A016933 -id:A016933 - cp's OEIS frontend

A016940 a(n) = (6*n + 2)^8.

256, 16777216, 1475789056, 25600000000, 208827064576, 1099511627776, 4347792138496, 14048223625216, 39062500000000, 96717311574016, 218340105584896, 457163239653376, 899194740203776, 1677721600000000, 2992179271065856, 5132188731375616, 8507630225817856

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016784, A016933, A016934, A016935, A016936, A016937, A016938, A016939.

[(6*n+2)^8: n in [0..20]]; // Vincenzo Librandi, May 04 2011

(6*Range[0,20]+2)^8 (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{256,16777216,1475789056,25600000000,208827064576,1099511627776,4347792138496,14048223625216,39062500000000},20] (* Harvey P. Dale, Sep 06 2020 *)

From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^8 = A016934(n)^4 = A016936(n)^2.
a(n) = 2^8*A016784(n).
Sum_{n>=0} 1/a(n) = PolyGamma(7, 1/3)/8465264640. (End)

A016941 a(n) = (6*n + 2)^9.

Original entry on oeis.org

512, 134217728, 20661046784, 512000000000, 5429503678976, 35184372088832, 165216101262848, 618121839509504, 1953125000000000, 5416169448144896, 13537086546263552, 31087100296429568, 66540410775079424, 134217728000000000, 257327417311663616, 472161363286556672

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016785, A016933, A016934, A016935, A016936, A016937, A016938, A016939, A016940.

[(6*n+2)^9: n in [0..25]]; // Vincenzo Librandi, May 05 2011

(6*Range[0,20]+2)^9 (* or *) LinearRecurrence[ {10,-45,120,-210,252,-210,120,-45,10,-1},{512,134217728,20661046784,512000000000,5429503678976,35184372088832,165216101262848,618121839509504,1953125000000000,5416169448144896},20] (* Harvey P. Dale, Sep 21 2013 *)

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Harvey P. Dale, Sep 21 2013
From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^9 = A016935(n)^3.
a(n) = 2^9*A016785(n).
Sum_{n>=0} 1/a(n) = 809*Pi^9/(14285134080*sqrt(3)) + 9841*zeta(9)/10077696. (End)

A016942 a(n) = (6*n + 2)^10.

Original entry on oeis.org

1024, 1073741824, 289254654976, 10240000000000, 141167095653376, 1125899906842624, 6278211847988224, 27197360938418176, 97656250000000000, 303305489096114176, 839299365868340224, 2113922820157210624, 4923990397355877376, 10737418240000000000, 22130157888803070976

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A016786, A016933, A016934, A016935, A016936, A016937, A016938, A016939, A016940, A016941.

[(6*n+2)^10: n in [0..25]]; // Vincenzo Librandi, May 05 2011

(6*Range[0,20]+2)^10 (* Harvey P. Dale, Apr 24 2017 *)

From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016933(n)^10 = A016934(n)^5 = A016937(n)^2.
a(n) = 2^10*A016786(n).
Sum_{n>=0} 1/a(n) = PolyGamma(9, 1/3)/21941965946880. (End)

A155156 Triangle T(n, k) = 4nk + 2n + 2k, read by rows.

Original entry on oeis.org

8, 14, 24, 20, 34, 48, 26, 44, 62, 80, 32, 54, 76, 98, 120, 38, 64, 90, 116, 142, 168, 44, 74, 104, 134, 164, 194, 224, 50, 84, 118, 152, 186, 220, 254, 288, 56, 94, 132, 170, 208, 246, 284, 322, 360, 62, 104, 146, 188, 230, 272, 314, 356, 398, 440, 68, 114, 160, 206, 252, 298, 344, 390, 436, 482, 528

Offset: 1

Vincenzo Librandi, Jan 21 2009

First column: A016933, second column: A017317, third column: A063151, fourth column: 2*A017209. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   8;
  14,  24;
  20,  34,  48;
  26,  44,  62,  80;
  32,  54,  76,  98, 120;
  38,  64,  90, 116, 142, 168;
  44,  74, 104, 134, 164, 194, 224;
  50,  84, 118, 152, 186, 220, 254, 288;
  56,  94, 132, 170, 208, 246, 284, 322, 360;
  62, 104, 146, 188, 230, 272, 314, 356, 398, 440;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A016933, A017317, A063151, A017209. A083487, A162254, A163832 (row sums).

[4*n*k + 2*n + 2*k : k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

seq(seq( 2*(2*n*k +n+k), k=1..n), n=1..15); # G. C. Greubel, Mar 20 2021

T[n_,k_]:=4*n*k +2*n +2*k; Table[T[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*(2*n*k +n+k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 20 2021

T(n, k) = 2*A083487(n, k). - R. J. Mathar, Jan 05 2011

Sum_{k=0..n} T(n,k) = n*(2*n^2 + 5*n + 1) = 2*A162254(n) = A163832(n). - G. C. Greubel, Mar 20 2021

Edited by Robert Hochberg, Jun 21 2010

A278481 Number of neighbors of the n-th term in a full isosceles triangle read by rows.

Original entry on oeis.org

2, 4, 4, 4, 6, 4, 4, 6, 6, 4, 4, 6, 6, 6, 4, 4, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the left border and the right border, the rest of the elements are 6's.

For the same idea but for a right triangle see A278480; for a square array see A278545, for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The sequence written as an isosceles triangle begins:
.
.                     2;
.                   4,  4;
.                 4,  6,  4;
.               4,  6,  6,  4;
.             4,  6,  6,  6,  4;
.           4,  6,  6,  6,  6,  4;
.         4,  6,  6,  6,  6,  6,  4;
.       4,  6,  6,  6,  6,  6,  6,  4;
.     4,  6,  6,  6,  6,  6,  6,  6,  4;
.   4,  6,  6,  6,  6,  6,  6,  6,  6,  4;
...

Row sums give A016933.
Left border gives A040002, the same as the right border.
Middle column gives the elements > 1 of A134201, also twice A122553.
Cf. A010722, A010731, A275015, A278317, A278480, A278545.

A296050 Number of permutations p of [n] such that min_{j=1..n} |p(j)-j| = 1.

Original entry on oeis.org

0, 0, 1, 2, 8, 40, 236, 1648, 13125, 117794, 1175224, 12903874, 154615096, 2007498192, 28075470833, 420753819282, 6726830163592, 114278495205524, 2055782983578788, 39039148388975552, 780412763620655061, 16381683795665956242, 360258256118419518680, 8283042472303599966974

Offset: 0

Alois P. Heinz, Jan 21 2019

nonn

			a(2) = 1: 21.
a(3) = 2: 231, 312.
a(4) = 8: 2143, 2341, 2413, 3142, 3421, 4123, 4312, 4321.
a(5) = 40: 21453, 21534, 23154, 23451, 23514, 24153, 24513, 24531, 25134, 25413, 25431, 31254, 31452, 31524, 34152, 34251, 35124, 35214, 35412, 35421, 41253, 41523, 41532, 43152, 43251, 43512, 43521, 45132, 45213, 45231, 51234, 51423, 51432, 53124, 53214, 53412, 53421, 54132, 54213, 54231.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Column k=1 of A299789.

Cf. A000142, A000166, A001883, A002467, A016933, A047252, A323671.

b:= proc(s, k) option remember; (n-> `if`(n=0, `if`(k=1, 1, 0), add(
      `if`(n=j, 0, b(s minus {j}, min(k, abs(n-j)))), j=s)))(nops(s))
    end:
a:= n-> b({$1..n}, n):
seq(a(n), n=0..14);
# second Maple program:
a:= n-> (f-> f(1)-f(2))(k-> `if`(n=0, 1, LinearAlgebra[Permanent](
        Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))):
seq(a(n), n=0..14);
# third Maple program:
g:= proc(n) g(n):= `if`(n<2, 1-n, (n-1)*(g(n-1)+g(n-2))) end:
h:= proc(n) h(n):= `if`(n<7, [1, 0$3, 1, 4, 29][n+1], n*h(n-1)+4*h(n-2)
      -3*(n-3)*h(n-3)+(n-4)*h(n-4)+2*(n-5)*h(n-5)-(n-7)*h(n-6)-h(n-7))
    end:
a:= n-> g(n)-h(n):
seq(a(n), n=0..25);

g[n_] := g[n] = If[n < 2, 1-n, (n-1)(g[n-1] + g[n-2])];
h[n_] := h[n] = If[n < 7, {1, 0, 0, 0, 1, 4, 29}[[n+1]],
     n h[n-1] + 4h[n-2] - 3(n-3)h[n-3] + (n-4)h[n-4] +
     2(n-5)h[n-5] - (n-7)h[n-6] - h[n-7]];
a[n_] := g[n] - h[n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after third Maple program *)

a(n) = A000142(n) - A001883(n) - A002467(n).
a(n) = A000166(n) - A001883(n).
a(n) = Sum_{k=1..n} A323671(n,k).
a(n) is odd <=> n in { A016933 }.
a(n) is even <=> n in { A047252 }.

A016943 a(n) = (6*n + 2)^11.

Original entry on oeis.org

2048, 8589934592, 4049565169664, 204800000000000, 3670344486987776, 36028797018963968, 238572050223552512, 1196683881290399744, 4882812500000000000, 16985107389382393856, 52036560683837093888, 143746751770690322432, 364375289404334925824, 858993459200000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A016787, A016933, A016934, A016935, A016936, A016937, A016938, A016939, A016940, A016941, A016942.

Subsequence of A008455.

[(6*n+2)^11: n in [0..25]]; // Vincenzo Librandi, May 05 2011

(6*Range[0,10]+2)^11 (* Harvey P. Dale, Sep 30 2019 *)

a(n) = A016787(n)*2^11. - Zerinvary Lajos, Jun 22 2009
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016933(n)^9 = A016935(n)^3.
Sum_{n>=0} 1/a(n) = 1847*Pi^11/(1285662067200*sqrt(3)) + 88573*zeta(11)/362797056. (End)

A080063 a(n) = n mod (spf(n)+1), where spf(n) is the smallest prime dividing n (A020639).

Original entry on oeis.org

1, 2, 3, 1, 5, 0, 7, 2, 1, 1, 11, 0, 13, 2, 3, 1, 17, 0, 19, 2, 1, 1, 23, 0, 1, 2, 3, 1, 29, 0, 31, 2, 1, 1, 5, 0, 37, 2, 3, 1, 41, 0, 43, 2, 1, 1, 47, 0, 1, 2, 3, 1, 53, 0, 1, 2, 1, 1, 59, 0, 61, 2, 3, 1, 5, 0, 67, 2, 1, 1, 71, 0, 73, 2, 3, 1, 5, 0, 79, 2, 1, 1, 83, 0, 1, 2, 3, 1, 89, 0, 3, 2, 1, 1, 5, 0

Offset: 1

Reinhard Zumkeller, Jan 24 2003

a(n) = 0 iff n mod 6 = 0 (A008588);
a(n) = 2 iff n mod 6 = 2 (A016933);
for n>1: a(n)=n iff n is prime (A000040, A008578).

Cf. A080064, A080065, A010872, A010873, A010875, A010877, A010881.

A140164 Binomial transform of [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, ...].

Original entry on oeis.org

1, 2, 4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326

Offset: 1

Gary W. Adamson, May 10 2008

Sum of antidiagonal terms of the following arithmetic array:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  2,  3,  4,  5,  6,  7,  8, ...
  1,  3,  5,  7,  9, 11, 13, 15, ...
  1,  4,  7, 10, 13, 16, 19, 22, ...
  1,  5,  9, 13, 17, 21, 25, 29, ...
  1,  6, 11, 16, 21, 26, 31, 36, ...
  1,  7, 13, 19, 25, 31, 37, 43, ...
  1,  8, 15, 22, 29, 36, 43, 50, ...
  ...
For [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, -55, ...], see A??????.

			a(4) = 8 = (1, 3, 3, 1) dot (1, 1, 1, 1) = (1 + 3 + 3 + 1).
a(5) = 14 = (4 + 5 + 4 + 1).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A028387.

Concatenation([1,2,4], List([4..60], n-> 6*n-16)); # G. C. Greubel, May 12 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( x*(1+x^2+2*x^3+2*x^4)/(1-x)^2 )); // G. C. Greubel, May 12 2019

From R. J. Mathar, May 03 2010: (Start)
A028387 := proc(n) option remember; if n <= 2 then op(n+1,[1,5,11]) ; else 3*procname(n-1)-3*procname(n-2)+procname(n-3) ; end if; end proc:
read("transforms") ; L := [1,1,1,1,-1, seq((-1)^(n+1)*A028387(n), n=0..60)]; BINOMIAL(L) ; (End)

Table[If[n < 4, 2^(n - 1), 6 n - 16], {n, 60}] (* or *)
Rest@CoefficientList[Series[x*(1+x^2+2x^3+2x^4)/(1-x)^2, {x, 0, 60}], x] (* Michael De Vlieger, Jul 18 2016 *)

a(n)=if(n<4,2^(n-1),6*n-16) \\ Charles R Greathouse IV, Jul 17 2016

(x*(1+x^2+2*x^3+2*x^4)/(1-x)^2).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Binomial transform of [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, -55,...]; where A028387 = (1, 5, 11, 19, 29, 41,...), such that A028387(n) = (2*T(n) - 1).
From R. J. Mathar, May 03 2010: (Start)
G.f.: x*(1+x^2+2*x^3+2*x^4)/(1-x)^2. [G.f. amended by Georg Fischer, May 12 2019]
a(n) = A016933(n-2), n>2. (End)
a(n) = 2*(3n-5), n >= 3, if offset is 0 instead of 1. - Daniel Forgues, Jul 17 2016

More terms from R. J. Mathar, May 03 2010

A168286 a(n) = (6n + 3(-1)^n + 1)/2.

Original entry on oeis.org

2, 8, 8, 14, 14, 20, 20, 26, 26, 32, 32, 38, 38, 44, 44, 50, 50, 56, 56, 62, 62, 68, 68, 74, 74, 80, 80, 86, 86, 92, 92, 98, 98, 104, 104, 110, 110, 116, 116, 122, 122, 128, 128, 134, 134, 140, 140, 146, 146, 152, 152, 158, 158, 164, 164, 170, 170, 176, 176, 182, 182

Offset: 1

Vincenzo Librandi, Nov 22 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. A016933, A168233, A168301.

[n le 1 select n+1 else 6*n-Self(n-1)-2: n in [1..70]]; // Vincenzo Librandi, Sep 17 2013

Table[3 n + 3 (-1)^n/2 + 1/2, {n, 70}] (* Bruno Berselli, Sep 17 2013 *)
CoefficientList[Series[(2 + 6 x - 2 x^2)/((1 + x) (1 - x)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 17 2013 *)

a(n) = 6*n - a(n-1) - 2, with n>1, a(1)=2.
From Vincenzo Librandi, Sep 17 2013: (Start)
a(n) = a(n-1) +a(n-2) -a(n-3).
G.f.: 2*x*(1 + 3*x - x^2)/((1+x)*(1-x)^2).
a(n) = 2*A168233(n) = A168301(n) + 1. (End)
E.g.f.: (1/2)*(3 - 4*exp(x) + (6*x + 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 17 2016

New definition by Bruno Berselli, Sep 17 2013

cp's OEIS Frontend

A016940 a(n) = (6*n + 2)^8.

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Magma

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A016941 a(n) = (6*n + 2)^9.

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Magma

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A016942 a(n) = (6*n + 2)^10.

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Magma

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A155156 Triangle T(n, k) = 4*n*k + 2*n + 2*k, read by rows.

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A278481 Number of neighbors of the n-th term in a full isosceles triangle read by rows.

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A296050 Number of permutations p of [n] such that min_{j=1..n} |p(j)-j| = 1.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A016943 a(n) = (6*n + 2)^11.

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Magma

Mathematica

Formula

A080063 a(n) = n mod (spf(n)+1), where spf(n) is the smallest prime dividing n (A020639).

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A155156 Triangle T(n, k) = 4nk + 2n + 2k, read by rows.

A168286 a(n) = (6n + 3(-1)^n + 1)/2.