A000930 -id:A000930 - cp's OEIS frontend

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A227834 2^a(n) is the highest power of 2 dividing A000930(n).

0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 3, 0, 3, 7, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 1, 0, 4, 1, 0, 0, 0, 4, 0, 4, 6, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 3, 0, 3, 6, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 1, 0, 5, 1, 0, 0, 0, 5, 0, 5, 7, 0

Offset: 0

N. J. A. Sloane, Aug 04 2013

nonn

This is the 2-adic valuation of A000930.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000930, A007814, A227835.

A000930:=func; [Valuation(A000930(n),2): n in [0..120]]; // Bruno Berselli, Aug 05 2013

Mathematica

A000930[n_] := Sum[Binomial[n - 2*i, i], {i, 0, Floor[n/3]}]; Table[IntegerExponent[A000930[n], 2], {n, 0, 100}] (* G. C. Greubel, Apr 26 2017 *)

A231621 a(n) = A000930(n*(n+1)/2), where A000930 is Narayana's cows sequence.

Original entry on oeis.org

1, 1, 2, 6, 28, 189, 1873, 27201, 578949, 18059374, 825604416, 55315679788, 5431645680297, 781666575692345, 164861247948842305, 50959194632488457965, 23085190353310504913320, 15326793132326730009566200, 14913379277290330452859885202, 21267074956884103635776195255433, 44447403127130268192387935737712641

Offset: 0

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Author

Paul D. Hanna, Nov 13 2013

Keywords

nonn

Links

G. C. Greubel, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000930, A231620.

Programs

Mathematica

Table[SeriesCoefficient[1/(1 - x - x^3), {x, 0, n*(n + 1)/2}], {n,0,50}] (* G. C. Greubel, Apr 26 2017 *)

PARI

{a(n) = polcoeff(1/(1-x-x^3 + x*O(x^(n*(n+1)/2))), n*(n+1)/2)} for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = [x^(n*(n+1)/2)] 1 / (1 - x - x^3) for n>=0.

A274452 Products of distinct Narayana's cow numbers (A000930).

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 12, 13, 18, 19, 24, 26, 27, 28, 36, 38, 39, 41, 48, 52, 54, 56, 57, 60, 72, 76, 78, 82, 84, 88, 104, 108, 112, 114, 117, 120, 123, 129, 144, 152, 156, 162, 164, 168, 171, 176, 180, 189, 216, 224, 228, 234, 240, 246, 247, 252, 258, 264, 277

Offset: 1

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Author

Clark Kimberling, Jun 23 2016

Keywords

nonn
easy

Examples

The Narayana's cow numbers numbers are 1, 2, 3, 4, 6, 9, 13, 19, 28, ..., so that the sequence of all products of distinct members, in increasing order, is (2, 3, 4, 6, 8, 9, 12, 13, 18, 19, 24, ...).

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A160009, A274280, A274432.

Programs

Mathematica

r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = r[n - 1] + r[n - 3] s = {1}; z = 60; f = Map[r, Range[z]]; Take[f, 20] (*A000930*) Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; Take[s, 2 z] (* A274452 *)

A170933 a(n) = A000930(n) + A000930(n+3) + 4.

Original entry on oeis.org

7, 8, 9, 12, 16, 21, 29, 41, 58, 83, 120, 174, 253, 369, 539, 788, 1153, 1688, 2472, 3621, 5305, 7773, 11390, 16691, 24460, 35846, 52533, 76989, 112831, 165360, 242345, 355172, 520528, 762869, 1118037

Offset: 0

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Author

N. J. A. Sloane, Feb 10 2010

Keywords

nonn
easy

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Z. Skupien, Sparse Hamiltonian 2-decompositions together with exact count of numerous Hamiltonian cycles, Discr. Math., 309 (2009), 6382-6390.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Programs

Mathematica

CoefficientList[Series[(7 - 6*x - 5*x^3)/(1 - 2*x + x^2 - x^3 + x^4), {x, 0, 50}], x] (* G. C. Greubel, Apr 26 2017 *) LinearRecurrence[{2,-1,1,-1},{7,8,9,12},40] (* Harvey P. Dale, Aug 07 2021 *)

PARI

x='x+O('x^50); Vec((7 - 6*x - 5*x^3)/(1 - 2*x + x^2 - x^3 + x^4)) \\ G. C. Greubel, Apr 26 2017

Formula

G.f.: (7-6*x-5*x^3)/(1-2*x+x^2-x^3+x^4). - Colin Barker, Dec 30 2011

a(n) = a(n-1) + a(n-3) - 4 for n>2. - Greg Dresden, Feb 09 2020

A170934 a(n) = b(n) + b(n+1) + 2, where b() = A000930().

Original entry on oeis.org

4, 4, 5, 7, 9, 12, 17, 24, 34, 49, 71, 103, 150, 219, 320, 468, 685, 1003, 1469, 2152, 3153, 4620, 6770, 9921, 14539, 21307, 31226, 45763, 67068, 98292, 144053, 211119, 309409, 453460, 664577, 973984, 1427442, 2092017, 3065999, 4493439, 6585454, 9651451

Offset: 0

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Author

N. J. A. Sloane, Feb 10 2010

Keywords

nonn
easy

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Z. Skupien, Sparse Hamiltonian 2-decompositions together with exact count of numerous Hamiltonian cycles, Discr. Math., 309 (2009), 6382-6390.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Programs

Mathematica

CoefficientList[Series[-(3*x^3 - x^2 + 4*x - 4)/((x - 1)*(x^3 + x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Apr 26 2017 *) LinearRecurrence[{2,-1,1,-1},{4,4,5,7},50] (* Harvey P. Dale, Jul 10 2020 *)

PARI

x='x+O('x^50); Vec(-(3*x^3 - x^2 + 4*x - 4)/((x - 1)*(x^3 + x - 1))) \\ G. C. Greubel, Apr 26 2017

Formula

From Colin Barker, Jul 25 2013: (Start)

a(n) = 2*a(n-1)-a(n-2)+a(n-3)-a(n-4).

G.f.: -(3*x^3-x^2+4*x-4) / ((x-1)*(x^3+x-1)). (End)

a(n) = a(n-1) + a(n-3) - 2, for n > 2. - Greg Dresden, Feb 09 2020

Extensions

More terms from Colin Barker, Jul 25 2013

A170935 b(n)*b(n+1), where b() = A000930().

Original entry on oeis.org

1, 1, 2, 6, 12, 24, 54, 117, 247, 532, 1148, 2460, 5280, 11352, 24381, 52353, 112462, 241570, 518840, 1114416, 2393694, 5141385, 11043135, 23719608, 50947336, 109429624, 235043840, 504850560, 1084367865, 2329112625, 5002699050, 10745292366, 23079802756, 49573086472

Offset: 0

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Author

N. J. A. Sloane, Feb 11 2010

Keywords

nonn
easy

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,3,1,-1,-1).

Programs

Mathematica

CoefficientList[Series[1/((x^3 - x^2 - 1)*(x^3 + 2*x^2 + x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Apr 26 2017 *) LinearRecurrence[{1,1,3,1,-1,-1},{1,1,2,6,12,24},40] (* Harvey P. Dale, Aug 27 2019 *)

PARI

x='x+O('x^50); Vec(1/((x^3-x^2-1)*(x^3+2*x^2+x-1))) \\ G. C. Greubel, Apr 26 2017

Formula

G.f.: 1 / ((x^3-x^2-1)*(x^3+2*x^2+x-1)). - Colin Barker, Jul 25 2013

A294688 Palindromes in Narayana's cows sequence (A000930).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 88, 595

Offset: 1

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Author

Chai Wah Wu, Nov 06 2017

Keywords

nonn
base
more
hard

Comments

595 is probably the last term.

Crossrefs

Cf. A000930, A002113, A045504, A055391.

Programs

Maple

a := proc(n) if n < 4 then n else simplify(hypergeom([-n/3, (1-n)/3, (-1-n)/3], [-n/2, (-1-n)/2], -27/4)); convert(%, base, 10); ListTools:-Reverse(%); if %% = % then add(%[k]*10^(k-1), k=1..nops(%)) else NULL fi fi end: seq(a(n), n=1..200); # Peter Luschny, Nov 07 2017

A137433 Coefficients of A000930 expansion similar to that given for Fibonacci numbers in Roman's Umbral Calculus.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 8, 3, 1, 0, 30, 35, 6, 1, 0, 144, 230, 95, 10, 1, 0, 1200, 1954, 945, 205, 15, 1, 0, 10800, 19824, 11494, 2835, 385, 21, 1, 0, 105840, 216012, 149212, 45409, 7000, 658, 28, 1, 0, 1249920, 2692080, 2055500, 740124, 140889, 15120, 1050, 36, 1

Offset: 1

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Author

Roger L. Bagula, Apr 17 2008

Keywords

nonn
tabl
uned

Comments

Row sums:

{1, 1, 2, 12, 72, 480, 4320, 45360, 524160, 6894720, 101606400}

Row_sum(n)/n!=A000930(n)

Examples

{1}, {0, 1}, {0, 1, 1}, {0, 8, 3, 1}, {0, 30, 35, 6, 1}, {0, 144, 230, 95, 10, 1}, {0, 1200, 1954, 945, 205, 15, 1}, {0, 10800, 19824, 11494, 2835, 385, 21, 1}, {0, 105840, 216012, 149212, 45409, 7000, 658, 28, 1}, {0, 1249920, 2692080, 2055500, 740124, 140889, 15120, 1050, 36, 1}, {0, 16692480, 37802736, 31266540, 12628160, 2814525, 370713, 29610, 1590, 45, 1}

References

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150

Crossrefs

Cf. A000045, A000930.

Programs

Mathematica

Clear[p, g]; p[t_] = 1/(1 - t - t^3)^x; Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

Formula

Coefficients expansion of p(x,n) in f(x,t)=1/(1-t-t^3)^x=Sum[p(x,n)*t^n/n!m{n,1,Infinity}]

A301877 Group the natural numbers into groups (1),(2),(3),(4),(5,6),(7,8,9),... so that the n-th group contains N(n) terms, where N(n) is the Narayana's cows sequence (A000930). Sequence contains the sum of the terms in the n-th group.

Original entry on oeis.org

1, 2, 3, 4, 11, 24, 46, 99, 216, 455, 969, 2086, 4469, 9570, 20548, 44118, 94689, 203318, 436653, 937720, 2013884, 4325391, 9290080, 19953405, 42857019, 92051300, 197714721, 424668244, 912140480, 1959179226, 4208109535, 9038581200, 19413940167, 41699153408, 89565528714, 192377651011, 413207678264

Offset: 1

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Author

Frank M Jackson, Mar 28 2018

Keywords

nonn

Comments

a(n) is the sum of all nodes at height n-1 within a binary tree structure recursively built from the Hofstadter H-sequence (see comments for A005374).

Examples

a(8) = 14 + 15 + 16 + 17 + 18 + 19 = (2N(8)+N(6)+1)*N(6)/2 = 99.

Links

J. Ramírez, V. Sirvent, A note on the k-Narayana sequence, Annales Mathematicae et Informaticae 45 (2015) pp. 91-105.

Wikipedia, Hofstadter sequence.

Crossrefs

Cf. A000930, A005374.

Programs

Mathematica

Narayana[n_] := Which[n==0, 1, n==1, 1, n==2, 1, True, Narayana[n]=Narayana[n-3] + Narayana[n-1]]; a[n_] := If[n==1, 1, (2 Narayana[n]+Narayana[n-2]+1)Narayana[n-2]/2]; Array[a, 50]

Formula

a(1) = 1 and for n > 1, a(n) = (2N(n)+N(n-2)+1)*N(n-2)/2, where N(n) is the Narayana's cows sequence (A000930).

Conjectures from Colin Barker, Mar 28 2018: (Start)

G.f.: x*(1 - x)*(1 + x - 5*x^3 - 5*x^4 - 3*x^5 + x^6 + 2*x^7 + x^8) / ((1 + x^2 - x^3)*(1 - x - x^3)*(1 - x - 2*x^2 - x^3)).

a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 6*a(n-4) - 6*a(n-5) + 5*a(n-6) - a(n-7) for n>10.

(End)

A368396 Prime numbers in Narayana's cows sequence A000930.

Original entry on oeis.org

2, 3, 13, 19, 41, 277, 1873, 8641, 17098272199297

Offset: 1

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Author

José E. Solsona, Dec 22 2023

Keywords

nonn

Comments

The next terms are too large to show here: a(10) and a(11) have 1036 and 4274 digits, respectively (see A170954). - Jon E. Schoenfield, Dec 29 2023

Crossrefs

Primes in A000930.

Cf. A000040, A170954.

Programs

Mathematica

a[0] = 1; a[1] = a[2] = 1; a[n_] := a[n] = a[n-1] + a[n-3]; Select[Table[a[n], {n, 0, 400}], PrimeQ]

Formula

a(n) = A000930(A170954(n)).

A000040 INTERSECT A000930.

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

A227834 2^a(n) is the highest power of 2 dividing A000930(n).

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Magma

Mathematica

A231621 a(n) = A000930(n*(n+1)/2), where A000930 is Narayana's cows sequence.

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Mathematica

PARI

Formula

A274452 Products of distinct Narayana's cow numbers (A000930).

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Examples

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Crossrefs

Programs

Mathematica

A170933 a(n) = A000930(n) + A000930(n+3) + 4.

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Mathematica

PARI

Formula

A170934 a(n) = b(n) + b(n+1) + 2, where b() = A000930().

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Mathematica

PARI

Formula

Extensions

A170935 b(n)*b(n+1), where b() = A000930().

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Programs

Mathematica

PARI

Formula

A294688 Palindromes in Narayana's cows sequence (A000930).

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Comments

Crossrefs

Programs

Maple

A137433 Coefficients of A000930 expansion similar to that given for Fibonacci numbers in Roman's Umbral Calculus.

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Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

A301877 Group the natural numbers into groups (1),(2),(3),(4),(5,6),(7,8,9),... so that the n-th group contains N(n) terms, where N(n) is the Narayana's cows sequence (A000930). Sequence contains the sum of the terms in the n-th group.

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