A008599 -id:A008599 - cp's OEIS frontend

A138199 a(n) = 14^(2n+1) + 3^(2n+1).

17, 2771, 538067, 105415691, 20661066467, 4049565346811, 793714774848467, 155568095572161131, 30491346729460336067, 5976303958950076658651, 1171355575953997682201267, 229585692886981589625399371, 44998795805848373961803836067, 8819763977946281138070581903291

Offset: 0

Reinhard Zumkeller, Mar 05 2008

Subsequence of A008599; A138200(n) = a(n)/17.

F. Ischebeck, Einladung zur Zahlentheorie (BI Wissenschaftsverlag 1992), Aufgabe 7, p. 18.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (205, -1764).

Cf. A008599, A076311, A138200.

[14^(2*n+1)+3^(2*n+1): n in [0..20]]; // Vincenzo Librandi, Dec 27 2010

Table[c=2n+1;14^c+3^c,{n,0,20}] (* or *) LinearRecurrence[{205,-1764},{17,2771},30] (* Harvey P. Dale, Aug 13 2013 *)
CoefficientList[Series[14/(1 - 196 x) + 3/(1 - 9 x), {x, 0, 25}], x] (* Vincenzo Librandi, May 23 2014 *)

a(0)=17, a(1)=2771; a(n) = 205*a(n-1)-1764*a(n-2). - Harvey P. Dale, Aug 13 2013

G.f.: 14/(1-196*x)+3/(1-9*x). -Harvey P. Dale, Aug 13 2013

Corrected and extended by Harvey P. Dale, Aug 13 2013

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

A154612 a(n) = 17*n + 7.

Original entry on oeis.org

7, 24, 41, 58, 75, 92, 109, 126, 143, 160, 177, 194, 211, 228, 245, 262, 279, 296, 313, 330, 347, 364, 381, 398, 415, 432, 449, 466, 483, 500, 517, 534, 551, 568, 585, 602, 619, 636, 653, 670, 687, 704, 721, 738, 755, 772, 789, 806, 823, 840, 857, 874, 891

Offset: 0

Vincenzo Librandi, Jan 15 2009

a(n)^4 = Sum_{j=0..(16*n*(17*n+14)+46)} (-1)^j*(119*n^2 + 98*n + 20 + j)^2. - Bruno Berselli, Apr 30 2010

			For n=5, a(5)^4 = 92^4 = 71639296 = 3485^2-3486^2+3487^2-..+11449^2-11450^2+11451^2. - _Bruno Berselli_, Apr 30 2010

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

Cf. A138629, A138630.

Sequences of the form 17*n+q: A361692 (q=-1), A008599 (q=0), A215137 (q=1), this sequence (q=7).

I:=[7, 24]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 21 2012

Range[7, 1000, 17] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

for(n=0, 50, print1(17*n + 7", ")); \\ Vincenzo Librandi, Feb 26 2012

[17*n+7 for n in range(61)] # G. C. Greubel, May 31 2024

G.f.: (7+10*x)/(1-x)^2. - Colin Barker, Jan 09 2012
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Feb 26 2012
E.g.f.: (7 + 17*x)*exp(x). - G. C. Greubel, May 31 2024

Offset corrected by Bruno Berselli, Aug 16 2010

A217558 Split-and-swap on 8-bit integers.

Original entry on oeis.org

0, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 1, 17, 33, 49, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 2, 18, 34, 50, 66, 82, 98, 114, 130, 146, 162, 178, 194, 210, 226, 242, 3, 19, 35, 51, 67, 83, 99, 115, 131, 147, 163, 179, 195, 211, 227, 243

Offset: 0

Jon Perry, Oct 06 2012

Split-and-swap consists of spliting a binary word into two halves and swapping the parts over to form a new word, for example 11001010 becomes 10101100.
There are 256 terms to the sequence. - Harvey P. Dale, Jul 14 2015
Fixed points are the multiples of 17 (A008599) in {0..255}. - Alois P. Heinz, May 02 2024

			a(17)=17 because 17 is 00010001 which is invariant over the SaS rule.
a(19)=49 because 00010011 becomes 00110001.

Harvey P. Dale, Table of n, a(n) for n = 0..255

Cf. A008599.

for (i=0;i<16;i++)
for (j=0;j<16;j++)
document.write(j*16+i+", ");

FromDigits[#,2]&/@(Join[Take[#,-4],Take[#,4]]&/@Select[Tuples[{0,1},8], Length[#] ==8 &]) (* Harvey P. Dale, Jul 14 2015 *)

A217558(n)=[1,16]*divrem(n,16) \\ - M. F. Hasler, Oct 07 2012

A361692 a(n) = 17*n - 1.

Original entry on oeis.org

16, 33, 50, 67, 84, 101, 118, 135, 152, 169, 186, 203, 220, 237, 254, 271, 288, 305, 322, 339, 356, 373, 390, 407, 424, 441, 458, 475, 492, 509, 526, 543, 560, 577, 594, 611, 628, 645, 662, 679, 696, 713, 730, 747, 764, 781, 798, 815, 832, 849, 866, 883, 900, 917, 934, 951, 968, 985, 1002, 1019

Offset: 1

Leo Tavares, Mar 20 2023

Leo Tavares, Illustration: Double Diamonds.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008590, A008599, A017257.

17*Range[100] - 1 (* Paolo Xausa, Aug 30 2024 *)
LinearRecurrence[{2,-1},{16,33},90] (* Harvey P. Dale, Jun 03 2025 *)

a(n) = 17*n - 1 = A008599(n) - 1.
a(n) = 2*A008590(n) + n - 1.
a(n) = A008590(n) + A017257(n-1).
From Elmo R. Oliveira, Apr 03 2025: (Start)
G.f.: x*(16 + x)/(x - 1)^2.
E.g.f.: exp(x)*(17*x - 1) + 1.
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A166391 Multiples of 17 whose reversal + 1 is also a multiple of 17.

Original entry on oeis.org

187, 238, 1870, 1921, 2380, 2431, 2754, 2805, 3264, 3315, 3587, 3638, 4097, 4148, 5661, 5712, 5984, 6171, 6222, 6494, 6545, 6868, 6919, 7055, 7106, 7378, 7429, 8891, 8942, 9452, 9503, 9775, 9826

Offset: 1

Claudio Meller, Oct 13 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Subsequence of A008599.

Select[17 Range[6!], Divisible[FromDigits[Reverse[IntegerDigits[#]]] + 1, 17] &] (* G. C. Greubel, May 12 2016 *)

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)

A346904 Numbers with sum of digits equaling 17, divisible by 17, and containing the string "17".

Original entry on oeis.org

13175, 15317, 17153, 17306, 17612, 21743, 30617, 41174, 51731, 61217, 101762, 107117, 110177, 111707, 117062, 117215, 117521, 122417, 125171, 131750, 153017, 153170, 170153, 170306, 170612, 171071, 171224, 171530, 172142, 172601, 173060, 173213, 174131, 175202, 176120, 214217

Offset: 1

Tanya Khovanova, Aug 06 2021

			13175 contains 17 as a substring; the sum of digits of 13175 is 17, and 13175 is divisible by 17. Thus, 13175 is in this sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..14300 (all terms < 10^11)

Cf. A121669 (for 19 instead of 17).

Intersection of A008599, A166370, and A293877

d17Q[n_] := Module[{idn = IntegerDigits[n]}, Total[idn] == 17 && MemberQ[Partition[idn, 2, 1], {1, 7}]]; Select[17*Range[20000], d17Q]

def ok(n): s = str(n); return n%17==0 and '17' in s and sum(map(int, s))==17
print(list(filter(ok, range(214218)))) # Michael S. Branicky, Aug 06 2021

A144604 Christoffel word of slope 6/11.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1

Offset: 0

N. J. A. Sloane, Jan 13 2009

The path is on the slope after 0, 17, 34, 51, 68, 85,... steps (see A008599), which gives the C-finite recurrence. - R. J. Mathar, May 28 2025

See A144595 for further details.

a(n) = a(n-17). - R. J. Mathar, May 28 2025

G.f.: -x^2*(x^2+1)*(x^12+x^7-x^5+x^4+x^3-x^2+1)/ (x-1) /(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 +x^8 +x^9 +x^10 +x^11 +x^12 +x^13 +x^14 +x^15 +x^16) . - R. J. Mathar, May 28 2025

cp's OEIS Frontend

A138199 a(n) = 14^(2*n+1) + 3^(2*n+1).

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A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

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A154612 a(n) = 17*n + 7.

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A217558 Split-and-swap on 8-bit integers.

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A361692 a(n) = 17*n - 1.

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A166391 Multiples of 17 whose reversal + 1 is also a multiple of 17.

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

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A138199 a(n) = 14^(2n+1) + 3^(2n+1).