A188947 -id:A188947 - cp's OEIS frontend

A056106 Second spoke of a hexagonal spiral.

1, 3, 11, 25, 45, 71, 103, 141, 185, 235, 291, 353, 421, 495, 575, 661, 753, 851, 955, 1065, 1181, 1303, 1431, 1565, 1705, 1851, 2003, 2161, 2325, 2495, 2671, 2853, 3041, 3235, 3435, 3641, 3853, 4071, 4295, 4525, 4761, 5003, 5251, 5505, 5765, 6031, 6303

Offset: 0

Henry Bottomley, Jun 09 2000

First differences of A027444. - J. M. Bergot, Jun 04 2012
Numbers of the form ((h^2+h+1)^2+(-h^2+h+1)^2+(h^2+h-1)^2)/(h^2-h+1) for h=n-1. - Bruno Berselli, Mar 13 2013
For n > 0: 2*a(n) = A058331(n) + A001105(n) + A001844(n-1) = A251599(3*n-2) + A251599(3*n-1) + A251599(3*n). - Reinhard Zumkeller, Dec 13 2014
For all n >= 6, a(n+1) expressed in base n is "353". - Mathew Englander, Jan 06 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Henry Bottomley, Spokes of a hexagonal spiral (illustration of initial terms).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A053698, A027444, and A188947.
Cf. A113524 (semiprime terms), A002061.
Cf. A001105, A001844, A058331, A251599.
Other spokes: A003215, A056105, A056107, A056108, A056109.
Other spirals: A054552.

a056106 n = n * (3 * n - 1) + 1  -- Reinhard Zumkeller, Dec 13 2014

I:=[1,3]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2)+6: n in [1..50]]; // Vincenzo Librandi, Nov 14 2011

Table[3*n^2 - n + 1, {n,0,50}] (* G. C. Greubel, Jul 19 2017 *)

PARI
```
a(n) = 3*n^2-n+1;
```

a(n) = 3*n^2 - n + 1.
a(n) = a(n-1) + 6*n - 4 = 2*a(n-1) - a(n-2) + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (1+2*x+3*x^2)*exp(x). - Paul Barry, Mar 13 2003
a(n) = A096777(3*n) for n>0. - Reinhard Zumkeller, Dec 29 2007
G.f.: (1+5*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
a(n) = n*A002061(n+1) - (n-1)*A002061(n). - Bruno Berselli, Jan 15 2013
a(-n) = A056108(n). - Bruno Berselli, Mar 13 2013

A188377 a(n) = n^3 - 4n^2 + 6n - 2.

Original entry on oeis.org

7, 22, 53, 106, 187, 302, 457, 658, 911, 1222, 1597, 2042, 2563, 3166, 3857, 4642, 5527, 6518, 7621, 8842, 10187, 11662, 13273, 15026, 16927, 18982, 21197, 23578, 26131, 28862, 31777, 34882, 38183, 41686, 45397, 49322, 53467, 57838, 62441, 67282, 72367

Offset: 3

Adeniji, Adenike & Makanjuola, Samuel (somakanjuola(AT)unilorin.edu.ng) Apr 14 2011

Number of nilpotent elements in the identity difference partial one - one transformation semigroup, denoted by N(IDI_n). For n=3, #N(IDI_n) = 7.

a(n+1) is also the Moore lower bound on the order of an (n,7)-cage. - Jason Kimberley, Oct 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.
Gordon Royle, Cages of higher valency
R. P. Sullivan, Semigroups generated by nilpotent transformations, Journal of Algebra 110 (1987), 324-345.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A188716, A188947.

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), this sequence (g=7). - Jason Kimberley, Oct 30 2011

[n^3 - 4*n^2 + 6*n - 2: n in [3..50]]; // Vincenzo Librandi, May 01 2011

[SequenceToInteger([2^^3,1],n-2):n in [5..50]]; // Jason Kimberley, Oct 20 2011

Table[n^3 - 4*n^2 + 6*n - 2, {n, 3, 80}] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)
LinearRecurrence[{4,-6,4,-1},{7,22,53,106},50] (* Harvey P. Dale, May 29 2019 *)

a(n)=n^3-4*n^2+6*n-2 \\ Charles R Greathouse IV, Apr 06 2012

a(n+1) = (n+1)^3 - 4*(n+1)^2 + 6*(n+1) - 2
       = (n-1)^3 + 2*(n-1)^2 + 2*(n-1) + 2
       = 1222 read in base n-1.
- Jason Kimberley, Oct 20 2011
From Colin Barker, Apr 06 2012: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x^3*(7 - 6*x + 7*x^2 - 2*x^3)/(1-x)^4. (End)
E.g.f.: 2 - x - x^2 + exp(x)*(x^3 - x^2 + 3*x - 2). - Stefano Spezia, Apr 09 2022

Edited by N. J. A. Sloane, Apr 23 2011

A189890 a(n) = (n^3 - 2n^2 + 3n + 2)/2.

Original entry on oeis.org

2, 4, 10, 23, 46, 82, 134, 205, 298, 416, 562, 739, 950, 1198, 1486, 1817, 2194, 2620, 3098, 3631, 4222, 4874, 5590, 6373, 7226, 8152, 9154, 10235, 11398, 12646, 13982, 15409, 16930, 18548, 20266, 22087, 24014, 26050, 28198, 30461, 32842, 35344, 37970, 40723, 43606, 46622

Offset: 1

Adeniji, Adenike and Samuel Makanjuola(somakanjuola(AT)unilorin.edu.ng), Apr 30 2011

Order preserving identity difference partial one - one transformation semigroup, OIDI_n is defined if for each transformation, alpha, x<= y implies xalpha <= yalpha, for all x,y in X_n (set of natural numbers) and also the absolute value of the difference between max(Im(alpha)) and min(Im(alpha)) is less than or equal to one with non-isolation property.

			For n = 4, a(4) = (4^3-2*4^2+3*4+2)/2 = 46/2 = 23.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A188947, A188377.

[(n^3-2*n^2+3*n+2)/2: n in [1..50]]; // Vincenzo Librandi, May 07 2011

Table[(n^3-2*n^2+3*n+2)/2, {n,1,50}] (* or *) LinearRecurrence[{4,-6,4, -1}, {2,4,10,23}, 50] (* G. C. Greubel, Jan 13 2018 *)

a(n)=(n^3-2*n^2+3*n+2)/2 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: -x*(-2+4*x-6*x^2+x^3) / (x-1)^4. - R. J. Mathar, Jun 20 2011
E.g.f.: 4*(-2 + (2 + 2*x + x^2 + x^3)*exp(x)). - G. C. Greubel, Jan 13 2018
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Apr 23 2021

A188716 a(n) = n + (n-1)*(2^n-2).

Original entry on oeis.org

1, 1, 4, 15, 46, 125, 316, 763, 1786, 4089, 9208, 20471, 45046, 98293, 212980, 458739, 983026, 2097137, 4456432, 9437167, 19922926, 41943021, 88080364, 184549355, 385875946, 805306345, 1677721576, 3489660903, 7247757286, 15032385509, 31138512868, 64424509411, 133143986146, 274877906913, 566935683040, 1168231104479

Offset: 0

Adeniji, Adenike and Samuel Makanjuola (somakanjuola(AT)unilorin.edu.ng) Apr 14 2011

Number of elements in the semigroup IDT_n.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A000337, A188377, A188947.

[n + (n-1)*(2^n-2): n in [0..50]]; // Vincenzo Librandi, May 01 2011

Table[n+(n-1)(2^n-2),{n,0,40}] (* or *) LinearRecurrence[{6,-13,12,-4},{1,1,4,15},40] (* Harvey P. Dale, Aug 03 2024 *)

a(n)=(n-1)<Charles R Greathouse IV, Apr 06 2012

From Colin Barker, Apr 06 2012: (Start)
a(n) = 6*a(n-1)-13*a(n-2)+12*a(n-3)-4*a(n-4).
G.f.: (1-5*x+11*x^2-8*x^3)/((1-x)^2*(1-2*x)^2). (End)
a(n) = A000337(n) - (n-1). - Andrew Penland , Mar 24 2016
E.g.f.: exp(x)*(2 - x + exp(x)*(2*x - 1)). - Stefano Spezia, Apr 10 2022

Edited by N. J. A. Sloane, Apr 23 2011

Offset changed from 1 to 0 by Vincenzo Librandi, May 01 2011

A191821 a(n) = n(2^n - n + 1) + 2^(n-1)(n^2 - 3*n + 2).

Original entry on oeis.org

2, 6, 26, 100, 332, 994, 2774, 7368, 18872, 47014, 114578, 274300, 647012, 1507146, 3473198, 7929616, 17956592, 40369870, 90177194, 200277636, 442498652, 973078066, 2130705926, 4647288280, 10099883432, 21877489014, 47244639554, 101737037068

Offset: 1

Adeniji, Adenike, Jun 17 2011

Conjecture: generating function = -((2 (-1+6 x-19 x^2+31 x^3-22 x^4+4 x^5))/(1-3 x+2 x^2)^3) - Harvey P. Dale, May 10 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Cf. A188377, A188947, A189890.

[n*(2^n-n+1)+2^(n-1)*(n^2-3*n+2): n in [1..40]]; // Vincenzo Librandi, Nov 25 2011

LinearRecurrence[{9,-33,63,-66,36,-8},{2,6,26,100,332,994},50] (* Vincenzo Librandi, Nov 25 2011 *)
Table[n(2^n-n+1)+2^(n-1) (n^2-3n+2),{n,30}] (* Harvey P. Dale, May 10 2021 *)

a(n)=(n^2-n+2)<<(n-1)-n*(n-1) \\ Charles R Greathouse IV, Jul 13 2011

G.f.: -2*x*(-1 + 6*x - 19*x^2 + 31*x^3 - 22*x^4 + 4*x^5) / ( (2*x-1)^3*(x-1)^3 ). - R. J. Mathar, Aug 26 2011

cp's OEIS Frontend

A056106 Second spoke of a hexagonal spiral.

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A188377 a(n) = n^3 - 4n^2 + 6n - 2.

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A189890 a(n) = (n^3 - 2*n^2 + 3*n + 2)/2.

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Magma

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A188716 a(n) = n + (n-1)*(2^n-2).

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Magma

Mathematica

PARI

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A191821 a(n) = n*(2^n - n + 1) + 2^(n-1)*(n^2 - 3*n + 2).

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Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A189890 a(n) = (n^3 - 2n^2 + 3n + 2)/2.

A191821 a(n) = n(2^n - n + 1) + 2^(n-1)(n^2 - 3*n + 2).