cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A048506 a(n) = T(0,n), array T given by A048505.

Original entry on oeis.org

1, 2, 7, 25, 81, 241, 673, 1793, 4609, 11521, 28161, 67585, 159745, 372737, 860161, 1966081, 4456449, 10027009, 22413313, 49807361, 110100481, 242221057, 530579457, 1157627905, 2516582401, 5452595201, 11777605633, 25367150593, 54492397569, 116769423361
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (1, 4, 9, 16, 25, ...).
Similar to A000697 in so far as it can be seen as the transform of 1, 1, 4, 9, 16, ... by a variant of the boustrophedon algorithm (see the Sage implementation). - Peter Luschny, Oct 30 2014

Links

Crossrefs

Cf. A048505, A000697.

Programs

  • Magma
    [n*(n+1)*2^(n-2) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • Mathematica
    LinearRecurrence[{7,-18,20,-8}, {1,2,7,25}, 30] (* Jean-François Alcover, Jun 11 2019 *)
  • PARI
    Vec(-(8*x^3-11*x^2+5*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Nov 26 2014
  • Sage
    def sq():
    yield 1
    for n in PositiveIntegers():
        yield n*n
def bous_variant(f):
    k = 0
    am = next(f)
    a = [am]
    while True:
        yield am
        am = next(f)
        a.append(am)
        for j in range(k,-1,-1):
            am += a[j]
            a[j] = am
        k += 1
b = bous_variant(sq())
print([next(b) for  in range(26)]) # _Peter Luschny, Oct 30 2014

Formula

a(n) = n*(n+1)*2^(n-2) + 1 = A001788(n) + 1. - Ralf Stephan, Jan 16 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Nov 26 2014
G.f.: -(8*x^3-11*x^2+5*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Nov 26 2014

A048507 a(n) = T(2,n), array T given by A048505.

Original entry on oeis.org

1, 10, 35, 101, 269, 685, 1693, 4093, 9725, 22781, 52733, 120829, 274429, 618493, 1384445, 3080189, 6815741, 15007741, 32899069, 71827453, 156237821, 338690045, 731906045, 1577058301, 3388997629, 7264534525, 15535702013, 33151778813, 70598524925
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (9, 16, 25, 36, 49, ...).

Links

Programs

  • Magma
    [(n^2+9*n+16) * 2^(n-2) - 3: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • Mathematica
    LinearRecurrence[{7,-18,20,-8},{1,10,35,101},30] (* Harvey P. Dale, Jan 21 2021 *)
  • PARI
    Vec((16*x^3-17*x^2+3*x+1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Nov 27 2014

Formula

a(n) = (n^2+9*n+16) * 2^(n-2) - 3. - Ralf Stephan, Feb 05 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Nov 27 2014
G.f.: (16*x^3-17*x^2+3*x+1) / ((x-1)*(2*x-1)^3). - Colin Barker, Nov 27 2014

A048508 a(n) = T(3,n), array T given by A048505.

Original entry on oeis.org

1, 17, 58, 160, 408, 1000, 2392, 5624, 13048, 29944, 68088, 153592, 344056, 765944, 1695736, 3735544, 8191992, 17891320, 38928376, 84410360, 182452216, 393215992, 845152248, 1811939320, 3875536888, 8271167480, 17616076792
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (16, 25, 36, 49, 64, ...).

Links

Programs

  • Magma
    [(n+4)*(n+9) * 2^(n-2) - 8: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • PARI
    Vec((40*x^3-43*x^2+10*x+1) / ((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Feb 25 2015

Formula

a(n) = (n+4)*(n+9) * 2^(n-2) - 8. - Ralf Stephan, Feb 05 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Feb 25 2015
G.f.: (40*x^3-43*x^2+10*x+1) / ((x-1)*(2*x-1)^3). - Colin Barker, Feb 25 2015

A048509 a(n) = T(4,n), array T given by A048505.

Original entry on oeis.org

1, 26, 87, 233, 577, 1377, 3217, 7409, 16881, 38129, 85489, 190449, 421873, 929777, 2039793, 4456433, 9699313, 21037041, 45481969, 98041841, 210763761, 451936241, 966787057, 2063597553, 4395630577, 9344909297, 19830669297, 42010148849, 88852135921
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (25, 36, 49, 64, 81...).

Links

Programs

  • Magma
    [(n^2+17*n+64) * 2^(n-2) - 15: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • Mathematica
    LinearRecurrence[{7,-18,20,-8},{1,26,87,233},30] (* Harvey P. Dale, Jul 08 2023 *)
  • PARI
    Vec((72*x^3-77*x^2+19*x+1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Mar 04 2015

Formula

a(n) = (n^2+17*n+64) * 2^(n-2) - 15. - Ralf Stephan, Feb 05 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Mar 04 2015
G.f.: (72*x^3-77*x^2+19*x+1) / ((x-1)*(2*x-1)^3). - Colin Barker, Mar 04 2015

A048510 a(n) = T(5,n), array T given by A048505.

Original entry on oeis.org

1, 37, 122, 320, 776, 1816, 4168, 9448, 21224, 47336, 104936, 231400, 507880, 1109992, 2416616, 5242856, 11337704, 24444904, 52559848, 112721896, 241172456, 514850792, 1096810472, 2332033000, 4949278696, 10485759976, 22179479528, 46841987048, 98784247784
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (36, 49, 64, 81, ...).

Links

Programs

  • Magma
    [(n^2+21*n+100) * 2^(n-2) - 24: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • Mathematica
    LinearRecurrence[{7,-18,20,-8},{1,37,122,320},30] (* Harvey P. Dale, Sep 24 2016 *)
  • PARI
    Vec((112*x^3-119*x^2+30*x+1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Mar 04 2015

Formula

a(n) = (n^2+21*n+100) * 2^(n-2) - 24. - Ralf Stephan, Feb 05 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Mar 04 2015
G.f.: (112*x^3-119*x^2+30*x+1) / ((x-1)*(2*x-1)^3). - Colin Barker, Mar 04 2015

A048511 a(n) = T(6,n), array T given by A048505.

Original entry on oeis.org

1, 50, 163, 421, 1005, 2317, 5245, 11741, 26077, 57565, 126429, 276445, 602077, 1306589, 2826205, 6094813, 13107165, 28114909, 60162013, 128450525, 273678301, 581959645, 1235222493, 2617245661, 5536481245, 11693719517
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (49, 64, 81, 100, ...).

Links

Programs

  • Magma
    [(n+9)*(n+16) * 2^(n-2) - 35: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • PARI
    Vec((160*x^3-169*x^2+43*x+1) / ((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Feb 25 2015

Formula

a(n) = (n+9)*(n+16) * 2^(n-2) - 35. - Ralf Stephan, Feb 05 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Feb 25 2015
G.f.: (160*x^3-169*x^2+43*x+1) / ((x-1)*(2*x-1)^3). - Colin Barker, Feb 25 2015

A048512 a(n) = T(7,n), array T given by A048505.

Original entry on oeis.org

1, 65, 210, 536, 1264, 2880, 6448, 14288, 31440, 68816, 149968, 325584, 704464, 1519568, 3268560, 7012304, 15007696, 32047056, 68288464, 145227728, 308281296, 653262800, 1382023120, 2919235536, 6157238224, 12968787920
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (64, 81, 100, 121, ...).

Links

Programs

  • Magma
    [(n^2+29*n+196) * 2^(n-2) - 48: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • PARI
    Vec((216*x^3-227*x^2+58*x+1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Feb 25 2015

Formula

a(n) = (n^2+29*n+196) * 2^(n-2) - 48. - Ralf Stephan, Feb 05 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Feb 25 2015
G.f.: (216*x^3-227*x^2+58*x+1) / ((x-1)*(2*x-1)^3). - Colin Barker, Feb 25 2015

A048513 a(n) = T(8,n), array T given by A048505.

Original entry on oeis.org

1, 82, 263, 665, 1553, 3505, 7777, 17089, 37313, 81089, 175553, 378817, 815041, 1748929, 3743681, 7995329, 17039297, 36241345, 76939201, 163053505, 344981441, 728760257, 1537212353, 3238002625, 6811549633, 14310965185
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (8^2, 9^2, 10^2, ...).

Links

Crossrefs

Cf. A048505.

Programs

  • Magma
    [(n^2+33*n+256) * 2^(n-2) - 63: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • PARI
    Vec((280*x^3-293*x^2+75*x+1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Feb 25 2015

Formula

a(n) = (n^2+33*n+256) * 2^(n-2) - 63. - Ralf Stephan, Feb 05 2004
a(n) = 7*a(n-1)-18*a(n-2)+20*a(n-3)-8*a(n-4). - Colin Barker, Feb 25 2015
G.f.: (280*x^3-293*x^2+75*x+1) / ((x-1)*(2*x-1)^3). Colin Barker, Feb 25 2015

A048515 a(n) = T(n,n), array T given by A048505.

Original entry on oeis.org

1, 5, 35, 160, 577, 1816, 5245, 14288, 37313, 94384, 232861, 563080, 1339249, 3141464, 7282493, 16711456, 38010625, 85786336, 192282301, 428342936, 948960881, 2091908680, 4590665245, 10032774640, 21843934657
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Magma
    [(9*n^2 + n) * 2^(n-2) - n^2 + 1: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011
  • PARI
    Vec(-(8*x^5-43*x^4+53*x^3-23*x^2+4*x-1)/((x-1)^3*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Feb 25 2015

Formula

a(n) = (9*n^2 + n) * 2^(n-2) - n^2 + 1. - Ralf Stephan, Feb 05 2004
G.f.: -(8*x^5-43*x^4+53*x^3-23*x^2+4*x-1) / ((x-1)^3*(2*x-1)^3). - Colin Barker, Feb 25 2015

A048514 a(n) = T(0,n)+T(1,n-1)+...+T(n,0), array T given by A048505.

Original entry on oeis.org

1, 3, 13, 54, 190, 587, 1659, 4412, 11244, 27785, 67089, 159106, 371930, 859159, 1964855, 4454968, 10025240, 22411221, 49804909, 110097630, 242217766, 530575683, 1157623603, 2516577524, 5452589700, 11777599457
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{10,-42,96,-129,102,-44,8},{1,3,13,54,190,587,1659},30] (* Harvey P. Dale, Aug 01 2022 *)
  • PARI
    T(k, n) = (n^2 + (4*k+1)*n + (2*k)^2) * 2^(n-2) - k^2 + 1
a(n) = sum(k=0, n, T(k,n-k)) \\ Colin Barker, Feb 25 2015

Formula

G.f.: (8*x^5-37*x^4+46*x^3-25*x^2+7*x-1) / ((x-1)^4*(2*x-1)^3). - Colin Barker, Feb 25 2015

Extensions

Typo in a(25) fixed by Colin Barker, Feb 25 2015
Showing 1-10 of 12 results. Next

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