Adeniji, Adenike - cp's OEIS frontend

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

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

A190531 Number of idempotents in Identity Difference Partial Transformation semigroup.

Original entry on oeis.org

2, 5, 17, 57, 185, 593, 1901, 6121, 19793, 64161, 208085, 674105, 2179001, 7023409, 22566269, 72268809, 230696609, 734153537, 2329503653, 7371475033, 23267249417, 73268609745, 230224239437, 721965697577, 2259855722225

Offset: 1

Adeniji, Adenike, Jun 04 2011

nonn

IDP_n is a semigroup with the non-isolation property and E(IDP_n) denotes the set of idempotents (satisfying e^2 = e) in IDP_n.

#E(IDP_n) is the number of idempotent elements in the semigroup IDP_n for each n in N. E(IDP_n) is a subset of partial transformation semigroup having the property that the difference in the image, Im(alpha), is not greater than 1 and e^2 = e for each e in IDP_n.

			Example: For n=4, #IDP_n = 3*9 + 4*8 - 4 + 2 = 27 + 32 - 2 = 57

Index entries for linear recurrences with constant coefficients, signature (12,-58,144,-193,132,-36).

Cf. A189890.

LinearRecurrence[{12,-58,144,-193,132,-36},{2,5,17,57,185,593},30] (* Harvey P. Dale, Apr 11 2020 *)

#IDP_n = (n-1)*3^(n-2) + n*2^(n-1) - n + 2.

G.f.: -x*(-2+19*x-73*x^2+145*x^3-153*x^4+68*x^5) / ( (x-1)^2*(3*x-1)^2*(2*x-1)^2 ). - R. J. Mathar, Jun 19 2011

A189826 a(n) = (3^n-n)(n-1) - 2^n(n-2).

Original entry on oeis.org

2, 7, 40, 199, 856, 3359, 12440, 44335, 153808, 523159, 1752928, 5804759, 19041608, 61981807, 200458504, 644783071, 2064276256, 6581953703, 20911793168, 66230028871, 209167217752, 658918365247, 2070973772920, 6495510239759, 20334154874096, 63545035094839

Offset: 1

Adeniji, Adenike Apr 28 2011

nonn

Previous name was: Identity difference partial transformation semigroup, IDP_n is obtained by taking the absolute value of the difference between the max(Im(alpha)) and min(Im(alpha)) <= 1. The number of elements for each n is denoted by #IDP_n.

			For n=4, #IDP_n = 199.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-70,202,-337,325,-168,36).

[(3^n-n)*(n-1)-2^n*(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 19 2011

LinearRecurrence[{13,-70,202,-337,325,-168,36},{2,7,40,199,856,3359,12440},30] (* Harvey P. Dale, Apr 03 2016 *)

a(n) = (3^n-n)*(n-1)-2^n*(n-2); \\ Altug Alkan, Sep 20 2018

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

G.f.: -x*(2 - 19*x + 89*x^2 - 235*x^3 + 329*x^4 - 210*x^5 + 36*x^6) / ( (3*x-1)^2 *(2*x-1)^2 *(x-1)^3 ). - R. J. Mathar, Jun 20 2011

Simpler name using formula from Joerg Arndt, Sep 20 2018

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

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

A188947 a(n) = n^3 - 2n^2 + 2n + 1.

Original entry on oeis.org

2, 5, 16, 41, 86, 157, 260, 401, 586, 821, 1112, 1465, 1886, 2381, 2956, 3617, 4370, 5221, 6176, 7241, 8422, 9725, 11156, 12721, 14426, 16277, 18280, 20441, 22766, 25261, 27932, 30785, 33826, 37061, 40496, 44137, 47990, 52061, 56356, 60881, 65642, 70645

Offset: 1

Adeniji, Adenike, Apr 14 2011

The original definition was "Identity difference partial one - one transformation semigroup is a semigroup having the property that the difference between max im(alpha) and min im(alpha) is not greater than 1. This is denoted by S = IDI_n for each n." [Needs editing.]

For all n >= 3, a(n) expressed in base n has the three digits n-2, 2, and 1; for example, a(16) in hexadecimal is "E21". For all n >= 3, a(n+1) expressed in base n is "1112". For all n >= 7, a(n+2) expressed in base n is "1465". - Mathew Englander, Jan 07 2021

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. A027444, A053698, A056106 (first differences), A060354, A162607, A188377, A188716.

[n^3 - 2*n^2 + 2*n + 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A188947[n_] := n^3 - 2*n^2 + 2*n + 1; Table[A188947[n], {n, 1, 42}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n)=n^3-2*n^2+2*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (n+1) + n*(n-1)^2 = n^3 - 2*n^2 + 2*n + 1 = 1 + A053698(n-1).
G.f.: ( -x*(-2 + 3*x - 8*x^2 + x^3) ) / ( (x-1)^4 ). - R. J. Mathar, Apr 14 2011
a(n) = A060354(n) + A162607(n+1). - Lechoslaw Ratajczak, Sep 24 2020
E.g.f.: exp(x)*(1 + x)*(1 + x^2) - 1. - Stefano Spezia, Apr 10 2022

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

A188626 a(n) = n + (n-1)*2^(n-2).

Original entry on oeis.org

1, 3, 7, 16, 37, 86, 199, 456, 1033, 2314, 5131, 11276, 24589, 53262, 114703, 245776, 524305, 1114130, 2359315, 4980756, 10485781, 22020118, 46137367, 96469016, 201326617, 419430426, 872415259, 1811939356, 3758096413

Offset: 1

Adeniji, Adenike & Makanjuola, Samuel, Apr 14 2011

Number of idempotent elements in IDT_n (Identity Difference Full Transformation Semigroup), denoted by E(IDT_n).

			For n = 4, #E(IDT_n)= 16.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

[n + (n-1)*2^(n-2): n in [1..50]]; // G. C. Greubel, Nov 01 2018

Table[n + (n-1)*2^(n-2), {n,1,50}] (* G. C. Greubel, Nov 01 2018 *)
LinearRecurrence[{6,-13,12,-4},{1,3,7,16},40] (* Harvey P. Dale, Dec 31 2018 *)

a(n) = n+(n-1)*2^(n-2) \\ Michel Marcus, Jun 29 2013

a(n) =  n + (n-1)*2^(n-2).
G.f. x*(1-3*x+2*x^2+x^3) / ( (2*x-1)^2*(x-1)^2 ). - R. J. Mathar, Apr 14 2011
E.g.f.: (2*exp(2*x)*x + 4*exp(x)*x - exp(2*x) + 1)/4. - Stefano Spezia, Dec 23 2021

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User: Adeniji, Adenike

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

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A190531 Number of idempotents in Identity Difference Partial Transformation semigroup.

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A189826 a(n) = (3^n-n)*(n-1) - 2^n*(n-2).

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

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

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

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Magma

Magma

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Formula

Extensions

A188947 a(n) = n^3 - 2*n^2 + 2*n + 1.

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

A189826 a(n) = (3^n-n)(n-1) - 2^n(n-2).

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

A188947 a(n) = n^3 - 2n^2 + 2n + 1.