A017066 -id:A017066 - cp's OEIS frontend

A174312 a(n) = 32*n.

0, 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864, 896, 928, 960, 992, 1024, 1056, 1088, 1120, 1152, 1184, 1216, 1248, 1280, 1312, 1344, 1376, 1408, 1440, 1472, 1504, 1536, 1568, 1600

Offset: 0

Paul Curtz, Nov 27 2010

Subsequence of squares is A017066 (see 2nd formula). - Bernard Schott, Mar 03 2023

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

Cf. A001105, A008598, A017066, A135628, A135631, A152691.

[32*n: n in [0..60]]; // Vincenzo Librandi, Aug 04 2011

Range[0, 3000, 32] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)

a(n)=n<<5 \\ Charles R Greathouse IV, Dec 21 2011

G.f.: 32*x/(1-x)^2.
a(A001105(n)) = A017066(n). - Bernard Schott, Mar 05 2023
From Elmo R. Oliveira, Apr 07 2025: (Start)
E.g.f.: 32*x*exp(x).
a(n) = 2*A008598(n) = A152691(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A157951 a(n) = 128*n + 1.

Original entry on oeis.org

129, 257, 385, 513, 641, 769, 897, 1025, 1153, 1281, 1409, 1537, 1665, 1793, 1921, 2049, 2177, 2305, 2433, 2561, 2689, 2817, 2945, 3073, 3201, 3329, 3457, 3585, 3713, 3841, 3969, 4097, 4225, 4353, 4481, 4609, 4737, 4865, 4993, 5121, 5249, 5377, 5505, 5633, 5761

Offset: 1

Vincenzo Librandi, Mar 10 2009

The identity (128*n + 1)^2 - (64*n^2 + n)*16^2 = 1 can be written as a(n)^2 - (A017066(n) + n)*16^2 = 1. - Vincenzo Librandi, Feb 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Edward J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 14 in the first table at p. 85, case d(t) = t*(8^2*t+1)).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017066.

A157951:=n->128*n + 1: seq(A157951(n), n=1..80); # Wesley Ivan Hurt, Jan 24 2017

128Range[50]+1  (* Harvey P. Dale, Mar 15 2011 *)
LinearRecurrence[{2, -1}, {129, 257}, 50] (* Vincenzo Librandi, Feb 10 2012 *)

for(n=1, 50, print1(128*n + 1", ")); \\ Vincenzo Librandi, Feb 10 2012

From Vincenzo Librandi, Feb 10 2012: (Start)
G.f.: x*(129-x)/(1-x)^2.
a(n) = 2*a(n-1) - a(n-2). (End)
E.g.f.: exp(x)*(128*x + 1) - 1. - Elmo R. Oliveira, Apr 04 2025

A303302 a(n) = 34*n^2.

Original entry on oeis.org

0, 34, 136, 306, 544, 850, 1224, 1666, 2176, 2754, 3400, 4114, 4896, 5746, 6664, 7650, 8704, 9826, 11016, 12274, 13600, 14994, 16456, 17986, 19584, 21250, 22984, 24786, 26656, 28594, 30600, 32674, 34816, 37026, 39304, 41650, 44064, 46546, 49096, 51714, 54400, 57154, 59976, 62866, 65824, 68850, 71944

Offset: 0

Omar E. Pol, May 13 2018

Sequence found by reading the line from 0, in the direction 0, 34, ..., in the square spiral whose vertices are the generalized 19-gonal numbers A303813.

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

Cf. A005843, A008599, A303813.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30), A244082 (k=32), this sequence (k=34), A016910 (k=36), A016982 (k=49), A017066 (k=64), A017162 (k=81), A017270 (k=100), A017390 (k=121), A017522 (k=144).

[34*n^2: n in [0..50]]; // Vincenzo Librandi Jun 07 2018

Table[34 n^2, {n, 0, 40}]
LinearRecurrence[{3,-3,1},{0,34,136},50] (* Harvey P. Dale, Jul 23 2018 *)

PARI
```
a(n) = 34*n^2;
```

concat(0, Vec(34*x*(1 + x) / (1 - x)^3 + O(x^40))) \\ Colin Barker, Jun 12 2018

a(n) = 34*A000290(n) = 17*A001105(n) = 2*A244630(n).
G.f.: 34*x*(1 + x)/(1 - x)^3. - Vincenzo Librandi, Jun 07 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 34*x*(1 + x)*exp(x).
a(n) = A005843(n)*A008599(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

cp's OEIS Frontend

A174312 a(n) = 32*n.

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A157951 a(n) = 128*n + 1.

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A303302 a(n) = 34*n^2.

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