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

A027164 a(n) = T(n,0) + T(n,1) + ... + T(n,n), T given by A027157.

Original entry on oeis.org

1, 3, 10, 22, 67, 145, 436, 940, 2821, 6079, 18238, 39298, 117895, 254029, 762088, 1642072, 4926217, 10614523, 31843570, 68613358, 205840075, 443523721, 1330571164, 2866982404, 8600947213, 18532465591, 55597396774
Offset: 0

Views

Author

Clark Kimberling

Keywords

Programs

  • Mathematica
    CoefficientList[Series[(1+2x+x^2)/(1-x-6x^2+6x^3-3x^4 + 3x^5),{x,0,30}],x]  (* Harvey P. Dale, Mar 20 2011 *)

Formula

G.f.: x(x^2+2x+1)/[(1-x)(1-6x^2-3x^4)].

A027166 a(n) = Sum_{0<=j<=i<=n} A027157(i, j).

Original entry on oeis.org

1, 4, 14, 36, 103, 248, 684, 1624, 4445, 10524, 28762, 68060, 185955, 439984, 1202072, 2844144, 7770361, 18384884, 50228454, 118841812, 324681887, 768205608, 2098776772, 4965759176, 13566706389, 32099171980, 87696568754, 207492309516, 566879531803
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Partial sums of A027164.

Programs

  • Mathematica
    LinearRecurrence[{2,5,-12,9,-6,3},{1,4,14,36,103,248},30] (* Harvey P. Dale, Apr 18 2019 *)
  • PARI
    Vec((1+x)^2/((1-x)^2*(1-6*x^2-3*x^4)) + O(x^40)) \\ Colin Barker, Feb 20 2016

Formula

From Colin Barker, Feb 20 2016: (Start)
a(n) = 2*a(n-1)+5*a(n-2)-12*a(n-3)+9*a(n-4)-6*a(n-5)+3*a(n-6) for n>5.
G.f.: (1+x)^2 / ((1-x)^2*(1-6*x^2-3*x^4)).
(End)

A027158 a(n) = T(2n,n), T given by A027157.

Original entry on oeis.org

1, 3, 24, 76, 672, 2208, 20224, 67776, 632320, 2144768, 20238336, 69203968, 658079744, 2263351296, 21643526144, 74764042240, 717975060480, 2488551997440, 23977645834240, 83334108020736, 805090608807936, 2804309838266368
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027159 a(n) = T(2n,n-1), T given by A027157.

Original entry on oeis.org

3, 10, 76, 264, 2208, 7744, 67776, 238720, 2144768, 7572480, 69203968, 244715520, 2263351296, 8012201984, 74764042240, 264873738240, 2488551997440, 8821831106560, 83334108020736, 295559851147264, 2804309838266368
Offset: 1

Views

Author

Clark Kimberling

Keywords

A027160 a(n) = T(2n,n-2), T given by A027157.

Original entry on oeis.org

5, 21, 176, 680, 5808, 21840, 190976, 708480, 6310400, 23220736, 209756160, 767676416, 7010676736, 25558732800, 235443322880, 855858053120, 7939647995904, 28796047982592, 268690773770240, 972740537352192, 9120760620646400
Offset: 2

Views

Author

Clark Kimberling

Keywords

A027161 T(2n-1,n-1), T given by A027157.

Original entry on oeis.org

2, 6, 40, 152, 1088, 4416, 32384, 135552, 1006592, 4289536, 32100352, 138407936, 1041203200, 4526702592, 34182430720, 149528084480, 1132376883200, 4977103994880, 37776580935680, 166668216041472, 1267318675996672
Offset: 1

Views

Author

Clark Kimberling

Keywords

A027162 a(n) = T(2n-1,n-2), T given by A027157.

Original entry on oeis.org

4, 15, 112, 440, 3328, 13552, 103168, 429696, 3282944, 13882880, 106307584, 454471680, 3485499392, 15022878720, 115345653760, 500317061120, 3844727111680, 16761479102464, 128891635105792, 564250624917504, 4341301000536064
Offset: 2

Views

Author

Clark Kimberling

Keywords

A027163 a(n) = T(n,[ n/2 ]), T given by A027157.

Original entry on oeis.org

1, 2, 3, 6, 24, 40, 76, 152, 672, 1088, 2208, 4416, 20224, 32384, 67776, 135552, 632320, 1006592, 2144768, 4289536, 20238336, 32100352, 69203968, 138407936, 658079744, 1041203200, 2263351296, 4526702592, 21643526144, 34182430720
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027165 T(n,0) + T(n,1) + ... + T(n,[ n/2 ]), T given by A027157.

Original entry on oeis.org

1, 2, 6, 10, 39, 61, 180, 300, 1157, 1823, 5502, 9282, 35911, 57101, 174312, 296536, 1147657, 1837307, 5643506, 9657582, 37349067, 60103305, 185313180, 318533124, 1230555661, 1988344759, 6145979174, 10601082970, 40907870479
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027167 a(n) = Sum_{k=0..floor(n/2)} A027157(n-k, k).

Original entry on oeis.org

1, 2, 4, 7, 15, 24, 50, 81, 165, 266, 544, 875, 1787, 2876, 5870, 9445, 19281, 31022, 63324, 101887, 207975, 334624, 683050, 1099001, 2243325, 3609426, 7367704, 11854355, 24197587, 38932996, 79471590, 127866765, 261006761
Offset: 0

Views

Author

Clark Kimberling

Keywords

Formula

Empirical g.f.: (1+x)^2*(1-x+x^2) / ((1-x)*(1-2*x^2-3*x^4-4*x^6)). - Colin Barker, Feb 20 2016
Showing 1-10 of 12 results. Next

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