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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A124799 Number of base 30 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 30, 88, 204, 548, 1460, 4006, 11090, 31036, 87468, 248018, 706670, 2021738, 5804010, 16711552, 48241364, 139572076, 404612780, 1175026834, 3417771710, 9955368238, 29035695998, 84784671532, 247838482400, 725183659570
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1
a(n) = T(n, 30) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,30}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

A124803 Number of base 31 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 31, 91, 211, 567, 1511, 4147, 11483, 32143, 90607, 256971, 732323, 2095527, 6016951, 17327779, 50028971, 144768703, 419747711, 1219179643, 3546768563, 10332747607, 30141046727, 88025807059, 257351710523, 753131995951
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1
a(n) = T(n, 31) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,31}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

A124804 Number of base 32 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 32, 94, 218, 586, 1562, 4288, 11876, 33250, 93746, 265924, 757976, 2169316, 6229892, 17944006, 51816578, 149965330, 434882642, 1263332452, 3675765416, 10710126976, 31246397456, 91266942586, 266864938646, 781080332332
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1
a(n) = T(n, 32) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,32}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

A136493 Triangle of coefficients of characteristic polynomials of symmetrical pentadiagonal matrices of the type (1,-1,1,-1,1).

Original entry on oeis.org

1, -1, 1, 1, -2, 0, -1, 3, 0, 0, 1, -4, 1, 2, 0, -1, 5, -3, -5, 1, 1, 1, -6, 6, 8, -5, -2, 1, -1, 7, -10, -10, 14, 4, -4, 0, 1, -8, 15, 10, -29, -4, 12, 0, 0, -1, 9, -21, -7, 50, -4, -30, 4, 4, 0, 1, -10, 28, 0, -76, 28, 61, -20, -15, 2, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 21 2008

Keywords

Comments

From Georg Fischer, Mar 29 2021: (Start)
The pentadiagonal matrices have 1 in the main diagonal, -1 in the first lower and upper diagonal, 1 in the second lower and upper diagonal, and 0 otherwise.
The linear recurrences that yield A124805, A124806, A124807 and similar can be derived from the rows of this triangle (the first element of a row must be removed and multiplied onto the remaining elements).
This observation extends to other sequences. For example the linear recurrence signature (5,-6,2,4,0) of A124698 "Number of base 5 circular n-digit numbers with adjacent digits differing by 1 or less" can be derived from the coefficients of the characteristic polynomial of a tridiagonal (type -1,1,-1) 5 X 5 matrix.
(End)

Examples

			Triangle begins:
   1;
  -1,   1;
   1,  -2,   0;
  -1,   3,   0,   0;
   1,  -4,   1,   2,   0;
  -1,   5,  -3,  -5,   1,  1;
   1,  -6,   6,   8,  -5, -2,   1;
  -1,   7, -10, -10,  14,  4,  -4,   0;
   1,  -8,  15,  10, -29, -4,  12,   0,   0;
  -1,   9, -21,  -7,  50, -4, -30,   4,   4,  0;
   1, -10,  28,   0, -76, 28,  61, -20, -15,  2,  1;

References

  • Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.

Links

Crossrefs

Cf. A011658, A087509, A124805 ff., A124696 ff., A124999 ff., A161680, A181983.

Programs

  • Mathematica
    T[n_, m_]:= Piecewise[{{-1, 1+m==n || m==1+n}, {1, 2+m==n || m==n || m==2+n}}];
MO[d_]:= Table[T[n, m], {n,d}, {m,d}];
CL[n_]:= CoefficientList[CharacteristicPolynomial[MO[n], x], x];
Join[{{1}}, Table[Reverse[CL[n]], {n,10}]]//Flatten
(* For the signature of A124698 added by Georg Fischer, Mar 29 2021 : *)
Reverse[CoefficientList[CharacteristicPolynomial[{{1,-1,0,0,0}, {-1, 1,-1,0,0}, {0,-1,1,-1,0}, {0,0,-1,1,-1}, {0,0,0,-1,1}}, x], x]]

Formula

Sum_{k=1..n} T(n, k) = (-1)^(n mod 3) * A087509(n+1) + [n=1].
From G. C. Greubel, Aug 01 2023: (Start)
T(n, n) = A011658(n+2).
T(n, 1) = (-1)^(n-1).
T(n, 2) = A181983(n-1).
T(n, 3) = (-1)^(n-3)*A161680(n-3). (End)

Extensions

Edited by Georg Fischer, Mar 29 2021

A124701 Number of base 8 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 8, 22, 50, 130, 338, 904, 2444, 6682, 18410, 51052, 142304, 398380, 1119308, 3154558, 8914010, 25246282, 71644298, 203665054, 579841286, 1653025900, 4718011460, 13479908926, 38548802570, 110327691316, 315985475588
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1
a(n) = T(n, 8) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,8}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

Formula

G.f.: (1 - 21*x^2 + 28*x^3 + 60*x^4 - 96*x^5 - 15*x^6 + 36*x^7) / ((1 - 2*x)*(1 - 3*x + x^3)*(1 - 3*x + 3*x^3)) (conjectured). - Colin Barker, Jun 03 2017

A124702 Number of base 9 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 9, 25, 57, 149, 389, 1045, 2837, 7789, 21549, 60005, 167957, 472169, 1332249, 3770785, 10701617, 30442909, 86779229, 247817845, 708837797, 2030401509, 5823331109, 16720830525, 48060737357, 138268935049, 398126270889
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1
a(n) = T(n, 9) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,9}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

Formula

G.f.: (1 - 28*x^2 + 56*x^3 + 63*x^4 - 196*x^5 + 30*x^6 + 108*x^7 - 21*x^8 - 8*x^9) / ((1 - x)*(1 - 3*x + x^2)*(1 - x - x^2)*(1 - 4*x + x^2 + 6*x^3 + x^4)) (conjectured). - Colin Barker, Jun 02 2017

A124703 Number of base 10 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 10, 28, 64, 168, 440, 1186, 3230, 8896, 24688, 68958, 193610, 545958, 1545190, 4387012, 12489224, 35639536, 101914160, 291970654, 837834650, 2407780858, 6928681418, 19961961014, 57573920446, 166216938550, 480300958390
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1
a(n) = T(n, 10) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,10}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

Formula

G.f.: (1 - 36*x^2 + 96*x^3 + 42*x^4 - 336*x^5 + 175*x^6 + 216*x^7 - 126*x^8 - 32*x^9 + 9*x^10) / ((1 - 6*x + 10*x^2 - x^3 - 6*x^4 + x^5)*(1 - 4*x + 2*x^2 + 5*x^3 - 2*x^4 - x^5)) (conjectured). - Colin Barker, Jun 01 2017

A124704 Number of base 11 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 11, 31, 71, 187, 491, 1327, 3623, 10003, 27827, 77911, 219263, 619747, 1758131, 5003239, 14276831, 40836163, 117049091, 336123463, 966831503, 2785160227, 8034032147, 23203096519, 67087148063, 194165268355, 562477731491
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1
a(n) = T(n, 11) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,11}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

Formula

G.f.: (1 - 45*x^2 + 150*x^3 - 18*x^4 - 504*x^5 + 490*x^6 + 300*x^7 - 420*x^8 - 32*x^9 + 72*x^10) / ((1 - x)*(1 - 2*x)*(1 - 2*x - x^2)*(1 - 2*x - 2*x^2)*(1 - 4*x + 2*x^2 + 4*x^3 - 2*x^4)) (conjectured). - Colin Barker, Jun 03 2017

A124705 Number of base 12 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 12, 34, 78, 206, 542, 1468, 4016, 11110, 30966, 86864, 244916, 693536, 1971072, 5619466, 16064438, 46032790, 132184022, 380276272, 1095828356, 3162539596, 9139382876, 26444232046, 76600376186, 222113604712, 644654567192
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1.
a(n) = T(n, 12) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,12}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

Formula

G.f.: (1 - 55*x^2 + 220*x^3 - 135*x^4 - 672*x^5 + 1050*x^6 + 216*x^7 - 1015*x^8 + 160*x^9 + 270*x^10 - 40*x^11 - 11*x^12) / ((1 - 5*x + 5*x^2 + 6*x^3 - 7*x^4 - 2*x^5 + x^6)*(1 - 7*x + 15*x^2 - 6*x^3 - 11*x^4 + 6*x^5 + x^6)) (conjectured). - Colin Barker, Jul 17 2017

A124706 Number of base 13 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 13, 37, 85, 225, 593, 1609, 4409, 12217, 34105, 95817, 270569, 767325, 2184013, 6235693, 17852045, 51229417, 147318953, 424429081, 1224825209, 3539918965, 10244733605, 29685367573, 86113604309, 250061941093, 726831403493
Offset: 0

Views

Author

R. H. Hardin, Dec 28 2006

Keywords

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1
a(n) = T(n, 13) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,13}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

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