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.

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A026578 T(2n-1,n-2), T given by A026568.

Original entry on oeis.org

1, 3, 13, 60, 225, 1148, 4235, 22296, 82425, 440308, 1634435, 8809736, 32819839, 178029138, 665162897, 3625521728, 13577768505
Offset: 2

Views

Author

Clark Kimberling

Keywords

A026579 T(n,[ n/2 ]), T given by A026568.

Original entry on oeis.org

1, 1, 1, 2, 7, 8, 19, 34, 103, 121, 341, 606, 1809, 2155, 6336, 11306, 33899, 40717, 121483, 217666, 656199, 792351, 2381512, 4280046, 12957809, 15703156, 47419503, 85410872, 259395664, 315180458, 954961034, 1722900242, 5245001951
Offset: 0

Views

Author

Clark Kimberling

Keywords

A026580 T(n,0) + T(n,1) + ... + T(n,n), T given by A026568.

Original entry on oeis.org

1, 2, 5, 12, 30, 75, 189, 478, 1211, 3076, 7818, 19901, 50680, 129179, 329383, 840336, 2144540, 5474889, 13980493, 35709394, 91228015, 233108456, 595740568, 1522725379, 3892624055, 9952119054, 25446913766, 65072487645
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A026568.

Formula

Conjecture: n*a(n) +(n-2)*a(n-1) +(-17*n+28)*a(n-2) +5*(-n+4)*a(n-3) +2*(34*n-107)*a(n-4) +16*(n-5)*a(n-5) +32*(-2*n+9)*a(n-6)=0. - R. J. Mathar, Jun 23 2013

A026582 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026568.

Original entry on oeis.org

1, 3, 8, 20, 50, 125, 314, 792, 2003, 5079, 12897, 32798, 83478, 212657, 542040, 1382376, 3526916, 9001805, 22982298, 58691692, 149919707, 383028163, 978768731, 2501494110, 6394118165, 16346237219, 41793150985, 106865638630
Offset: 0

Views

Author

Clark Kimberling

Keywords

Formula

Conjecture: n*a(n) +2*(-2*n+1)*a(n-1) +2*(-n+3)*a(n-2) +20*(n-2)*a(n-3) +(-7*n+6)*a(n-4) +2*(-12*n+41)*a(n-5) +8*(2*n-7)*a(n-6)=0. - R. J. Mathar, Jun 23 2013

A027278 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026568.

Original entry on oeis.org

2, 8, 60, 312, 2030, 11752, 74174, 450292, 2829898, 17572000, 110733420, 695885000, 4403631638, 27874477216, 177101756296, 1126506215424, 7182056699722, 45847087891480, 293141734241436, 1876495434801400, 12026885684352138, 77161051350902456, 495534832203481122
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027279 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026568.

Original entry on oeis.org

7, 44, 273, 1696, 10573, 66151, 415277, 2615004, 16512355, 104526093, 663143211, 4215574482, 26846320149, 171243444982, 1093897673897, 6997031842554, 44809862773473, 287283383574927, 1843675415666053, 11842954734864350, 76138777668602115, 489884818519692190
Offset: 2

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027280 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026568.

Original entry on oeis.org

26, 164, 1224, 7664, 51488, 326456, 2126376, 13574046, 87554326, 560845908, 3607790136, 23158545126, 148945375022, 957564479960, 6162874994108, 39672627184498, 255578754020002, 1647166273816172, 10622076015026288, 68529892093271816, 442357250742603624
Offset: 3

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027281 a(n) = Sum_{k=0..2n} (k+1) * A026568(n, k).

Original entry on oeis.org

1, 6, 21, 76, 235, 738, 2177, 6424, 18423, 52590, 147917, 413796, 1147523, 3166842, 8689305, 23745584, 64621199, 175277574, 473909989, 1277864060, 3436938267, 9223203858, 24699503281, 66020102472, 176160084775, 469293598878
Offset: 0

Views

Author

Clark Kimberling

Keywords

Formula

G.f.: (2x^2+4x+1)/(1-x-4x^2)^2.

A026552 Irregular triangular array T read by rows: T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor(n/2 + 1), for even n >= 2, T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), otherwise T(n, k) = T(n-1, k-2) + T(n-1, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 4, 12, 24, 40, 52, 58, 52, 40, 24, 12, 4, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 5, 18, 45, 93, 156, 226, 278
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i)-s(i-1)|<=1 if i is even or i = 1, |s(i)-s(i-1)| = 1 if i is odd and i >= 3.

Examples

			First 5 rows:
  1;
  1, 1, 1;
  1, 2, 3,  2,  1;
  1, 2, 4,  4,  4,  2,  1;
  1, 3, 7, 10, 12, 10,  7,  3,  1;

Links

Crossrefs

Cf. A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.
Cf. A026519, A026536, A026568, A026584, A027926.
Cf. A026565.

Programs

  • Mathematica
    z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[n/2 + 1]; t[n_, k_] := Floor[n/2 + 1] /; k == 2 n - 1; t[n_, k_] := t[n, k] = If[EvenQ[n], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k], t[n - 1, k - 2] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u] (* A026552 array *)
v = Flatten[u] (* A026552 sequence *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+2)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-2)
flatten([[T(n,k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Dec 17 2021

Formula

Sum_{k=0..2*n} T(n,k) = A026565(n). - G. C. Greubel, Dec 17 2021

Extensions

Updated by Clark Kimberling, Aug 28 2014

A026519 Irregular triangular array T read by rows: T(n, k) = T(n-1, k-2) + T(n-1, k) if (n mod 2) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), with T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor((n+1)/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 2, 5, 6, 8, 6, 5, 2, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 3, 9, 16, 27, 33, 38, 33, 27, 16, 9, 3, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 4, 14, 32, 65, 104, 150, 180, 196, 180, 150, 104, 65, 32, 14, 4, 1
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i) - s(i-1)| = 1 if i is even, |s(i) - s(i-1)| <= 1 if i is odd.

Examples

			First 5 rows:
1
1 ... 1 ... 1
1 ... 1 ... 2 ... 1 ... 1
1 ... 2 ... 4 ... 4 ... 4 ... 2 ... 1
1 ... 2 ... 5 ... 6 ... 8 ... 6 ... 5 ... 2 ... 1

Links

Crossrefs

Cf. A026552, A026536, A026568, A026584, A027926.
Cf. A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Programs

  • Mathematica
    z = 12; t[n_, 0]:= 1; t[n_, k_]:= 1/; k==2n; t[n_, 1]:= Floor[(n+1)/2]; t[n_, k_] := Floor[(n+1)/2] /; k==2n-1; t[n_, k_]:= t[n, k]= If[EvenQ[n], t[n-1, k-2] + t[n-1, k], t[n-1, k-2] + t[n-1, k-1] + t[n-1, k]];
u = Table[t[n, k], {n, 0, z}, {k, 0, 2n}];
TableForm[u]  (* A026519 array *)
Flatten[u] (* A026519 sequence *)
  • Sage
    @CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Dec 19 2021

Formula

T(n, k) = T(n-1, k-2) + T(n-1, k) if (n mod 2) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), with T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor((n+1)/2).

Extensions

Updated by Clark Kimberling, Aug 29 2014
Offset changed to 0 by G. C. Greubel, Dec 19 2021
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