A084990 -id:A084990 - cp's OEIS frontend

A212668 a(n) = (16/3)(n+1)n(n-1) + 8n^2 + 1.

9, 65, 201, 449, 841, 1409, 2185, 3201, 4489, 6081, 8009, 10305, 13001, 16129, 19721, 23809, 28425, 33601, 39369, 45761, 52809, 60545, 69001, 78209, 88201, 99009, 110665, 123201, 136649, 151041, 166409, 182785, 200201, 218689, 238281, 259009, 280905, 304001

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

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, 32*n^5).

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 (4,-6,4,-1).

Cf. A038754, A084990, A091042, A212500, A212592.

[(16/3)*(n+1)*n*(n-1)+8*n^2+1: n in [1..40]]; // Vincenzo Librandi, Dec 01 2015

LinearRecurrence[{4, -6, 4, -1}, {9, 65, 201, 449}, 40] (* Vincenzo Librandi, Dec 01 2015 *)
CoefficientList[Series[x (9+29x-5x^2-x^3)/(1-x)^4,{x,0,40}],x] (* Harvey P. Dale, Mar 29 2023 *)

a(n)=16*(n+1)*n*(n-1)/3+8*n^2+1 \\ Charles R Greathouse IV, Oct 07 2015

Vec(x*(9+29*x-5*x^2-x^3)/(1-x)^4 + O(x^100)) \\ Colin Barker, Nov 30 2015

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^5(Pi*i/(2*n+1)) * sin(2*Pi*i/(2*n+1)).

G.f.: x*(9+29*x-5*x^2-x^3) / (1-x)^4. - Colin Barker, Nov 30 2015

A064043 Number of length 3 walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.

Original entry on oeis.org

0, 3, 18, 51, 108, 195, 318, 483, 696, 963, 1290, 1683, 2148, 2691, 3318, 4035, 4848, 5763, 6786, 7923, 9180, 10563, 12078, 13731, 15528, 17475, 19578, 21843, 24276, 26883, 29670, 32643, 35808, 39171, 42738, 46515, 50508, 54723, 59166, 63843, 68760, 73923, 79338

Offset: 0

Henry Bottomley, Aug 23 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Number of walks length 0, 1 and 2 are A000012, A001477 and A002378.

Cf. A084990.

seq(sum(3*n+n^2-1, k=1..n), n=0..39); # Zerinvary Lajos, Jan 28 2008

Table[n*(n^2 + 3n -1), {n,0,50}] (* G. C. Greubel, Jul 20 2017 *)

a(n) = { n*(n^2 + 3*n - 1) } \\ Harry J. Smith, Sep 06 2009

a(n) = n*(n^2 + 3*n - 1) = n*A014209(n) = A064044(n, 3).
a(n) = a(n-1) + 3*A002378(n-1) + 6*A001477(n-1) + 3*A000012(n-1).
G.f.: 3*x*(1+2*x-x^2)/(1-x)^4. - Colin Barker, Apr 19 2012
E.g.f.: (x^3 + 6*x^2 + 3*x)*exp(x). - G. C. Greubel, Jul 20 2017
a(n) = A084990(n)/3. - Alois P. Heinz, Jul 21 2017

A212669 a(n) = 2/15 * (32n^5 + 80n^4 + 40n^3 - 20n^2 + 3*n).

Original entry on oeis.org

18, 340, 2022, 7400, 20602, 48060, 99022, 186064, 325602, 538404, 850102, 1291704, 1900106, 2718604, 3797406, 5194144, 6974386, 9212148, 11990406, 15401608, 19548186, 24543068, 30510190, 37585008, 45915010, 55660228, 66993750, 80102232, 95186410, 112461612

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

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, 64*n^6).

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 (6,-15,20,-15,6,-1).

Cf. A038754, A084990, A091042, A212500, A212592, A212592, A212592, A212668.

Vec(2*x*(9+116*x+126*x^2+4*x^3+x^4)/(1-x)^6 + O(x^50)) \\ Colin Barker, Dec 01 2015

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^6(Pi*i/(2*n+1)).

G.f.: 2*x*(9+116*x+126*x^2+4*x^3+x^4) / (1-x)^6. - Colin Barker, Dec 01 2015

A212670 a(n) = 1/15(128n^5 + 320n^4 + 80n^3 - 200n^2 + 92n - 15).

Original entry on oeis.org

27, 615, 3843, 14351, 40363, 94711, 195859, 368927, 646715, 1070727, 1692195, 2573103, 3787211, 5421079, 7575091, 10364479, 13920347, 18390695, 23941443, 30757455, 39043563, 49025591, 60951379, 75091807, 91741819, 111221447, 133876835, 160081263, 190236171

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

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, 128*n^7).

Colin Barker, Table of n, a(n) for n = 1..1000
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 (6,-15,20,-15,6,-1).

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

Table[(1/15) (8 n^2 - 4 n + 1) (16 n^3 + 48 n^2 + 32 n - 15), {n, 29}] (* Bruno Berselli, May 24 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{27,615,3843,14351,40363,94711},30] (* Harvey P. Dale, Apr 30 2018 *)

Vec(x*(27+453*x+558*x^2-22*x^3+7*x^4+x^5)/(1-x)^6 + O(x^50)) \\ Colin Barker, Dec 01 2015

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^7(Pi*i/(2*n+1))*sin(2*Pi*i/(2*n+1)).

G.f.: x*(27+453*x+558*x^2-22*x^3+7*x^4+x^5)/(1-x)^6. [Bruno Berselli, May 24 2012]

A014205 (1/12)(n+5)(n+1)*n^2.

Original entry on oeis.org

0, 1, 7, 24, 60, 125, 231, 392, 624, 945, 1375, 1936, 2652, 3549, 4655, 6000, 7616, 9537, 11799, 14440, 17500, 21021, 25047, 29624, 34800, 40625, 47151, 54432, 62524, 71485, 81375, 92256, 104192, 117249, 131495, 147000, 163836, 182077, 201799, 223080, 246000

Offset: 0

N. J. A. Sloane

Partial sums of A084990. - Arkadiusz Wesolowski, Jan 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A084990.

[(1/12)*(n+5)*(n+1)*n^2: n in [0..50]]; // Vincenzo Librandi, Aug 11 2014

seq(2*binomial(n+3, 4)-binomial(n+1, 2), n=0..32); # Zerinvary Lajos, May 02 2007

Table[((n+5)(n+1)n^2)/12,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,7,24,60},50] (* Harvey P. Dale, Aug 10 2014 *)
CoefficientList[Series[x (x^2 - 2 x - 1)/(x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 11 2014 *)

a(n)=n^2*(n+1)*(n+5)/12 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = 2*C(n+3,4) - C(n+1,2). - Zerinvary Lajos, May 02 2007
G.f.: x*(x^2-2*x-1)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(0)=0, a(1)=1, a(2)=7, a(3)=24, a(4)=60, a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)- 5*a(n-4)+a(n-5). - Harvey P. Dale, Aug 10 2014

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

A128226 Triangle, A004736 * A127899 (unsigned).

Original entry on oeis.org

1, 4, 2, 7, 7, 3, 10, 12, 10, 4, 13, 17, 17, 13, 5, 16, 22, 24, 22, 16, 6, 19, 27, 31, 31, 27, 19, 7, 22, 32, 38, 40, 38, 32, 22, 8, 25, 37, 45, 49, 49, 45, 37, 25, 9, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10

Offset: 1

Gary W. Adamson, Feb 19 2007

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

A128225 = A127899(unsigned) * A004736.

			First few row of the triangle are:
   1;
   4,  2;
   7,  7,  3;
  10, 12, 10,  4;
  13, 17, 17, 13,  5;
  16, 22, 24, 22, 16,  6;
  19, 27, 31, 31, 27, 19,  7;
  ...

Cf. A004736, A084990, A127899, A128225.

(* a127899U computes the unsigned version of A127899 *)
a127899U[n_, k_] := If[n==k||n-1==k, n, 0]/;(1<=k<=n)
a004736[n_, k_] := n-k+1/;(1<=k<=n+1)
a128225[n_, k_] := a127899U[n, n](a004736[n, k] + a004736[n-1, k])/;(1<=k<=n)
a128225[r_] := Table[a128225[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128225[7]] (* triangle *)
Flatten[a128225[10]] (* data *) (* Hartmut F. W. Hoft, Mar 13 2017 *)

A004736 * A127899 (unsigned). By columns, (3k+1), (5k+2), (7k+3), ...; k=0,1,2...

A131782 Triangle read by rows: (A004736 * A000012) + (A000012 * A004736) - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 5, 1, 11, 5, 1, 19, 11, 5, 1, 29, 19, 11, 5, 1, 41, 29, 19, 11, 5, 1, 55, 41, 29, 19, 11, 5, 1, 71, 55, 41, 29, 19, 11, 5, 1, 89, 71, 55, 41, 29, 19, 11, 5, 1, 109, 89, 71, 55, 41, 29, 19, 11, 5, 1, 131, 109, 89, 71, 55, 41, 29, 19, 11, 5, 1, 155, 131, 109, 89, 71, 55, 41, 29, 19, 11, 5, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

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

			First few rows of the triangle:
   1;
   5,  1;
  11,  5,  1;
  19, 11,  5,  1;
  29, 19, 11,  5,  1;
  41, 29, 19, 11,  5,  1;
  55, 41, 29, 19, 11,  5,  1;
  ...

Cf. A004736, A028387, A084990.

n-th row = n descending terms of A028387: (1, 5, 11, 19, 29, 41, 55, ...). By columns, each column = A028387: (1, 5, 11, 19, 29, ...).

Sign in definition and a(41) corrected; more terms from Georg Fischer, Jun 05 2023

A206492 Sums of rows of the sequence of triangles with nonnegative integers and row widths defined by A004738.

Original entry on oeis.org

0, 3, 3, 9, 21, 19, 11, 25, 45, 74, 66, 49, 26, 55, 90, 134, 190, 170, 138, 97, 50, 103, 162, 230, 310, 405, 365, 310, 243, 167, 85, 173, 267, 370, 485, 615, 763, 693, 605, 502, 387, 263, 133, 269, 411, 562, 725, 903, 1099, 1316, 1204, 1071, 920, 754, 576, 389

Offset: 1

Alex Ratushnyak, Jun 28 2012

Row widths: A004738(n): 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5...
Pits: A051925(n+1): 0, 3, 11, 26, 50, 85, 133, 196, 276, 375, 495, 638...
Peak tops: A007290(n+3): 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, 572...
Peak bases: A084990(n+1): 1, 6, 17, 36, 65, 106, 161, 232, 321, 430, 561...

			The sequence of triangles begins:
0
1 2
3
4 5
6 7 8
9 10
11
12 13
14 15 16
17 18 19 20
21 22 23
24 25
26
27 28
29 30 31
32 33 34 35
36 37 38 39 40
41 42 43 44
45 46 47
48 49
50
51 52

Cf. A004738, A051925, A007290, A084990.

Cf. A027480: sums of rows of a triangle with increasing row widths: 0; 1,2; 3,4,5; 6,7,8,9; ...

curSign=-1
curLength=sum=0
rowLength=topLength=1
for n in range(1232):
    sum += n
    curLength += 1
    if curLength==rowLength:
        print(sum, end=',')
        curLength = sum = 0
        if rowLength==1 or rowLength==topLength:
            curSign = -curSign
        topLength += (rowLength==1)
        rowLength += curSign

cp's OEIS Frontend

A212668 a(n) = (16/3)*(n+1)*n*(n-1) + 8*n^2 + 1.

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A064043 Number of length 3 walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.

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Maple

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A212669 a(n) = 2/15 * (32*n^5 + 80*n^4 + 40*n^3 - 20*n^2 + 3*n).

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A212670 a(n) = 1/15*(128*n^5 + 320*n^4 + 80*n^3 - 200*n^2 + 92*n - 15).

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A014205 (1/12)*(n+5)*(n+1)*n^2.

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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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A128226 Triangle, A004736 * A127899 (unsigned).

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A212668 a(n) = (16/3)(n+1)n(n-1) + 8n^2 + 1.

A212669 a(n) = 2/15 * (32n^5 + 80n^4 + 40n^3 - 20n^2 + 3*n).

A212670 a(n) = 1/15(128n^5 + 320n^4 + 80n^3 - 200n^2 + 92n - 15).

A014205 (1/12)(n+5)(n+1)*n^2.

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).