A132583 -id:A132583 - cp's OEIS frontend

A298699 Primes of the form A132583(k)*42 - 43.

419, 5039, 51239, 513239, 5133239, 51333239, 5133333333333239, 513333333333333239, 5133333333333333333239, 5133333333333333333333239, 513333333333333333333333239, 513333333333333333333333333333333333333333333333239

Offset: 1

Paolo Galliani, Jan 27 2018

The corresponding values of k are 0, 1, 2, 3, 4, 5, 13, 15, 19, 22, 24, 48, 59, 187, 215, 232. - Bruno Berselli, Jan 29 2018

Further values of k (below 4000): 394, 441, 506, 541, 569, 1456, 2136, 3510. - Daniel Starodubtsev, Jan 05 2020

			5039 is prime and 5039 = 121*42 - 43, hence it is in the sequence.
51239 is prime and 51239 = 1221*42 - 43, hence it is in the sequence.
513333239 = 12222221*42 - 43 = 61*1747*4817 is not prime, therefore it is not in the sequence.

Cf. A003020, A132583.

Select[42 NestList[10 # + 11 &, 11, 50] - 43, PrimeQ] (* Michael De Vlieger, Feb 01 2018, after Harvey P. Dale at A132583 *)

a(8)-a(12) from Bruno Berselli, Jan 29 2018

A352341 Numbers whose maximal Pell representation (A352339) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 10, 20, 27, 40, 49, 54, 58, 63, 68, 88, 93, 119, 136, 150, 167, 221, 238, 288, 300, 310, 322, 334, 338, 360, 372, 382, 394, 406, 508, 530, 542, 696, 737, 771, 812, 833, 867, 908, 942, 983, 1242, 1276, 1317, 1392, 1681, 1710, 1734, 1763, 1792, 1802

Offset: 1

Amiram Eldar, Mar 12 2022

A000129(n) - 2 is a term for n > 1. The maximal Pell representations of these numbers are 0, 11, 121, 1221, 12221, ... (0 and A132583).
A048739 is a subsequence since these are the repunit numbers in the maximal Pell representation.
A065113 is a subsequence since the maximal Pell representation of A065113(n) is 2*n 2's.

			The first 10 terms are:
   n  a(n)  A352339(a(n))
  --  ----  -------------
   1    0               0
   2    1               1
   3    3              11
   4    6              22
   5    8             111
   6   10             121
   7   20            1111
   8   27            1221
   9   40            2222
  10   49           11111

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000129, A048739, A065113, A132583, A352339.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; lazy[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Select[Range[0, 2000], PalindromeQ[lazy[#]] &]

A146884 a(n) = 7*Sum_{k=0..n} 6^k.

Original entry on oeis.org

7, 49, 301, 1813, 10885, 65317, 391909, 2351461, 14108773, 84652645, 507915877, 3047495269, 18284971621, 109709829733, 658258978405, 3949553870437, 23697323222629, 142183939335781, 853103636014693, 5118621816088165

Offset: 0

Roger L. Bagula, Nov 02 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A003464, A005843, A068156, A100774, A132583, A146882, A146883, A146885.

[n le 2 select 7^n else 7*Self(n-1) -6*Self(n-2): n in [1..31]]; // G. C. Greubel, Oct 12 2022

a[n_]:= Sum[7*6^m, {m,0,n}]; Table[a[n], {n,0,30}]
Accumulate[7*6^Range[0,20]] (* Harvey P. Dale, Dec 18 2021 *)

[(7/5)*(6^(n+1)-1) for n in range(41)] # G. C. Greubel, Oct 12 2022

From G. C. Greubel, Oct 12 2022: (Start)
a(n) = (7/5)*(6^(n+1) - 1).
a(n) = 7*A003464(n+1).
a(n) = 7*a(n-1) - 6*a(n-2).
G.f.: 7/((1-x)*(1-6*x)).
E.g.f.: (7/5)*(6*exp(6*x) - exp(x)). (End)

A146885 a(n) = 8*Sum_{k=0..n} 7^k.

Original entry on oeis.org

8, 64, 456, 3200, 22408, 156864, 1098056, 7686400, 53804808, 376633664, 2636435656, 18455049600, 129185347208, 904297430464, 6330082013256, 44310574092800, 310174018649608, 2171218130547264, 15198526913830856

Offset: 0

Roger L. Bagula, Nov 02 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A005843, A023000, A068156, A100774, A132583, A146882, A146883, A146885.

[n le 2 select 8^n else 8*Self(n-1) -7*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 12 2022

a[n_]:= Sum[8*7^m, {m,0,n}]; Table[a[n], {n,0,30}]
LinearRecurrence[{8,-7}, {8,64}, 41] (* G. C. Greubel, Oct 12 2022 *)

[(4/3)*(7^(n+1)-1) for n in range(41)] # G. C. Greubel, Oct 12 2022

From G. C. Greubel, Oct 12 2022: (Start)
a(n) = (4/3)*(7^(n+1) - 1).
a(n) = 8*A023000(n+1).
a(n) = 8*a(n-1) - 7*a(n-2).
G.f.: 8/((1-x)*(1-7*x)).
E.g.f.: (4/3)*(7*exp(7*x) - exp(x)). (End)

A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 11, 2, 0, 0, 111, 22, 3, 0, 0, 1111, 222, 33, 4, 0, 0, 11111, 2222, 333, 44, 5, 0, 0, 111111, 22222, 3333, 444, 55, 6, 0, 0, 1111111, 222222, 33333, 4444, 555, 66, 7, 0, 0, 11111111, 2222222, 333333, 44444, 5555, 666, 77, 8, 0

Offset: 0

Stefano Spezia, Sep 14 2023

			The array begins:
  0,     0,     0,     0,     0,     0, ...
  0,     1,     2,     3,     4,     5, ...
  0,    11,    22,    33,    44,    55, ...
  0,   111,   222,   333,   444,   555, ...
  0,  1111,  2222,  3333,  4444,  5555, ...
  0, 11111, 22222, 33333, 44444, 55555, ...
  ...

Cf. A000004 (n=0 or k=0), A001477 (n=1), A002275 (k=1), A002276 (k=2), A002277 (k=3), A002278 (k=4), A002279 (k=5), A002280 (k=6), A002281 (k=7), A002282 (k=8), A002283 (k=9), A008593 (n=2), A053422 (main diagonal), A105279 (k=10), A132583, A177769 (n=3), A365645 (antidiagonal sums), A365646.

A[n_,k_]:=k(10^n-1)/9; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten

O.g.f.: x*y/((1 - x)*(1 - 10*x)*(1 - y)^2).
E.g.f.: y*exp(x+y)*(exp(9*x) - 1)/9.
A(n, 11) = A132583(n-1) for n > 0.
A(n, 12) = A073551(n+1) for n > 0.

A185123 a(n) = n 9's sandwiched between two 1's.

Original entry on oeis.org

11, 191, 1991, 19991, 199991, 1999991, 19999991, 199999991, 1999999991, 19999999991, 199999999991, 1999999999991, 19999999999991, 199999999999991, 1999999999999991, 19999999999999991, 199999999999999991, 1999999999999999991, 19999999999999999991

Offset: 0

José María Grau Ribas, Jan 20 2012

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf.A132583

H[n_]:=10^n+1+Sum[10^i*9,{i,1,n-1}];Array[H,100]
CoefficientList[Series[(11 + 70*x)/( (1-x)*(1-10*x) ), {x,0,50}], x] (* G. C. Greubel, Jun 23 2017 *)
Table[10FromDigits[PadRight[{1},n,9]]+1,{n,20}] (* or *) LinearRecurrence[ {11,-10},{11,191},20] (* Harvey P. Dale, May 18 2021 *)

a(n)=20*10^n-9 \\ Charles R Greathouse IV, Jan 20 2012

a(n) = 20*10^n - 9. - Charles R Greathouse IV, Jan 20 2012
a(0)=11, a(n) = 10*a(n-1) + 81.
From G. C. Greubel, Jun 22 2017: (Start)
G.f.: (11 + 70*x)/( (1-x)*(1-10*x) ).
E.g.f.: 20*exp(10*x) - 9*exp(x). (End)

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A298699 Primes of the form A132583(k)*42 - 43.

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A352341 Numbers whose maximal Pell representation (A352339) is palindromic.

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A146884 a(n) = 7*Sum_{k=0..n} 6^k.

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A146885 a(n) = 8*Sum_{k=0..n} 7^k.

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A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

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A185123 a(n) = n 9's sandwiched between two 1's.

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