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 1-6 of 6 results.

A073947 Number of strings over Z_3 of length n with trace 0 and subtrace 0.

Original entry on oeis.org

1, 1, 3, 9, 21, 63, 225, 729, 2187, 6561, 19845, 59535, 177633, 531441, 1594323, 4782969, 14344533, 43033599, 129127041, 387420489, 1162261467, 3486784401, 10460471301, 31381413903, 94143533121, 282429536481, 847288609443, 2541865828329, 7625594296341
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over GF(3) of length n with trace 0 and subtrace 0.

Examples

			a(3;0,0)=3 since the three ternary strings of trace 0, subtrace 0 and length 3 are { 000, 111, 222 }.

Links

Crossrefs

Cf. A073948, A073949, A073950, A073951, A073952.

Programs

  • Mathematica
    LinearRecurrence[{6,-15,27,-36,27},{1,1,3,9,21},40] (* Harvey P. Dale, Nov 29 2024 *)

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: q*(21*q^4-21*q^3+12*q^2-5*q+1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013

A281270 a(n) is the number of closed BCK (a.k.a. affine) lambda terms of size n.

Original entry on oeis.org

0, 0, 1, 2, 3, 9, 30, 81, 242, 838, 2799, 9365, 33616, 122937, 449698, 1696724, 6558855, 25559806, 101294687, 409363758, 1673735259, 6928460475, 29115833976, 123835124242, 532449210893, 2317382872404, 10199542298725, 45345006540851, 203704505953902, 924427259637953, 4234544300812834
Offset: 0

Views

Author

Gheorghe Coserea, Apr 02 2017

Keywords

Comments

It appears that for n >= 1, a(n + 5) == a(n) (mod 5), a(n + 38*7) == a(n) (mod 7), a(n + 30*11) == a(n) (mod 11) and a(n + 288*17) == a(n) (mod 17). - Peter Bala, Apr 11 2022

Examples

			A(x) = x^2 + 2*x^3 + 3*x^4 + 9*x^5 + 30*x^6 + 81*x^7 + 242*x^8 + ...

Links

Crossrefs

Cf. A062980, A262301, A267827.

Programs

  • Mathematica
    a[0] = a[1] = 0; a[n_] := a[n] = 1 + a[n - 1] + 2 Sum[ k a[k], {k, 2, n - 3}] + Sum[a[k] a[n - 1 - k], {k, 2, n - 3}]; Table[a@ n, {n, 0, 30}] (* Michael De Vlieger, Apr 02 2017 *)
  • PARI
    seq(N) = {
  my(a = vector(N));
  for (n=2, N, my(s1 = sum(k=2, n-3, k*a[k]));
    a[n] = 1 + a[n-1] + 2*s1 + sum(k=2, n-3, a[k]*a[n-1-k]));
  concat(0,a);
};
seq(30)
\\ test: y = Ser(seq(201)); 0 == 2*x^4*y' + (x-x^2)*y^2 - (1-x)^2*y + x^2

Formula

a(n) = 1 + a(n-1) + 2*Sum_{k=2..n-3} k*a(k) + Sum_{k=2..n-3} a(k)*a(n-1-k) for n>=2.
0 = 2*x^4*y' + (x-x^2)*y^2 - (1-x)^2*y + x^2, where y(x) is the g.f.
a(3*n+1) = Sum_{k=0..n-1} binomial(3*n,3*k+1)*A062980(k).

Extensions

Name clarified by Pierre Lescanne, May 19 2017

A073948 Number of strings over Z_3 of length n with trace 0 and subtrace 1.

Original entry on oeis.org

0, 0, 0, 6, 30, 90, 252, 756, 2268, 6642, 19602, 58806, 176904, 530712, 1592136, 4780782, 14351094, 43053282, 129146724, 387440172, 1162320516, 3486843450, 10460294154, 31380882462, 94143001680, 282429005040, 847287015120, 2541864234006, 7625599079310
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over GF(3) of length n with trace 0 and subtrace 1.

Links

Crossrefs

Cf. A073947, A073949, A073950, A073951, A073952.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: -6q^4(q-1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013

A073949 Number of strings over Z_3 of length n with trace 0 and subtrace 2.

Original entry on oeis.org

0, 2, 6, 12, 30, 90, 252, 702, 2106, 6480, 19602, 58806, 176904, 532170, 1596510, 4785156, 14351094, 43053282, 129146724, 387400806, 1162202418, 3486725352, 10460294154, 31380882462, 94143001680, 282430067922, 847290203766, 2541867422652, 7625599079310
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over GF(3) of length n with trace 0 and subtrace 2.

Links

Crossrefs

Cf. A073947, A073948, A073950, A073951, A073952.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: -2q^2(3*q^3-3*q^2+3*q-1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013

A073951 Number of strings over Z_3 of length n with trace 1 and subtrace 1.

Original entry on oeis.org

0, 1, 3, 6, 21, 81, 252, 729, 2187, 6642, 19845, 59049, 176904, 531441, 1594323, 4780782, 14344533, 43046721, 129146724, 387420489, 1162261467, 3486843450, 10460471301, 31381059609, 94143001680, 282429536481, 847288609443, 2541864234006, 7625594296341
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over Z_3 of length n with trace 2 and subtrace 1. Same as number of strings over GF(3) of length n with trace 1 and subtrace 1. Same as number of strings over GF(3) of length n with trace 2 and subtrace 1.

Examples

			a(2;2,1)=1 since the one ternary string of trace 2, subtrace 1 and length 2 is { 11 }.

Links

Crossrefs

Cf. A073947, A073948, A073949, A073950, A073952.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: q^2(3*q^3+3*q^2-3*q+1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013

A073952 Number of strings over Z_3 of length n with trace 1 and subtrace 2.

Original entry on oeis.org

0, 0, 3, 12, 30, 81, 252, 756, 2187, 6480, 19602, 59049, 176904, 530712, 1594323, 4785156, 14351094, 43046721, 129146724, 387440172, 1162261467, 3486725352, 10460294154, 31381059609, 94143001680, 282429005040, 847288609443, 2541867422652, 7625599079310
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over Z_3 of length n with trace 2 and subtrace 2. Same as number of strings over GF(3) of length n with trace 1 and subtrace 2. Same as number of strings over GF(3) of length n with trace 2 and subtrace 2.

Examples

			a(3;1,2)=3 since the three ternary strings of trace 1, subtrace 2 and length 3 are { 112, 121, 211 }.

Links

Crossrefs

Cf. A073947, A073948, A073949, A073950, A073951.

Programs

  • Mathematica
    LinearRecurrence[{6,-15,27,-36,27},{0,0,3,12,30},30] (* Harvey P. Dale, Oct 22 2019 *)

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: 3q^3(q^2-2*q+1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013
Showing 1-6 of 6 results.

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