A005191 -id:A005191 - cp's OEIS frontend

A208591 Number of n-bead necklaces labeled with numbers -2..2 not allowing reversal, with sum zero.

1, 3, 7, 23, 77, 297, 1163, 4783, 20041, 85735, 371955, 1634429, 7254547, 32486709, 146576693, 665720447, 3041096613, 13963651875, 64410421307, 298331882891, 1386933722321, 6469566027565, 30271114380919, 142037330038069, 668187211909327

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

			All solutions for n=3:
.-2...-1...-2...-2....0...-1...-1
..2...-1....1....0....0....1....0
..0....2....1....2....0....0....1

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 2 of A208597.

comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r - 1 - i*m, k - 1]*Binomial[k, i], {i, 0, Floor[(r - k)/m]}]; a[n_Integer, k_] := DivisorSum[n, EulerPhi[n/#] comps[#*(k + 1), 2 k + 1, #] &]/n; a[n_] = a[n, 2]; Array[a, 25] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A005191(d). - Andrew Howroyd, Mar 02 2017

a(22)-a(25) from Andrew Howroyd, Mar 02 2017

A104313 Numbers n such that the coefficient of x^(2n) in (x^4+x^3+x^2+x+1)^n is prime.

Original entry on oeis.org

2, 3, 28, 30, 31

Offset: 1

T. D. Noe, Mar 01 2005

n such that A005191(n) is prime. No other n<10000. The primes are in A104314. Only coefficients of the x, x^(2n) and x^(4n-1) terms can be prime; the coefficients of x and x^(4n-1) terms are prime whenever n is prime.

No other n<195316. Most likely this sequence is finite. Terms A005191(n) that are not a multiple of 5 have zero density, namely, there are fewer than n^(log(4)/log(5)) such terms among A005191(1..n). In particular, A005191(5k+2) and A005191(5k+4) are multiples of 5 for every k. - Max Alekseyev, Apr 25 2005

Cf. A005191 (pentanomial coefficients).

f=1; Do[f=Expand[f*(x^4+x^3+x^2+x+1)]; s=Coefficient[f, x, 2n]; If[PrimeQ[s], Print[{n, s}]], {n, 100}]

A124893 Number of base 15 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 15, 69, 255, 1097, 4765, 21237, 95873, 437537, 2012997, 9321899, 43397633, 202926209, 952396005, 4484003747, 21168581905, 100170213121, 474981809053, 2256307682541, 10735210220803, 51148724456627, 244007372314919

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (15, -78, 130, 176, -682, 44, 1128, -330, -818, 249, 261, -62, -30, 3, 1).

Cf. Row 15 of A285280.

A124949 Number of base 21 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 21, 99, 369, 1607, 7051, 31743, 144683, 666527, 3094947, 14463569, 67946603, 320593259, 1518250599, 7212838277, 34360479199, 164079145839, 785173661987, 3764380920489, 18077996769707, 86948519336037, 418760656546517

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (21, -171, 627, -595, -2635, 6891, 1703, -19279, 6565, 26454, -13848, -20145, 11191, 8590, -4346, -1925, 775, 190, -50, -5, 1).

Cf. Row 21 of A285280.

A124950 Number of base 22 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 22, 104, 388, 1692, 7432, 33494, 152818, 704692, 3275272, 15320514, 72038098, 340204434, 1612559698, 7667644032, 36559128748, 174730634628, 836872304392, 4015726470338, 19301794774522, 92915156135782, 447886264698862

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (23, -213, 973, -1924, -914, 10569, -11449, -15587, 33579, 4445, -38653, 8193, 21819, -7612, -6108, 2405, 765, -285, -35, 10).

Cf. Row 22 of A285280.

A124951 Number of base 23 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 23, 109, 407, 1777, 7813, 35245, 160953, 742857, 3455597, 16177459, 76129593, 359815609, 1706868797, 8122449787, 38757778297, 185382123417, 888570946797, 4267072020187, 20525592779337, 98881792935527, 477011872851207

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (23, -210, 910, -1406, -2888, 13004, -5440, -35735, 37373, 49139, -71097, -38050, 68300, 16925, -36185, -4145, 10375, 500, -1450, -25, 75).

Cf. Row 23 of A285280.

A124952 Number of base 24 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 24, 114, 426, 1862, 8194, 36996, 169088, 781022, 3635922, 17034404, 80221088, 379426784, 1801177896, 8577255542, 40956427846, 196033612206, 940269589202, 4518417570036, 21749390784152, 104848429735272, 506137481003552

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (24, -231, 1078, -1975, -2716, 16928, -12648, -44490, 67520, 55299, -131598, -32826, 137368, 4960, -82672, 4208, 28400, -2247, -5170, 395, 420, -25, -10).

Cf. Row 24 of A285280.

A124953 Number of base 25 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 25, 119, 445, 1947, 8575, 38747, 177223, 819187, 3816247, 17891349, 84312583, 399037959, 1895486995, 9032061297, 43155077395, 206685100995, 991968231607, 4769763119885, 22973188788967, 110815066535017, 535263089155897

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (25, -253, 1265, -2674, -2240, 21374, -23256, -51850, 111350, 49001, -222697, 3926, 248990, -47344, -166462, 41345, 66379, -16465, -14979, 3200, 1700, -265, -75, 5, 1).

Cf. Row 25 of A285280.

A124954 Number of base 26 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 26, 124, 464, 2032, 8956, 40498, 185358, 857352, 3996572, 18748294, 88404078, 418649134, 1989796094, 9486867052, 45353726944, 217336589784, 1043666874012, 5021108669734, 24196986793782, 116781703334762

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (26, -276, 1472, -3520, -1364, 26235, -38076, -55671, 171530, 18971, -349296, 102674, 410216, -187435, -296636, 150952, 132708, -65631, -35400, 15386, 5160, -1755, -340, 75, 6, -1).

Cf. Row 26 of A285280.

A124955 Number of base 27 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 27, 129, 483, 2117, 9337, 42249, 193493, 895517, 4176897, 19605239, 92495573, 438260309, 2084105193, 9941672807, 47552376493, 227988078573, 1095365516417, 5272454219583, 25420784798597, 122748340134507

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (27, -300, 1700, -4531, 23, 31338, -57950, -52991, 249983, -50491, -511207, 308325, 614063, -492688, -462734, 414962, 221420, -202415, -65687, 56707, 11333, -8470, -1000, 570, 36, -12).

Cf. Row 27 of A285280.

cp's OEIS Frontend

A208591 Number of n-bead necklaces labeled with numbers -2..2 not allowing reversal, with sum zero.

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A104313 Numbers n such that the coefficient of x^(2n) in (x^4+x^3+x^2+x+1)^n is prime.

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A124893 Number of base 15 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124949 Number of base 21 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124950 Number of base 22 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124951 Number of base 23 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124952 Number of base 24 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124953 Number of base 25 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124954 Number of base 26 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124955 Number of base 27 circular n-digit numbers with adjacent digits differing by 2 or less.

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