Henry W. Gould - cp's OEIS frontend

User: Henry W. Gould

Henry W. Gould has authored 4 sequences.

A227800 Number of different values the product p*q can have where p >= 1, q >= 1 with p+q < n.

0, 0, 1, 2, 4, 5, 8, 10, 13, 16, 19, 21, 26, 29, 34, 39, 44, 48, 53, 58, 65, 71, 78, 83, 91, 97, 104, 111, 118, 124, 134, 141, 150, 158, 167, 176, 186, 194, 204, 213, 224, 232, 245, 254, 267, 278, 290, 301, 315, 328, 339, 351, 366, 376, 391, 404, 419, 432, 446

Offset: 1

Henry W. Gould, Sep 23 2013

nonn

Game played often with n = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See p. 20.

A227800 := proc(n)
    local s, p, q ;
    s := {} ;
    for p from 1 to iquo(n-1, 2) do
    for q from p to n-1-p do
            s := s union {p*q} ;
    end do:
    end do:
    nops(s) ;
end proc:
seq(A227800(n), n=1..120) ; # R. J. Mathar, Nov 24 2013

A227800[n_] := Module[{s, p, q}, s = {}; For[p = 1, p <= Quotient[n-1, 2], p++, For[q = p, q <= n-1-p, q++, s = s ~Union~ {p*q}]] ; Length[s]]; Table[A227800[n], {n, 1, 120}] (* Jean-François Alcover, Feb 27 2014, after R. J. Mathar *)

A003100 Decimal Gray code for n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 79, 78, 77

Offset: 0

N. J. A. Sloane, Henry W. Gould

This permutation of the nonnegative integers is not self-inverse, as previously claimed. The first exception is a(100) = 190, but a(190) = 109. - Franklin T. Adams-Watters, Mar 05 2010

a(n) = A118757(n) for n<=100, = a(100)=A118757(100)=190, but a(101)=191, A118757(101)=180. - Reinhard Zumkeller, May 01 2006

M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000
A. J. Cole, Cyclic progressive number systems, Math. Gaz., 50 (1966), 122-131.
A. J. Cole, Cyclic progressive number systems, Math. Gaz., 50 (1966), 122-131. [Annotated scanned copy]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
D. E. Knuth, Gray Codes (Vol. 4 of TAOCP)
Index entries for sequences that are permutations of the natural numbers

Inverse is A174025.

A003100 :=proc(n)
    local s,i:
    s:=[op(convert(n,base,10)),0]:
    add(piecewise(s[i+1] mod 2=0,s[i],9-s[i])*10^(i-1),i=1..nops(s)-1) :
end proc:
seq(A003100(j),j=0..100); # Pab Ter, Oct 14 2005

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 14 2005

Incorrect comment replaced by Franklin T. Adams-Watters, Mar 05 2010

A003099 a(n) = Sum_{k=0..n} binomial(n,k^2).

Original entry on oeis.org

1, 2, 3, 4, 6, 11, 22, 43, 79, 137, 231, 397, 728, 1444, 3018, 6386, 13278, 26725, 51852, 97243, 177671, 320286, 579371, 1071226, 2053626, 4098627, 8451288, 17742649, 37352435, 77926452, 159899767, 321468048, 632531039, 1219295320, 2308910353, 4314168202

Offset: 0

N. J. A. Sloane, Henry W. Gould

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Henry W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]
Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.

Cf. A010052, A206849.

Partial sums of A103198.

[(&+[Binomial(n, j^2): j in [0..n]]): n in [0..50]]; // G. C. Greubel, Oct 26 2022

Table[Sum[Binomial[n, k^2], {k, 0, Sqrt[n]}], {n, 0, 50}] (* T. D. Noe, Sep 10 2011 *)

a(n)=sum(k=0,sqrtint(n),binomial(n,k^2)) \\ Charles R Greathouse IV, Mar 26 2013

def A003099(n): return sum( binomial(n,k^2) for k in range(isqrt(n)+1))
[A003099(n) for n in range(50)] # G. C. Greubel, Oct 26 2022

a(n)*sqrt(n)/2^n is bounded: lim sup a(n)*sqrt(n)/2^n = 0.82... and lim inf a(n)*sqrt(n)/2^n = 0.58... - Benoit Cloitre, Nov 14 2003 [These constants are sqrt(2/Pi) * JacobiTheta3(0,exp(-4)) = 0.827112271364145742... and sqrt(2/Pi) * JacobiTheta2(0,exp(-4)) = 0.587247586271786487... - Vaclav Kotesovec, Jan 15 2023]
Binomial transform of the characteristic function of squares A010052. - Carl Najafi, Sep 09 2011
G.f.: (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^(k^2). - Ilya Gutkovskiy, Jan 22 2024

More terms from Carl Najafi, Sep 09 2011

A003101 a(n) = Sum_{k = 1..n} (n - k + 1)^k.

Original entry on oeis.org

0, 1, 3, 8, 22, 65, 209, 732, 2780, 11377, 49863, 232768, 1151914, 6018785, 33087205, 190780212, 1150653920, 7241710929, 47454745803, 323154696184, 2282779990494, 16700904488705, 126356632390297, 987303454928972, 7957133905608836, 66071772829247409

Offset: 0

N. J. A. Sloane, Henry W. Gould

For n > 0: a(n) = sum of row n of triangles A051129 and A247358. - Reinhard Zumkeller, Sep 14 2014
a(n-1) is the number of set partitions of [n] into two or more blocks such that all absolute differences between least elements of consecutive blocks are 1. a(3) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, May 22 2017
Min_{n >= 1} a(n+1)/a(n) = 8/3. This is the answer to the 4th problem proposed during the first day of the final round of the 16th Austrian Mathematical Olympiad in 1985 (see link IMO Compendium). - Bernard Schott, Jan 07 2019
If n rings of different internal diameter can fit close together on a tapering column, a(n) is the number of different arrangements of at least one ring. For example, if the rings increasing in size are 1, 2 and 3, then a(3) = 8 corresponding to the possible arrangements from the point on the column of smallest diameter (1XX), (X2X), (XX3), (12X), (32X), (1X3), (X23) and (123), where X denotes a space on the column. - Ian Duff, Jun 23 2025

			For n = 3 we get a(3) = 3^1 + 2^2 + 1^3 = 8. For n = 4 we get a(4) = 4^1 + 3^2 + 2^3 + 1^4 = 22.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 0..598
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
Mathematics Stack Exchange, Asymptotics of 1^n+2^(n-1)+3^(n-2)+...+(n-1)^2+n^1, 2011.
Daniel Ropp, Problem 2 - 16th Austrian Mathematical Olympiad (Final round), Crux Mathematicorum, page 7, Vol. 14, Jun. 88.
The IMO compendium, Problem 2, 16th Austrian Mathematical Olympiad, 1985.
Index to sequences related to Olympiads.

Cf. A026898, A051129, A062810, A247358, A287215.

First differences are in A047970.

a003101 n = sum $ zipWith (^) [0 ..] [n + 1, n .. 1]
-- Reinhard Zumkeller, Sep 14 2014

[n eq 0 select 0 else (&+[(n-j+1)^j: j in [1..n]]): n in [0..50]]; // G. C. Greubel, Oct 26 2022

A003101 := n->add((n-k+1)^k, k=1..n);
a:= n-> add((n-j+1)^j, j=1..n): seq(a(n), n=0..30); # Zerinvary Lajos, Jun 07 2008

Table[Sum[(n-k+1)^k,{k,n}],{n,0,25}] (* Harvey P. Dale, Aug 14 2011 *)

a(n)=sum(k=1,n,(n-k+1)^k) \\ Charles R Greathouse IV, Oct 31 2011

def A003101(n): return sum( (n-k+1)^k for k in range(1,n+1))
[A003101(n) for n in range(50)] # G. C. Greubel, Oct 26 2022

a(n) = A026898(n) - 1.
G.f.: G(0)/x-1/(1-x)/x where G(k) = 1 + x*(2*k*x-1)/((2*k*x+x-1) - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
G.f.: Sum_{k>=1} x^k/(1 - (k + 1)*x). - Ilya Gutkovskiy, Oct 09 2018
a(n) = n^1 + (n-1)^2 + (n-2)^3 + ... + 3^(n-2) + 2^(n-1) + 1^n. - Bernard Schott, Jan 07 2019
log(a(n)) ~ (1 - 1/LambertW(exp(1)*n)) * n * log(1 + n/LambertW(exp(1)*n)). - Vaclav Kotesovec, Jun 15 2021
a(n) ~ sqrt(2*Pi/(n+1 + w(n))) * w(n)^(n+2 - w(n)), where w(n) = (n+1)/LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics Stack Exchange link.

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User: Henry W. Gould

A227800 Number of different values the product p*q can have where p >= 1, q >= 1 with p+q < n.

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A003100 Decimal Gray code for n.

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A003099 a(n) = Sum_{k=0..n} binomial(n,k^2).

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A003101 a(n) = Sum_{k = 1..n} (n - k + 1)^k.

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