A017449 -id:A017449 - cp's OEIS frontend

A017459 a(n) = (11*n + 5)^11.

48828125, 17592186044416, 5559060566555523, 238572050223552512, 3909821048582988049, 36279705600000000000, 231122292121701565271, 1127073856954876807168, 4501035456767426597157, 15394540563150776827904

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Powers of the form (11*n+5)^m: A017449 (m=1), A017450 (m=2), A017451 (m=3), A017452 (m=4), A017453 (m=5), A017454 (m=6), A017455 (m=7), A017456 (m=8), A017457 (m=9), A017458 (m=10), this sequence (m=11), A017460 (m=12).

List([0..20], n-> (11*n+5)^11); # G. C. Greubel, Sep 19 2019

[(11*n+5)^11: n in [0..10]]; // Vincenzo Librandi, Sep 03 2011

seq((11*n+5)^11, n=0..20); # G. C. Greubel, Sep 19 2019

(11Range[0,20]+5)^11 (* Harvey P. Dale, May 12 2011 *)

vector(20, n, (11*n-6)^11) \\ G. C. Greubel, Sep 19 2019

[(11*n+5)^11 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (48828125 +17591600106916*x +5347957556678781*x^2 + 173024396961630192*x^3 +1409984186533172778*x^4 +3893323100536505064*x^5 +4065965093212217778*x^6 +1612934439380337744*x^7 +220215589053761433* x^8 +7882270656385972*x^9 +34267542742961*x^10 +362797056*x^11)/(1-x)^12.
E.g.f.: (48828125 +17592137216291*x +2761938121647408*x^2 + 36991274172198511*x^3 +124534035099698400*x^4 +158840151787803530*x^5 + 93615574446397542*x^6 +28270098736853457*x^7 +4580560974275055*x^8 + 396972283518305*x^9 +17118700236660*x^10 +285311670611*x^11)*exp(x). (End)

A017460 a(n) = (11*n + 5)^12.

Original entry on oeis.org

244140625, 281474976710656, 150094635296999121, 9065737908494995456, 191581231380566414401, 2176782336000000000000, 16409682740640811134241, 92420056270299898187776, 418596297479370673535601, 1601032218567680790102016

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Powers of the form (11*n+5)^m: A017449 (m=1), A017450 (m=2), A017451 (m=3), A017452 (m=4), A017453 (m=5), A017454 (m=6), A017455 (m=7), A017456 (m=8), A017457 (m=9), A017458 (m=10), A017459 (m=11), this sequence (m=12).

List([0..20], n-> (11*n+5)^12); # G. C. Greubel, Sep 19 2019

[(11*n+5)^12: n in [0..10]]; // Vincenzo Librandi, Sep 03 2011

seq((11*n+5)^12, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[21] -6)^12 (* G. C. Greubel, Sep 19 2019 *)

vector(20, n, (11*n-6)^12) \\ G. C. Greubel, Sep 19 2019

[(11*n+5)^12 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (244140625 +281471802882531*x +146435479642729343*x^2 + 7136462627993219301*x^3 +85353518454518704170*x^4 +350628073514443644414 *x^5 +569002784856695826846*x^6 +380284494715132979466*x^7 + 101126771751016700469*x^8 +9408164121360836975*x^9 +224644345794247731* x^10 +582593939059393*x^11 +2176782336*x^12)/(1-x)^13.
E.g.f.: (244140625 +281474732570031*x +74765842793859217*x^2 + 1436049737881664906*x^3 +6509071735779405221*x^4 +10900283493364894200* x^5 +8393947455360064312*x^6 +3347919415332356436*x^7 + 736963256712968142*x^8 +91671288202929325*x^9 +6335345645917255*x^10 + 224254973100246*x^11 +3138428376721*x^12)*exp(x). (End)

A083487 Triangle read by rows: T(n,k) = 2nk + n + k (1 <= k <= n).

Original entry on oeis.org

4, 7, 12, 10, 17, 24, 13, 22, 31, 40, 16, 27, 38, 49, 60, 19, 32, 45, 58, 71, 84, 22, 37, 52, 67, 82, 97, 112, 25, 42, 59, 76, 93, 110, 127, 144, 28, 47, 66, 85, 104, 123, 142, 161, 180, 31, 52, 73, 94, 115, 136, 157, 178, 199, 220, 34, 57, 80, 103, 126, 149, 172, 195, 218, 241, 264

Offset: 1

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Jun 09 2003

T(n,k) gives number of edges (of unit length) in a k X n grid.

The values 2*T(n,k)+1 = (2*n+1)*(2*k+1) are nonprime and therefore in A047845.

			Triangle begins:
   4;
   7, 12;
  10, 17, 24;
  13, 22, 31, 40;
  16, 27, 38, 49,  60;
  19, 32, 45, 58,  71,  84;
  22, 37, 52, 67,  82,  97, 112;
  25, 42, 59, 76,  93, 110, 127, 144;
  28, 47, 66, 85, 104, 123, 142, 161, 180;

Vincenzo Librandi, Rows n = 1..100, flattened
Alain Kraus, Cours-Arithmétique et algèbre, 2016-2017, Université de Paris VI. See Exercice 6 p. 13.
OEIS Wiki, Odd composites

Cf. A000217, A016777, A016873, A017017, A017209, A017449, A033954.
Cf. A056220, A082652, A083003, A090288, A182868, A186113, A271625.
See A151890 for another version.

[(2*n*k + n + k): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Jun 01 2014

T[n_,k_]:= 2 n k + n + k; Table[T[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jun 01 2014 *)

def T(r, c): return 2*r*c + r + c
a = [T(r, c) for r in range(12) for c in range(1, r+1)]
print(a) # Michael S. Branicky, Sep 07 2022

flatten([[2*n*k +n +k for k in range(1,n+1)] for n in range(1,14)]) # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
T(n, 1) = A016777(n).
T(n, 2) = A016873(n).
T(n, 3) = A017017(n).
T(n, 4) = A017209(n).
T(n, 5) = A017449(n).
T(n, 6) = A186113(n).
T(n, n-1) = A056220(n).
T(n, n-2) = A090288(n-2).
T(n, n-3) = A271625(n-2).
T(n, n) = 4*A000217(n).
T(2*n, n) = A033954(n).
Sum_{k=1..n} T(n, k) = A162254(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A182868((n+1)/2) if n is odd otherwise A182868(n/2) + 1. (End)

Edited by N. J. A. Sloane, Jul 23 2009

Name edited by Michael S. Branicky, Sep 07 2022

A153384 Numbers n such that 24*n+1 is not prime.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 9, 11, 12, 15, 16, 20, 21, 22, 23, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 44, 45, 46, 49, 51, 53, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 68, 70, 71, 72, 76, 77, 79, 80, 81, 82, 85, 86, 91, 92, 93, 94, 96, 97, 98, 100, 101, 102

Offset: 1

Vincenzo Librandi, Dec 25 2008

Contains all numbers == 1 (mod 5), ==2 (mod 7), ==5 (mod 11), == 7 (mod 13), == 12 (mod 17), == 15 (mod 19), == 22 (mod 23), == 6 (mod 29) etc, so it is the union of A016861, A017005, A017449, A269044, etc. - R. J. Mathar, Jun 10 2020

Even terms of A153383, halved. - R. J. Mathar, Jun 10 2020

			Triangle begins:
*;
*,1;
*,*,2;
*,*,*,*;
*,*,*,*,5;
*,*,*,*,*,7;
*,*,*,*,*,*,*;
*,*,*,*,*,*,*,12;
*,*,*,*,*,*,*,*,15;
*,*,*,*,*,*,*,*,*,*;
*,*,*,*,*,*,*,*,*,*,22; etc.
where * marks the non-integer values of (2*h*k + k + h)/12 with h >= k >= 1. - _Vincenzo Librandi_, Jan 14 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001318, A111174 (complement).

[n: n in [0..150] | not IsPrime(24*n + 1)]; // Vincenzo Librandi, Jan 14 2013

Select[Range[0, 200], !PrimeQ[24 # + 1] &] (* Vincenzo Librandi, Jan 14 2013 *)

0 added by Arkadiusz Wesolowski, Aug 03 2011

A168463 a(n) = 5 + 11*floor(n/2).

Original entry on oeis.org

5, 16, 16, 27, 27, 38, 38, 49, 49, 60, 60, 71, 71, 82, 82, 93, 93, 104, 104, 115, 115, 126, 126, 137, 137, 148, 148, 159, 159, 170, 170, 181, 181, 192, 192, 203, 203, 214, 214, 225, 225, 236, 236, 247, 247, 258, 258, 269, 269, 280, 280, 291, 291, 302, 302, 313

Offset: 1

Vincenzo Librandi, Nov 26 2009

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

Cf. A017449.

[5+11*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013

Table[5 + 11 Floor[n/2], {n, 70}] (* or *) CoefficientList[Series[(5 + 11 x - 5 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)
LinearRecurrence[{1,1,-1},{5,16,16},80] (* Harvey P. Dale, Aug 28 2019 *)

a(n) = 11*n - a(n-1) - 1, with n>1, a(1)=5.
G.f.: x*(5 + 11*x - 5*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 19 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 19 2013
From G. C. Greubel, Jul 23 2016: (Start)
a(n) = (22*n + 11*(-1)^n + 9)/4.
E.g.f.: (1/4)*(11 - 20*exp(x) + (22*x + 9)*exp(2*x))*exp(-x). (End)

New definition by Vincenzo Librandi, Sep 19 2013

A355668 Array read by upwards antidiagonals T(n,k) = J(k) + n*J(k+1) where J(n) = A001045(n) is the Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 3, 3, 4, 3, 4, 4, 7, 8, 5, 5, 5, 10, 13, 16, 11, 6, 6, 13, 18, 27, 32, 21, 7, 7, 16, 23, 38, 53, 64, 43, 8, 8, 19, 28, 49, 74, 107, 128, 85, 9, 9, 22, 33, 60, 95, 150, 213, 256, 171, 10, 10, 25, 38, 71, 116, 193, 298, 427, 512, 341

Offset: 0

Paul Curtz, Jul 13 2022

			Row n=0 is A001045(k), then for further rows we successively add A001045(k+1).
       k=0  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9 k=10
  n=0:  0    1    1    3    5   11   21   43   85  171 ... = A001045
  n=1:  1    2    4    8   16   32   64  128  256  512 ... = A000079
  n=2:  2    3    7   13   27   53  107  213  427  853 ... = A048573
  n=3:  3    4   10   18   38   74  150  298  598 1194 ... = A171160
  n=4:  4    5   13   23   49   95  193  383  769 1535 ... = abs(A140683)
  ...

Cf. A001477, A000027, A016777, A016885, A017449, A321373.

Antidiagonal sums give A320933(n+1).

T[n_, k_] := (2^k - (-1)^k + n*(2^(k + 1) + (-1)^k))/3; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 13 2022 *)

T(n, k) = (2^k - (-1)^k + n*(2^(k + 1) + (-1)^k))/3.

G.f.: (x*(y-1) - y)/((x - 1)^2*(y + 1)*(2*y - 1)). - Stefano Spezia, Jul 13 2022

cp's OEIS Frontend

A017459 a(n) = (11*n + 5)^11.

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A017460 a(n) = (11*n + 5)^12.

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A083487 Triangle read by rows: T(n,k) = 2*n*k + n + k (1 <= k <= n).

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A153384 Numbers n such that 24*n+1 is not prime.

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Magma

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A168463 a(n) = 5 + 11*floor(n/2).

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A355668 Array read by upwards antidiagonals T(n,k) = J(k) + n*J(k+1) where J(n) = A001045(n) is the Jacobsthal numbers.

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A083487 Triangle read by rows: T(n,k) = 2nk + n + k (1 <= k <= n).