A048494 -id:A048494 - cp's OEIS frontend

A048504 a(n) = T(n,n), array T given by A048494.

1, 2, 7, 28, 101, 326, 967, 2696, 7177, 18442, 46091, 112652, 270349, 638990, 1490959, 3440656, 7864337, 17825810, 40108051, 89653268, 199229461, 440401942, 968884247, 2122317848, 4630511641

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (8, -25, 38, -28, 8).

[n*(n-1)*2^(n-1) + n + 1 : n in [0..30]]; // Vincenzo Librandi, Sep 25 2011

Table[n(n-1)2^(n-1)+n+1,{n,0,30}] (* or *) LinearRecurrence[{8,-25,38,-28,8},{1,2,7,28,101},30] (* Harvey P. Dale, Jul 29 2021 *)

a(n) = n*(n-1)*2^(n-1) + n + 1. - Ralf Stephan, Jan 15 2004

G.f.: (-1 - 4*x^4 + 16*x^3 - 16*x^2 + 6*x)/((x-1)^2*(2*x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 13 2009

A048483 Array read by antidiagonals: T(k,n) = (k+1)2^n - k.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 10, 5, 1, 32, 31, 22, 13, 6, 1, 64, 63, 46, 29, 16, 7, 1, 128, 127, 94, 61, 36, 19, 8, 1, 256, 255, 190, 125, 76, 43, 22, 9, 1, 512, 511, 382, 253, 156, 91, 50, 25, 10, 1, 1024, 1023, 766, 509, 316, 187

Offset: 0

Clark Kimberling

n-th difference of (T(k,n),T(k,n-1),...,T(k,0)) is k+1, for n=1,2,3,...; k=0,1,2,...

			1 2 4 8 16 32 ...
1 3 7 15 31 63 ...
1 4 10 22 46 94 ...
1 5 13 29 61 125 ...
1 6 16 36 76 156 ...

Rows are A000079 (k=0), A000225 (k=1), A033484 (k=2), A036563 (k=3), A048487 (k=4), A048488 (k=5), A048489 (k=6), A048490 (k=7), A048491 (k=8).

Main diagonal is A048493. Cf. A048494.

G.f.: (1-x+kx)/[(1-x)(1-2x)]. E.g.f.: (k+1)*exp(2x) - k*exp(x).

Recurrences: T(k, n) = 2T(k, n-1)+k = T(k-1, n)+2^n-1, T(k, 0) = 1.

Edited by Ralf Stephan, Feb 05 2004

A048495 a(n) = (n-1)*2^n + 2.

Original entry on oeis.org

1, 2, 6, 18, 50, 130, 322, 770, 1794, 4098, 9218, 20482, 45058, 98306, 212994, 458754, 983042, 2097154, 4456450, 9437186, 19922946, 41943042, 88080386, 184549378, 385875970, 805306370, 1677721602, 3489660930, 7247757314

Offset: 0

Clark Kimberling

Binomial transform of 1 followed by the odd numbers (2n-1+2*0^n, or abs(A060747)). Binomial transform is A084643. - Paul Barry, Jun 09 2003
Total number of bits of all binary numbers less than 2^n (see example).
Total number of zero bits of all binary numbers less than 2^(n+1). - Olivier Gérard, Feb 25 2014.
Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>2, 4>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second element, and the fourth element is larger than the third element. - Sergey Kitaev, Dec 08 2020

			a(1)=2 : 0 1
a(2)=6 : 0 1 10 11
a(3)=18 : 0 1 10 11 100 101 110 111
a(4)=50 : 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111
...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

a(n) = T(1, n), array T given by A048494.

Cf. A002064, A060747, A084643.

[(n-1)*2^n + 2: n in [0..30]]; // Vincenzo Librandi, Sep 25 2011

A048495:=n->(n-1)*2^n + 2; seq(A048495(n), n=0..30); # Wesley Ivan Hurt, Jun 29 2014

f[n_]:=(n-1)2^n+2;Array[f,29,0] (* Robert G. Wilson v, Jun 29 2014 *)
LinearRecurrence[{5,-8,4},{1,2,6},30] (* Harvey P. Dale, Jan 23 2015 *)

a(n) = (n-1)*2^n + 2; \\ Joerg Arndt, Feb 25 2014

Vec(-(4*x^2-3*x+1)/((x-1)*(2*x-1)^2) + O(x^100)) \\ Colin Barker, Jun 29 2014

a(n) - 1 = Sum_{i=0..n-1} (n-i) * 2^(n-i-1) = n*2^(n-1) + (n-1)*2^(n-2) + (n-2)*2^(n-3) + ... + 1*(2^0). - Matthew Erbst (matt(AT)erbst.org), Apr 19 2006
a(n) = 2 * A002064(n-1), n >= 1. - Omar E. Pol, Sep 30 2012
a(n) = a(n-1) + (2^n - 2^(n-1)) * n = a(n-1) + n*2^(n-1). - Olivier Gérard, Feb 25 2014
G.f.: -(4*x^2-3*x+1) / ((x-1)*(2*x-1)^2). - Colin Barker, Jun 29 2014
E.g.f.: exp(x)*(2 + exp(x)*(2*x - 1)). - Stefano Spezia, Feb 14 2025

Better description from John W. Layman, May 04 1999

A048496 a(n) = 2^(n-1)(3n-4) + 3.

Original entry on oeis.org

1, 2, 7, 23, 67, 179, 451, 1091, 2563, 5891, 13315, 29699, 65539, 143363, 311299, 671747, 1441795, 3080195, 6553603, 13893635, 29360131, 61865987, 130023427, 272629763, 570425347, 1191182339, 2483027971, 5167382531

Offset: 0

Clark Kimberling

a(n) = T(2, n), array T given by A048494.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5, -8, 4).

n-th difference of a(n), a(n-1), ..., a(0) is (1, 4, 7, 10, ...).

[2^(n-1)*(3*n-4)+3 : n in [0..30]]; // Vincenzo Librandi, Sep 26 2011

a(n) = A027992(n-1) + 1 = A053565(n) + 3.
From R. J. Mathar, Oct 31 2008: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3).
G.f.: (1 - 3*x + 5*x^2)/((1-x)(1-2*x)^2). (End)

Formula from Ralf Stephan, Jan 15 2004

A048497 a(n) = 2^(n-1)(4n - 6) + 4.

Original entry on oeis.org

1, 2, 8, 28, 84, 228, 580, 1412, 3332, 7684, 17412, 38916, 86020, 188420, 409604, 884740, 1900548, 4063236, 8650756, 18350084, 38797316, 81788932, 171966468, 360710148, 754974724, 1577058308, 3288334340, 6845104132

Offset: 0

Clark Kimberling

Equals binomial transform of A016813 preceded by a "1": (1, 1, 5, 9, 13, 21, ...). [Gary W. Adamson, Jan 13 2009]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

a(n) = T(3, n), array T given by A048494.

Cf. A016813. [Gary W. Adamson, Jan 13 2009]

[2^(n-1)*(4*n-6)+4: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

LinearRecurrence[{5,-8,4},{1,2,8},30] (* Harvey P. Dale, Apr 16 2019 *)

From Colin Barker, Oct 07 2012: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3).
G.f.: (1 - 3*x + 6*x^2)/((1-x)*(1-2*x)^2). (End)

Formula from Ralf Stephan, Jan 15 2004

A048498 2^(n-1)*(5n-8)+5.

Original entry on oeis.org

1, 2, 9, 33, 101, 277, 709, 1733, 4101, 9477, 21509, 48133, 106501, 233477, 507909, 1097733, 2359301, 5046277, 10747909, 22806533, 48234501, 101711877, 213909509, 448790533, 939524101, 1962934277, 4093640709, 8522825733

Offset: 0

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

a(n)=T(4, n), array T given by A048494.

seq(2^(n-1)*(5*n-8)+5, n=0..27); # Nathaniel Johnston, Jun 14 2011

Table[2^(n-1) (5n-8)+5,{n,0,35}] (* or *) LinearRecurrence[{5,-8,4},{1,2,9},35] (* Harvey P. Dale, Jun 14 2011 *)

a(n)= (5*n-8)<<(n-1)+5 \\ Charles R Greathouse IV, Jun 14 2011

a(0)=1, a(1)=2, a(2)=9, a(n)=5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale, Jun 14 2011

G.f.: (-1+3*x-7*x^2)/((-1+x)(-1+2*x)^2). - Harvey P. Dale, Jun 14 2011

Formula from Ralf Stephan, Jan 15 2004

A048499 a(n) = 2^(n-1)(6n-10)+6.

Original entry on oeis.org

1, 2, 10, 38, 118, 326, 838, 2054, 4870, 11270, 25606, 57350, 126982, 278534, 606214, 1310726, 2818054, 6029318, 12845062, 27262982, 57671686, 121634822, 255852550, 536870918, 1124073478, 2348810246, 4898947078, 10200547334, 21206401030, 44023414790

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A048494.

[2^(n-1)*(6*n-10)+6: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

Vec((1-3*x+8*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

a(n) = T(5, n), array T given by A048494.
a(0)=1, a(1)=2, a(2)=10, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale and Vincenzo Librandi, Sep 23 2011
G.f.: (1-3*x+8*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Aug 24 2016

Formula from Ralf Stephan, Jan 15 2004

A048500 a(n) = 2^(n-1)(7n-12)+7.

Original entry on oeis.org

1, 2, 11, 43, 135, 375, 967, 2375, 5639, 13063, 29703, 66567, 147463, 323591, 704519, 1523719, 3276807, 7012359, 14942215, 31719431, 67108871, 141557767, 297795591, 624951303, 1308622855, 2734686215, 5704253447, 11878268935, 24696061959, 51271172103

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A048494.

[2^(n-1)*(7*n-12)+7: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

Vec((1-3*x+9*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

a(n) = T(6, n), array T given by A048494.
a(0)=1, a(1)=2, a(2)=11, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale and Vincenzo Librandi, Sep 23 2011
G.f.: (1-3*x+9*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Aug 24 2016

Formula from Ralf Stephan, Jan 15 2004

A048501 a(n) = 2^(n-1)(8n-14)+8.

Original entry on oeis.org

1, 2, 12, 48, 152, 424, 1096, 2696, 6408, 14856, 33800, 75784, 167944, 368648, 802824, 1736712, 3735560, 7995400, 17039368, 36175880, 76546056, 161480712, 339738632, 713031688, 1493172232, 3120562184, 6509559816, 13555990536, 28185722888, 58518929416

Offset: 0

Clark Kimberling

a(n) = T(7, n), array T given by A048494.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

[2^(n-1)*(8*n-14)+8 : n in [0..30]]; // Vincenzo Librandi, Sep 25 2011

Table[2^(n-1) (8n-14)+8,{n,0,30}] (* or *) LinearRecurrence[{5,-8,4},{1,2,12},30] (* Harvey P. Dale, Nov 22 2014 *)

Vec((1-3*x+10*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

G.f.: (1-3*x+10*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Aug 24 2016

Formula from Ralf Stephan, Jan 15 2004

A048502 a(n) = 2^(n-1)(9n-16)+9.

Original entry on oeis.org

1, 2, 13, 53, 169, 473, 1225, 3017, 7177, 16649, 37897, 85001, 188425, 413705, 901129, 1949705, 4194313, 8978441, 19136521, 40632329, 85983241, 181403657, 381681673, 801112073, 1677721609, 3506438153, 7314866185, 15233712137, 31675383817, 65766686729

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A048494.

[2^(n-1)*(9*n-16)+9 : n in [0..30]]; // Vincenzo Librandi, Sep 25 2011

A048502:=n->2^(n-1)*(9*n-16)+9; seq(A048502(n), n=0..30); # Wesley Ivan Hurt, Dec 04 2013

Table[2^(n-1)*(9n-16)+9, {n,0,30}] (* Wesley Ivan Hurt, Dec 04 2013 *)

Vec((1-3*x+11*x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

a(n) = T(8, n), array T given by A048494.
a(n) = 2^(n-1)*(9n-16)+9 = A000079(n-1)*A017185(n-2)+9. - Wesley Ivan Hurt, Dec 04 2013
G.f.: (1-3*x+11*x^2) / ((1-x)*(1-2*x)^2). - Colin Barker, Aug 24 2016

Formula and more terms from Ralf Stephan, Jan 15 2004

cp's OEIS Frontend

A048504 a(n) = T(n,n), array T given by A048494.

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A048483 Array read by antidiagonals: T(k,n) = (k+1)2^n - k.

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A048495 a(n) = (n-1)*2^n + 2.

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A048496 a(n) = 2^(n-1)*(3*n-4) + 3.

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A048497 a(n) = 2^(n-1)*(4*n - 6) + 4.

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A048498 2^(n-1)*(5n-8)+5.

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A048499 a(n) = 2^(n-1)*(6*n-10)+6.

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A048496 a(n) = 2^(n-1)(3n-4) + 3.

A048497 a(n) = 2^(n-1)(4n - 6) + 4.

A048499 a(n) = 2^(n-1)(6n-10)+6.

A048500 a(n) = 2^(n-1)(7n-12)+7.

A048501 a(n) = 2^(n-1)(8n-14)+8.

A048502 a(n) = 2^(n-1)(9n-16)+9.