A212500 -id:A212500 - cp's OEIS frontend

A213828 Rectangular array: (row n) = b**c, where b(h) = 3h-2, c(h) = 3n-4+3*h, n>=1, h>=1, and ** = convolution.

2, 13, 5, 42, 28, 8, 98, 78, 43, 11, 190, 164, 114, 58, 14, 327, 295, 230, 150, 73, 17, 518, 480, 400, 296, 186, 88, 20, 772, 728, 633, 505, 362, 222, 103, 23, 1098, 1048, 938, 786, 610, 428, 258, 118, 26, 1505, 1449, 1324

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213829.
Antidiagonal sums: A213830.
Row 1, (1,4,7,10,...)**(2,5,8,11,...): (3*k^2 + k)/2.
Row 2, (1,4,7,10,...)**(5,8,11,14,...): (3*k^3 + 9*k^2 + 4*k)/2.
Row 3, (1,4,7,10,...)**(8,11,14,17,...): (3*k^3 + 18*k^2 + 7*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
2....13...42....98....190
5....28...78....164...295
8....43...114...230...400
11...58...150...296...505
14...73...186...362...610
17...88...222...428...715

Clark Kimberling, Antidiagonals n=1..60, flattened

Cf. A212500

b[n_]:=3n-2;c[n_]:=3n-1;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213828 *)
d=Table[t[n,n],{n,1,40}] (* A213829 *)
d/2 (* A005915 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213830 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*((6*n-4) - (3*n-8)*x - (3*n-5)*x^2) and g(x) = (1-x)^4.

A213836 Rectangular array: (row n) = b**c, where b(h) = 4*h-3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 2, 22, 13, 3, 50, 37, 19, 4, 95, 78, 52, 25, 5, 161, 140, 106, 67, 31, 6, 252, 227, 185, 134, 82, 37, 7, 372, 343, 293, 230, 162, 97, 43, 8, 525, 492, 434, 359, 275, 190, 112, 49, 9, 715, 678, 612, 525, 425, 320, 218, 127

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213837.
Antidiagonal sums: A071238.
Row 1, (1,5,9,13,...)**(1,2,3,4,...): A002412.
Row 2, (1,5,9,13,...)**(2,3,4,5,...): (4*k^3 + 15*k^2 - 7*k)/6.
Row 3, (1,5,9,13,...)**(3,4,5,6,...): (4*k^3 + 27*k^2 - 13*k)/6.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1...7....22...50....95
2...13...37...78....140
3...19...52...106...185
4...25...67...134...230
5...31...82...162...275
6...37...97...190...320

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A212500.

b[n_]:=4n-3;c[n_]:=n;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213836 *)
Table[t[n,n],{n,1,40}] (* A213837 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A071238 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(n + (2*n+1)*x - 3*(n-1)*x^2) and g(x) = (1-x)^4.

A213849 Rectangular array: (row n) = bc, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 8, 6, 4, 2, 14, 11, 9, 5, 3, 20, 17, 14, 10, 6, 3, 30, 26, 23, 17, 13, 7, 4, 40, 36, 32, 26, 20, 14, 8, 4, 55, 50, 46, 38, 32, 23, 17, 9, 5, 70, 65, 60, 52, 44, 35, 26, 18, 10, 5, 91, 85, 80, 70, 62, 50, 41, 29, 21, 11, 6

Offset: 1

Clark Kimberling, Jul 05 2012

Principal diagonal: A049778.
Antidiagonal sums: A213850.
Row 1, (1,1,2,2,3,3,...)**(1,1,2,2,3,3,...).
Row 2, (1,1,2,2,3,3,...)**(1,2,2,3,3,4,...).
Row 3, (1,1,2,2,3,3,...)**(2,2,3,3,4,4,...).
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1...2...5....8....14...20...30...40
1...3...6....11...17...26...36...50
2...4...9....14...23...32...46...60
2...5...10...17...26...38...52...70
3...6...13...20...32...44...62...80

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A212500.

b[n_]:=Floor[(n+1)/2];c[n_]:=Floor[(n+1)/2];
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}]  (* A213849 *)
d=Table[t[n,n],{n,1,50}] (* A049778 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
s1=Table[s[n],{n,1,50}] (* A213850 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(ceiling(n/2) + m(n)*x - floor(n/2)*x^2), where m(n) = (n+1 mod 2), and g(x) = (1+x)^2 *(1-x)^4.

A212594 a(n) is the difference between multiples of 11 with even and odd decimal digit sum in interval [0,10^n).

Original entry on oeis.org

1, 10, 19, 430, 841, 20602, 40363, 995710, 1951057, 48162410, 94373763, 2329795534, 4565217305, 112701782490, 220838347675, 5451852478622, 10682866609569, 263728727794378, 516774588979187, 12757653047779310, 24998531506579433, 617140623134480698

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 22 2012

Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (0,55,0,-330,0,462,0,-165,0,11).

Cf. A038754, A212500, A212592, A212593, A091042.

m:=23; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+10*x-36*x^2-120*x^3+126*x^4+252*x^5-84*x^6-120*x^7+9*x^8+10*x^9)/(1-55*x^2+330*x^4-462*x^6+165*x^8-11*x^10))); // Bruno Berselli, May 22 2012

LinearRecurrence[{0, 55, 0, -330, 0, 462, 0, -165, 0, 11}, {1, 10, 19, 430, 841, 20602, 40363, 995710, 1951057, 48162410}, 22] (* Bruno Berselli, May 22 2012 *)

For n>=11, a(n) = 55*a(n-2)-330*a(n-4)+462*a(n-6)-165*a(n-8)+11*a(n-10).

G.f.: x*(1+10*x-36*x^2-120*x^3+126*x^4+252*x^5-84*x^6-120*x^7+9*x^8+10*x^9)/(1-55*x^2+330*x^4-462*x^6+165*x^8-11*x^10). [Bruno Berselli, May 22 2012]

A212705 a(n) is the difference between numbers of nonnegative multiples of 2n+1 with even and odd digit sum in base 2n in interval [0, (2*n)^8).

Original entry on oeis.org

54, 3220, 38794, 237832, 995710, 3256540, 8954258, 21645200, 47366982, 95758500, 181475866, 325939096, 559444366, 923676652, 1474657570, 2286163232, 3453646934, 5098701492, 7374096042, 10469422120, 14617383838

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 24 2012

Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177v2 [math.NT], 2007
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A038754, A084990, A091042, A212500, A212592, A212593, A212594, A212668, A212669, A212670.

a(n) = 2/(2*n+1)*sum{i=1..n}tan^8(pi*i/(2*n+1)).
a(n) = 2/315*n*(1088*n^6+3808*n^5+3920*n^4+280*n^3-868*n^2+322n-45).
G.f.: 2*x*(27+1394*x+7273*x^2+7308*x^3+1373*x^4+34*x^5-x^6)/(1-x)^8. [Bruno Berselli, May 24 2012]

A212706 a(n) is the difference between numbers of nonnegative multiples of 2n+1 with even and odd digit sum in base 2n in interval [0, (2*n)^9).

Original entry on oeis.org

81, 5825, 73745, 461313, 1951057, 6418369, 17712657, 42921473, 94087249, 190446273, 361259537, 649305089, 1115101521, 1841932225, 2941740049, 4561961985, 6893373521, 10179012289, 14724250641, 20908086785, 29195724113, 40152508353, 54459292177, 72929296897

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 24 2012

Colin Barker, Table of n, a(n) for n = 1..1000
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A038754, A084990, A091042, A212500, A212592, A212592, A212592, A212668, A212669, A212670, A212705.

[1+n/315*(4352*n^6+15232*n^5+12992*n^4-5600*n^3- 5152*n^2+5488*n-2112): n in [1..25]]; // Vincenzo Librandi, Dec 02 2015

Table[1 + n/315 (4352 n^6 + 15232 n^5 + 12992 n^4 - 5600 n^3 - 5152 n^2 + 5488 n - 2112), {n, 30}] (* Vincenzo Librandi, Dec 02 2015 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{81,5825,73745,461313,1951057,6418369,17712657,42921473},30] (* Harvey P. Dale, Aug 05 2025 *)

Vec(x*(81+5177*x+29413*x^2+29917*x^3+4883*x^4+171*x^5-9*x^6-x^7)/(1-x)^8 + O(x^40)) \\ Colin Barker, Dec 01 2015

a(n) = 2/(2*n+1) * Sum_{i=1..n} tan^9(Pi*i/(2*n+1)) * sin(2*Pi*i/(2*n+1)).
a(n) = 1+n/315*(4352*n^6 + 15232*n^5 + 12992*n^4 - 5600*n^3 - 5152*n^2 + 5488*n - 2112).
G.f.: x*(81+5177*x+29413*x^2+29917*x^3+4883*x^4+171*x^5-9*x^6-x^7) / (1-x)^8. - Colin Barker, Dec 01 2015

Typo in data fixed by Colin Barker, Dec 01 2015

A214458 Let S_3(n) denote difference between multiples of 3 in interval [0,n) with even and odd binary digit sums. Then a(n)=(-1)^A000120(n)(S_3(n)-3S_3(floor(n/4))).

Original entry on oeis.org

0, -1, -1, 1, 1, -1, -1, 0, 0, 0, 1, -1, 1, -2, -2, 2, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 0, 1, -1, 1, -2, -2, 2, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 0, 1, -1, 1, -2, -2, 2, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, -1, 1, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jul 18 2012

In 1969, D. J. Newman (see the reference) proved L. Moser's conjecture that difference between numbers of multiples of 3 with even and odd binary digit sums in interval [0,x] is always positive. This fact is known as Moser-Newman phenomenon.

Theorem: The sequence is periodic with period of length 24.

J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107-115.
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
Vladimir Shevelev, Two algorithms for evaluation of the Newman digit sum, and a new proof of Coquet's theorem, arXiv:0709.0885 [math.NT], 2007-2012.

Cf. A091042, A212500.

Recursion for evaluation S_3(n): S_3(n)=3*S_3(floor(n/4))+(-1)^A000120(n)*a(n). As a corollary, we have |S_3(n)-3*S_3(n/4)|<=2.

A212822 Triangle of coefficients of polynomials concerning Newman-like phenomenon of multiples of b+1 in even base b in interval [0,b^n) (see comment).

Original entry on oeis.org

1, 2, -1, 1, 3, -1, 2, 6, -8, 3, 2, 10, 10, -10, 3, 4, 20, 10, -50, 46, -15, 17, 119, 245, 35, -217, 161, -45, 34, 238, 406, -350, -644, 1372, -1056, 315, 62, 558, 1722, 1638, -1092, -1008, 1828, -1188, 315, 124, 1116, 3138, 1134, -5838, 1134, 9452, -14724, 10134, -2835

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, May 28 2012

In 1969, D. J. Newman (see the reference) proved that difference between numbers of multiples of 3 with even and odd binary digit sums in interval [0,x] is always positive. This fact now is known as Newman phenomenon.
Consider difference between numbers of multiples of b+1 with even and odd digit sums in even base b in interval [0, b^n). It is a polynomial in b P_n(b) of degree n-1 and multiple of b, if n is even, and n-2, if n is odd, such that all polynomials Q_n(b):=A156769(n/2)*P_n(b)/b, if n is even, and Q_n(b):=A156769((n-1)/2)*P_n(b), if n is odd, presumably have integer coefficients and are of degree n-2. The sequence is triangle of coefficients of polynomials Q_n(b).
The r-th row contains r-1 entries.
Since, evidently, P_n(1)=1, then the row sums form sequence A156769 repeated.

			Triangle begins (r is the number of row or the number of polynomial; coefficients of b^k, k=r-2-i, i=0,1,..., r-2)
r/i.|..0......1......2.....3.....4......5......6.....7
======================================================
.2..|..1
.3..|..2.....-1
.4..|..1......3.....-1
.5..|..2......6.....-8.....3
.6..|..2.....10.....10...-10.....3
.7..|..4.....20.....10...-50....46....-15
.8..|.17....119....245....35..-217....161....-45
.9..|.34....238....406..-350..-644...1372..-1056....315
For example, if r=4, the polynomial
P_4(b)=b*(b^2+3*b-1)/A156769(4/2)=b/3*(b^2+3*b-1) (b==0 mod 2)
gives difference between multiples of b+1 with even and odd digit sums in  base b in interval [0, b^4). Note also that P_2(b)=b. Therefore, setting in the formula n=r=3, again for P_4(b) we have P_4(b)=b*C(b+1,2)-C(b,3)=b/3*(b^2+3*b-1).

D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177v2 [math.NT], 2007.

Cf. A156769, A038754, A084990, A091042, A212500, A212592, A212593, A212594, A212668, A212669, A212670, A212705, A212706.

A156769[n_] := Denominator[(2^(2*n-2)/Factorial[2*n-1])]; poly[1, b_] := 1; poly[2, b_] := b; poly[n_, b_] :=  poly[n, b] = If[OddQ[n], (-1)^((n - 1)/2) (FunctionExpand[Binomial[b - 1, n - 1]] - Sum[(-1)^(k/2) FunctionExpand[Binomial[b + 1, n - k - 1]] poly[k + 1, b], {k, 0, n - 2, 2}]), (-1)^((n - 2)/2) (FunctionExpand[Binomial[b, n - 1]] - Sum[(-1)^((k - 1)/2) FunctionExpand[Binomial[b + 1, n - k - 1]] poly[k + 1, b], {k, 1, n - 2, 2}])]; Table[If[EvenQ[z], Most[Reverse[CoefficientList[poly[z, b] A156769[z/2], b]]], Reverse[CoefficientList[poly[z, b] A156769[(z - 1)/2], b]]], {z, 2, 12}]

If n>=2 is even, then P_(n+1)(b) = (-1)^((n-2)/2)*(C(b+1,n)-C(b-1,n))-sum{i=2,4,...,n-2}(-1)^((n+i)/2)*C(b+1, n-i)*P_(i+1)(b), where P_n(b)=b*Q_n(b)/A156769(n/2);
if n>=3 is odd, then P_(n+1)(b) = (-1)^((n-1)/2)*(C(b,n)-b*C(b+1,n-1))+sum{i=3,5,...,n-2}(-1)^((n+i)/2)*C(b+1, n-i)*P_(i+1)(b), where
P_n(b) = Q_n(b)/A156769((n-1)/2).
P_n(b) = 2/(b+1)*Sum_{j=1..b/2}(tan(j*Pi/(b+1)))^n, if n is even, and
P_n(b) = 2/(b+1)*Sum_{j=1..b/2}(tan(j*Pi/(b+1)))^n*sin(j*Pi/(b+1)), if n is odd.

cp's OEIS Frontend

A213828 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

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A213836 Rectangular array: (row n) = b**c, where b(h) = 4*h-3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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A213849 Rectangular array: (row n) = b**c, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and ** = convolution.

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A212594 a(n) is the difference between multiples of 11 with even and odd decimal digit sum in interval [0,10^n).

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A212705 a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, (2*n)^8).

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A212706 a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, (2*n)^9).

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A214458 Let S_3(n) denote difference between multiples of 3 in interval [0,n) with even and odd binary digit sums. Then a(n)=(-1)^A000120(n)*(S_3(n)-3*S_3(floor(n/4))).

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A212822 Triangle of coefficients of polynomials concerning Newman-like phenomenon of multiples of b+1 in even base b in interval [0,b^n) (see comment).

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A213828 Rectangular array: (row n) = b**c, where b(h) = 3h-2, c(h) = 3n-4+3*h, n>=1, h>=1, and ** = convolution.

A213849 Rectangular array: (row n) = bc, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and = convolution.

A212705 a(n) is the difference between numbers of nonnegative multiples of 2n+1 with even and odd digit sum in base 2n in interval [0, (2*n)^8).

A212706 a(n) is the difference between numbers of nonnegative multiples of 2n+1 with even and odd digit sum in base 2n in interval [0, (2*n)^9).

A214458 Let S_3(n) denote difference between multiples of 3 in interval [0,n) with even and odd binary digit sums. Then a(n)=(-1)^A000120(n)(S_3(n)-3S_3(floor(n/4))).