A010684 -id:A010684 - cp's OEIS frontend

A272124 a(n) = 12n^4 + 16n^3 + 10n^2 + 4n + 1.

1, 43, 369, 1507, 4273, 9771, 19393, 34819, 58017, 91243, 137041, 198243, 277969, 379627, 506913, 663811, 854593, 1083819, 1356337, 1677283, 2052081, 2486443, 2986369, 3558147, 4208353, 4943851, 5771793, 6699619, 7735057, 8886123, 10161121, 11568643

Offset: 0

Vincenzo Librandi, Apr 21 2016

M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, Coefficients and roots of Ehrhart Polynomials, Contemp. Math. 374 (2005), 15-36, page 19.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A010684, A272039.

[12*n^4+16*n^3+10*n^2+4*n+1: n in [0..50]];

A272124:=n->(12*n^4 + 16*n^3 + 10*n^2 + 4*n + 1): seq(A272124(n), n=0..60); # Wesley Ivan Hurt, Apr 22 2016

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 43, 369, 1507, 4273}, 50]
CoefficientList[Series[(1 + 38*x + 164*x^2 + 82*x^3 + 3*x^4)/(1 - x)^5, {x, 0, 30}], x] (* Wesley Ivan Hurt, Apr 22 2016 *)

vector(100, n, n--; 12*n^4+16*n^3+10*n^2+4*n+1) \\ Altug Alkan, Apr 22 2016

O.g.f.: (1+38*x+164*x^2+82*x^3+3*x^4)/(1-x)^5.
E.g.f.: (1+42*x+142*x^2+88*x^3+12*x^4)*exp(x).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>4.
a(n) mod 4 = a(n) mod 8 = A010684(n). - Wesley Ivan Hurt, Apr 22 2016

A124846 Triangle read by rows: T(n,k) = (2 - (-1)^k)*binomial(n,k) (0 <= k <= n).

Original entry on oeis.org

1, 1, 3, 1, 6, 1, 1, 9, 3, 3, 1, 12, 6, 12, 1, 1, 15, 10, 30, 5, 3, 1, 18, 15, 60, 15, 18, 1, 1, 21, 21, 105, 35, 63, 7, 3, 1, 24, 28, 168, 70, 168, 28, 24, 1, 1, 27, 36, 252, 126, 378, 84, 108, 9, 3, 1, 30, 45, 360, 210, 756, 210, 360, 45, 30, 1, 1, 33, 55, 495, 330, 1386, 462, 990

Offset: 0

Gary W. Adamson, Nov 10 2006

			First few rows of the triangle:
  1;
  1,  3;
  1,  6,  1;
  1,  9,  3,  3;
  1, 12,  6, 12,  1;
  1, 15, 10, 30,  5,  3;
  ...
A046055(4) = 16 = sum of row 4 terms (1 + 9 + 3 + 3).

Cf. A010684, A046055, A124845.

T:=(n,k)->(2-(-1)^k)*binomial(n,k): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Edited by N. J. A. Sloane, Nov 24 2006

A135247 a(n) = 3a(n-1) + 2a(n-2) - 6*a(n-3).

Original entry on oeis.org

1, 3, 11, 33, 103, 309, 935, 2805, 8431, 25293, 75911, 227733, 683263, 2049789, 6149495, 18448485, 55345711, 166037133, 498111911, 1494335733, 4483008223, 13449024669, 40347076055, 121041228165, 363123688591, 1089371065773, 3268113205511, 9804339616533

Offset: 0

Paul Curtz, Feb 15 2008

This sequence interleaves A016133 and 3*A016133, see formulas. - Mathew Englander, Jan 08 2024

a(n) is the number of partitions of n into parts 1 (in three colors) and 2 (in two colors) where the order of colors matters. For example, the a(2)=11 such partitions (using parts 1, 1', 1'', 2, and 2') are 2, 2', 1+1, 1+1', 1+1'', 1'+1, 1'+1', 1'+1'', 1''+1, 1''+1', 1''+1''. For such partitions where the order of colors does not matter see A002624. - Joerg Arndt, Jan 18 2024

Sean A. Irvine, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,-6).

Cf. A016133.

a:=[1,3,11];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-6*a[n-3]; od; a; # G. C. Greubel, Nov 20 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-3*x-2*x^2+6*x^3) )); // G. C. Greubel, Nov 20 2019

seq(coeff(series(1/(1-3*x-2*x^2+6*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 20 2019

LinearRecurrence[{3,2,-6},{1,3,11},30] (* Harvey P. Dale, Jun 27 2015 *)

my(x='x+O('x^30)); Vec(1/(1-3*x-2*x^2+6*x^3)) \\ G. C. Greubel, Nov 20 2019

def A135247_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-3*x-2*x^2+6*x^3) ).list()
A135247_list(30) # G. C. Greubel, Nov 20 2019

G.f.: 1/((1-3*x)*(1-2*x^2)). - G. C. Greubel, Oct 04 2016
From Mathew Englander, Jan 08 2024: (Start)
a(n) = A010684(n) * A016133(floor(n/2)).
a(n) = 3*a(n-1) + A077957(n) for n >= 1.
a(n) = (A000244(n+2) - A164073(n+3))/7.
(End)

More terms from Harvey P. Dale, Jun 27 2015

Dropped two leading terms = 0. - Joerg Arndt, Jan 18 2024

A146562 'Erratic' numbers in A064353 [Kolakoski (1,3)].

Original entry on oeis.org

3, 6, 9, 15, 18, 21, 25, 29, 32, 35, 41, 44, 47, 51, 54, 57, 61, 64, 67, 73, 76, 79, 83, 87, 90, 93, 99, 102, 105, 109, 112, 115, 121, 124, 127, 131, 134, 137, 143, 146, 149, 153, 157, 160, 163, 169, 172, 175, 179, 182, 185, 189, 192, 195, 201, 204, 207, 211

Offset: 1

J. Perry (johnandruth(AT)jrperry.orangehome.co.uk), Nov 01 2008

nonn

A064353(n) is generally 1 if n is odd and 3 if n is even. An 'erratic' number breaks this rule.

			A064353(3) is predicted to be 1 but is 3 so 3 is in the list.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A064353.

Cf. A010684.

import Data.List (findIndices)
a146562 n = a146562_list !! (n-1)
a146562_list = map (+ 1) $
   findIndices (/= 0) $ zipWith (-) a064353_list $ cycle [1, 3]
-- Reinhard Zumkeller, Aug 03 2013

A173197 a(0)=1, a(n)= 2+2^n/6+4*(-1)^n/3, n>0.

Original entry on oeis.org

1, 1, 4, 2, 6, 6, 14, 22, 46, 86, 174, 342, 686, 1366, 2734, 5462, 10926, 21846, 43694, 87382, 174766, 349526, 699054, 1398102, 2796206, 5592406, 11184814, 22369622, 44739246, 89478486, 178956974, 357913942, 715827886, 1431655766, 2863311534, 5726623062, 11453246126, 22906492246, 45812984494, 91625968982, 183251937966, 366503875926, 733007751854

Offset: 0

Paul Curtz, Feb 12 2010

Linked to Jacobsthal numbers (expansion of tan(x), a.k.a. Zag numbers) A000182=1,2,16,272,...: a(n+1)-2a(n) = -(-1)^n*(A000182(n) mod 10) = (-1,2,-6,2,-6,2,-6,...).
Cf. A173114, related to Euler (or secant, or Zig) numbers, A000364. a(n+1)+A010684=A001045.
First differences: 0,3,-2,4,0,8,8,24,... = 0,A154879 (third differences of A001045).
Main diagonal: A003945; first upper diagonal: -A171449; second: 4*A011782.

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

a(n) = A093380(n+4), n>3.
a(n) = +2*a(n-1) +a(n-2) -2*a(n-3), n>3.
G.f.: 1-x*(-1-2*x+7*x^2)/((x-1)*(2*x-1)*(1+x)).
a(2n+2)+a(2n+3)=6*A047689.
a(2n)-a(2n-2) = 3,1,2,4,8,16,... = 3,A000079.

A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 5, 1, 3, 1, 1, 6, 1, 4, 1, 2, 1, 7, 1, 5, 1, 3, 1, 1, 8, 1, 6, 1, 4, 1, 2, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2

Offset: 2

Paul Curtz, Feb 14 2010

One may define another array B(n,0) = -1, B(n,k) = T(n,k-1) + 2*B(n,k-1), n>=2, which also starts in columns k>=0, as follows:
  -1, -1, 0, 1,  4,  9, 20,  41,  84, 169,  340,  681, 1364 ...: A084639;
  -1, -1, 1, 3,  9, 19, 41,  83, 169, 339,  681, 1363, 2729;
  -1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;
  -1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;
B(n,k) = (n-1)*A001045(k) - T(n,k).
First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).

			The array T(n,k) starts in row n=2 with columns k>=0 as:
  1,  2, 1,  2, 1,  2, 1,  2, 1,  2, 1,  2 ... A000034;
  1,  3, 1,  3, 1,  3, 1,  3, 1,  3, 1,  3 ... A010684;
  1,  4, 1,  4, 1,  4, 1,  4, 1,  4, 1,  4 ... A010685;
  1,  5, 1,  5, 1,  5, 1,  5, 1,  5, 1,  5 ... A010686;
  1,  6, 1,  6, 1,  6, 1,  6, 1,  6, 1,  6 ... A010687;
  1,  7, 1,  7, 1,  7, 1,  7, 1,  7, 1,  7 ... A010688;
  1,  8, 1,  8, 1,  8, 1,  8, 1,  8, 1,  8 ... A010689;
  1,  9, 1,  9, 1,  9, 1,  9, 1,  9, 1,  9 ... A010690;
  1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
Antidiagonal triangle begins as:
  1;
  1,  2;
  1,  3,  1;
  1,  4,  1,  2;
  1,  5,  1,  3,  1;
  1,  6,  1,  4,  1,  2;
  1,  7,  1,  5,  1,  3,  1;
  1,  8,  1,  6,  1,  4,  1,  2;
  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 13,  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 14,  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;

G. C. Greubel, Antidiagonals n = 0..50 of the array, flattened

Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.

Cf. A001045, A024206, A084639, A093178, A173114.

T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* G. C. Greubel, Dec 03 2021 *)

flatten([[(1/2)*((n-k+3) - (n-k+1)*(-1)^k) for k in (2..n)] for n in (2..17)]) # G. C. Greubel, Dec 03 2021

From G. C. Greubel, Dec 03 2021: (Start)
T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).
Sum_{k=0..n} T(n-k, k) = A024206(n).
Sum_{k=0..floor((n+2)/2)} T(n-2*k+2, k) = (1/16)*(2*n^2 4*n -5*(1 +(-1)^n) + 4*sin(n*Pi/2)) (diagonal sums).
T(2*n-2, n) = A093178(n). (End)

A190906 a(n) = gcd(n! / floor(n/2)!^2, 3^n).

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 1, 1, 1, 9, 9, 9, 3, 3, 3, 9, 9, 9, 1, 1, 1, 3, 3, 3, 1, 1, 1, 27, 27, 27, 9, 9, 9, 27, 27, 27, 3, 3, 3, 9, 9, 9, 3, 3, 3, 27, 27, 27, 9, 9, 9, 27, 27, 27, 1, 1, 1, 3, 3, 3, 1, 1, 1, 9, 9, 9, 3, 3, 3, 9, 9, 9, 1, 1, 1, 3, 3, 3, 1, 1, 1

Offset: 0

Peter Luschny, Jun 30 2011

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A060632.

A190906 := n -> igcd(n!/iquo(n,2)!^2,3^n): seq(A190906(n),n=0..80);

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := GCD[sf[n], 3^n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 29 2013 *)

a(n)=gcd(n!/(n\2)!^2, 3^n) \\ Charles R Greathouse IV, Jun 30 2011

a(n)=my(s);while(n\=3,s+=n%2);3^s \\ Charles R Greathouse IV, Jun 30 2011

a(n) = gcd(A056040(n), 3^n).
a(n) <= n. - Charles R Greathouse IV, Jun 30 2011
From Johannes W. Meijer, Jun 30 2011: (Start)
a(3*n) = a(3*n+1) = a(3*n+2) = A010684(n)*a(n) for n > 1 with a(0) = a(1) = a(2) = 1.
a(9*n+3) = a(9*n+4) = a(9*n+5) = 3*a(n).
a(9*n) = a(9*n+1) = a(9*n+2) = a(9*n+6) = a(9*n+7) = a(9*n+8) = A010690(n)*a(n). (End)

A235202 Numbers written in an alternating binary-then-quaternary base.

Original entry on oeis.org

1, 10, 11, 20, 21, 30, 31, 100, 101, 110, 111, 120, 121, 130, 131, 1000, 1001, 1010, 1011, 1020, 1021, 1030, 1031, 1100, 1101, 1110, 1111, 1120, 1121, 1130, 1131, 2000, 2001, 2010, 2011, 2020, 2021, 2030, 2031, 2100

Offset: 1

Jeremy Gardiner, Jan 04 2014

Mixed-radix number representation produced by a serial counter with generating sequence (1, 3, 1, 3, ...) = A010684.
Places reading from the right have values (1, 2, 8, 16, 64, 128, ...) = unsigned A094014.
Conjecture: This sequence interpreted as quaternary (base 4) numbers gives A126001 (hence a simplified scheme for computing that sequence).

			a(15) = 131 since 15 = 1*1+3*2+1*8.

Jeremy Gardiner, Table of n, a(n) for n = 1..1024

Cf. A109827 (Numbers written in an alternating binary-then-ternary base).

A271390 a(n) = (2n + 1)^(2floor((n-1)/2) + 1).

Original entry on oeis.org

1, 3, 5, 343, 729, 161051, 371293, 170859375, 410338673, 322687697779, 794280046581, 952809757913927, 2384185791015625, 4052555153018976267, 10260628712958602189, 23465261991844685929951, 59938945498865420543457, 177482997121587371826171875, 456487940826035155404146917

Offset: 0

Ilya Gutkovskiy, Apr 06 2016

All members are odd, therefore:
........................
|   k   |  a(n) mod k  |
|.......|..............|
|  n+1  |  A001477(n)  |
| 2*n+2 |  A005408(n)  |
|   2   |  A000012(n)  |
|   3   |  A080425(n+2)|
|   4   |  A010684(n)  |
|   6   |  A130793(n)  |
........................
Final digit of (2*n + 1)^(2*floor((n-1)/2) + 1) gives periodic sequence -> period 20: repeat [1,3,5,3,9,1,3,5,3,9,1,7,5,7,9,1,7,5,7,9], defined by the recurrence relation b(n) = b(n-2) - b(n-4) + b(n+5) + b(n+6) - b(n-7) - b(n-8) + b(n-9) - b(n-11) + b(n-13).

			a(0) =  1;
a(1) =  3^1 = 3;
a(2) =  5^1 = 5;
a(3) =  7^3 = 343;
a(4) =  9^3 = 729;
a(5) = 11^5 = 161051;
a(6) = 13^5 = 371293;
a(7) = 15^7 = 170859375;
a(8) = 17^7 = 410338673;
...
a(10000) = 1.644...*10^43006;
...
a(100000) = 8.235...*10^530097, etc.
This sequence can be represented as a binary tree:
                                    1
                 ................../ \..................
                3^1                                   5^1
     7^3......../ \......9^3                11^5....../ \.......13^5
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
15^7    17^7        19^9    21^9        23^11   25^11       27^13   29^13

Ilya Gutkovskiy, Table of n, a(n) for n = 0..75

Cf. A005408, A092503, A109613.

A271390:=n->(2*n + 1)^(n - 1/2 - (-1)^n/2): seq(A271390(n), n=0..30); # Wesley Ivan Hurt, Apr 10 2016

Table[(2 n + 1)^(2 Floor[(n - 1)/2] + 1), {n, 0, 18}]
Table[(2 n + 1)^(n - 1 + (1 + (-1)^(n - 1))/2), {n, 0, 18}]

a(n) = (2*n + 1)^(2*((n-1)\2) + 1); \\ Altug Alkan, Apr 06 2016

for n in range(0,10**3):print((int)((2*n+1)**(2*floor((n-1)/2)+1)))
# Soumil Mandal, Apr 10 2016

a(n) = (2*n + 1)^(n - 1 + (1 + (-1)^(n-1))/2).
a(n) = A005408(n)^A109613(n-1).
a(n) = (2*n + 1)^(n - 1/2 - (-1)^n/2). - Wesley Ivan Hurt, Apr 10 2016

A200439 Decimal expansion of constant arising in clubbed binomial approximation for the lightbulb process.

Original entry on oeis.org

2, 7, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3

Offset: 1

Jonathan Vos Post, Nov 17 2011

In the so-called lightbulb process, on days r = 1, ..., n, out of n lightbulbs, all initially off, exactly r bulbs selected uniformly and independent of the past have their status changed from off to on, or vice versa. With W_n the number of bulbs on at the terminal time n and C_n a suitable clubbed binomial distribution, d_{TV}(W_n,C_n) <= 2.7314 sqrt{n} e^{-(n+1)/3} for all n >= 1.

This is the value of the function g_1(9) after eq (16) of the preprint.

			2.731313... = 1352/495.

Larry Goldstein, Aihua Xia, Clubbed Binomial Approximation for the Lightbulb Process, arXiv:1111.3984v1 [math.PR], Nov 16, 2011.

Essentially the same as A176040, A153284 and A010684.

1352/495. \\ Charles R Greathouse IV, Nov 29 2011

Corrected by R. J. Mathar, Nov 29 2011

cp's OEIS Frontend

A272124 a(n) = 12*n^4 + 16*n^3 + 10*n^2 + 4*n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A124846 Triangle read by rows: T(n,k) = (2 - (-1)^k)*binomial(n,k) (0 <= k <= n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

A135247 a(n) = 3*a(n-1) + 2*a(n-2) - 6*a(n-3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A146562 'Erratic' numbers in A064353 [Kolakoski (1,3)].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A173197 a(0)=1, a(n)= 2+2^n/6+4*(-1)^n/3, n>0.

Views

Author

Keywords

Comments

Links

Formula

A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A190906 a(n) = gcd(n! / floor(n/2)!^2, 3^n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A235202 Numbers written in an alternating binary-then-quaternary base.

A272124 a(n) = 12n^4 + 16n^3 + 10n^2 + 4n + 1.

A135247 a(n) = 3a(n-1) + 2a(n-2) - 6*a(n-3).

A271390 a(n) = (2n + 1)^(2floor((n-1)/2) + 1).