 7n + 2.

2, 3, 20, 53, 102, 167, 248, 345, 458, 587, 732, 893, 1070, 1263, 1472, 1697, 1938, 2195, 2468, 2757, 3062, 3383, 3720, 4073, 4442, 4827, 5228, 5645, 6078, 6527, 6992, 7473, 7970, 8483, 9012, 9557, 10118, 10695, 11288, 11897, 12522, 13163, 13820, 14493, 15182

Offset: 0

James Propp, Feb 06 2023

This is one of the four quadratic sequences that, interleaved, yield A360418.

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

Cf. A051870, A125201, A360416, A360418.

Mathematica
```
Table[8*n^2 - 7*n + 2, {n, 0, 100}]
```

From Elmo R. Oliveira, Jan 28 2025: (Start)
G.f.: (17*x^2-3*x+2)/(1-x)^3.
E.g.f.: exp(x)*(2 + x + 8*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = A360418(4*n-1) for n>=1. (End)
a(n) = A125201(n) + 1, n >= 1. - Omar E. Pol, Jan 28 2025

a(0)=2 prepended and more terms from Alois P. Heinz, Jan 28 2025

A322161 Numbers k such that m = 8k^2 + 2k + 33 and 8m - 7 are both primes.

Original entry on oeis.org

1, 46, 58, 133, 145, 175, 208, 223, 241, 403, 430, 463, 526, 568, 808, 868, 985, 1015, 1021, 1105, 1120, 1360, 1465, 1501, 1600, 1918, 1978, 2236, 2350, 2413, 2908, 2965, 3043, 3211, 3265, 3523, 3556, 3568, 3601, 3721, 3811, 3868, 4066, 4291, 4300, 4336, 4831

Offset: 1

Amiram Eldar, Nov 29 2018

nonn

Rotkiewicz proved that if k is in this sequence, and m = 8k^2 + 2k + 33, then m*(8m - 7) is an octadecagonal Fermat pseudoprime to base 2 (A322160), and thus under Schinzel's Hypothesis H there are infinitely many decagonal Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 14491, 2326319101, 5858192341, 160881885091, 227198832571, 481700815831, 960833787841, ...

			1 is in the sequence since 8*1^2 + 2*1 + 33 = 43 and 8*43 - 7 = 337 are both primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A001567, A051870, A322160.

Select[Range[1000], PrimeQ[8#^2 + 2# + 33] && PrimeQ[64#^2 + 16# + 257]  &]

isok(n) = isprime(m = 8*n^2+2*n+33) && isprime(8*m-7); \\ Michel Marcus, Nov 29 2018

A160970 Indices of square numbers that are also 18-gonal numbers.

Original entry on oeis.org

0, 1, 10, 44, 341, 1495, 11584, 50786, 393515, 1725229, 13367926, 58607000, 454115969, 1990912771, 15426575020, 67632427214, 524049434711, 2297511612505, 17802254205154, 78047762397956, 604752593540525, 2651326409917999, 20543785926172696, 90067050174814010

Offset: 1

Sture Sjöstedt, Jun 01 2009, Jul 02 2009

Solving the Diophantine equation A051870(m) = m*(8*m-7) = k^2 leads to the entries.

k in the sequence and a list of associated m = 0, 1, 4, 16, 121, 529, 4096, 17956, 139129, 609961...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Cf. A006451, A006452.

Join[{0},LinearRecurrence[{0,34,0,-1},{1,10,44,340},23]] (* Ray Chandler, Aug 01 2015 *)

is(n)=ispolygonal(n^2,18) \\ Charles R Greathouse IV, Feb 14 2013

concat(0, Vec(x^2*(x+1)*(x^2+9*x+1)/((x^2-6*x+1)*(x^2+6*x+1)) + O(x^50))) \\ Colin Barker, Jun 24 2015

a(n) = 34*a(n-2) - a(n-4), n>5. - R. J. Mathar, Oct 04 2009
G.f.: x^2*(x+1)*(x^2 + 9*x + 1)/((x^2 - 6*x + 1)*(x^2 + 6*x + 1)). - Colin Barker, Oct 07 2012
For all values excepting the leading 0, a(n) = sqrt(8*A006452(n)^2 - 7)*A006452(n) = sqrt(A006451(n-1)*(A006451(n-1) + 1)/2 + 1)*(2*A006451(n-1) + 1). - Raphie Frank, Feb 11 2013

0 added in front and extended by R. J. Mathar, Oct 04 2009

A373646 Positive integers that cannot be written as a sum of a practical number and an 18-gonal number.

Original entry on oeis.org

10, 11, 14, 15, 23, 27, 35, 39, 44, 45, 47, 51, 62, 68, 70, 76, 77, 86, 92, 94, 95, 103, 110, 119, 125, 134, 137, 143, 149, 152, 153, 165, 170, 175, 182, 187, 202, 203, 206, 215, 223, 230, 233, 236, 239, 263, 283, 284, 287, 299, 314, 317, 319, 329, 332, 335

Offset: 1

Duc Van Khanh Tran, Jun 12 2024

Somu and Tran (2024) conjectured that there are finitely many such integers. It was also conjectured that 8541224 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..1020
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, A051870.

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.

cp's OEIS Frontend

A360417 a(n) = 8*n^2 - 7*n + 2.

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A322161 Numbers k such that m = 8k^2 + 2k + 33 and 8m - 7 are both primes.

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A160970 Indices of square numbers that are also 18-gonal numbers.

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PARI

PARI

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A373646 Positive integers that cannot be written as a sum of a practical number and an 18-gonal number.

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

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

Formula

A360417 a(n) = 8n^2 - 7n + 2.