A084990 -id:A084990 - cp's OEIS frontend

A271668 Triangle read by rows. The first column is A000217(n+1). From the second row we apply - A002262(n) for the following terms of the row.

1, 3, 3, 6, 6, 5, 10, 10, 9, 7, 15, 15, 14, 12, 9, 21, 21, 20, 18, 15, 11, 28, 28, 27, 25, 22, 18, 13, 36, 36, 35, 33, 30, 26, 21, 15, 45, 45, 44, 42, 39, 35, 30, 24, 17, 55, 55, 54, 52, 49, 45, 40, 34, 27, 19, 66, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21

Offset: 0

Paul Curtz, Apr 12 2016

Row sums: A084990(n+1).
A158405(n) = A002262(n) + A002260(n). See the formula.
(Without its first column, A094728 is A120070, which could be built from positive A005563 and -A158894.)

			a(0) = 1, a(1) = 3, a(2) =3-0 = 3,  a(3) = 6, a(4) =6-0= 6, a(5) =6-1= 5, ... .
Triangle:
1,
3,   3,
6,   6,  5,
10, 10,  9,  7,
15, 15, 14, 12,  9,
21, 21, 20, 18, 15, 11,
28, 28, 27, 25, 22, 18, 13,
36, 36, 35, 33, 30, 26, 21, 15,
etc.

Cf. A000096, A000217, A002260, A002262, A005408, A005563, A049780, A084990, A094728, A120070, A158405, A158894.

Table[(n^2 - n)/2 - Prepend[Accumulate@ Range[0, n - 3], 0], {n, 12}] // Flatten (* Michael De Vlieger, Apr 12 2016 *)

a(n) = A094728(n+1) - A049780(n).

A131783 A000012 * (A004736 + A002260 - I).

Original entry on oeis.org

1, 4, 2, 8, 6, 3, 13, 11, 8, 4, 19, 17, 14, 10, 5, 26, 24, 21, 17, 12, 6, 34, 32, 29, 25, 20, 14, 7, 43, 41, 38, 34, 29, 23, 16, 8, 53, 51, 48, 44, 39, 33, 26, 18, 9, 64, 62, 59, 55, 50, 44, 37, 29, 20, 10

Offset: 1

Gary W. Adamson, Jul 14 2007

Left column = A034856: (1, 4, 8, 13, 19, ...).

Row sums = A084990: (1, 6, 17, 36, 65, 106, ...).

			First few rows of the triangle:
   1;
   4,  2;
   8,  6,  3;
  13, 11,  8,  4;
  19, 17, 14, 10,  5;
  26, 24, 21, 17, 12,  6;
  34, 32, 29, 25, 20, 14,  7;
  ...

Cf. A004736, A002260, A034856, A084990.

A000012 * (A004736 + A002260 - I), I = Identity matrix; A004736 = (1; 2,1; 3,2,1; ...); A002260 = (1; 1,2; 1,2,3; ...).

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.

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A271668 Triangle read by rows. The first column is A000217(n+1). From the second row we apply - A002262(n) for the following terms of the row.

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A131783 A000012 * (A004736 + A002260 - I).

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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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