A058331 -id:A058331 - cp's OEIS frontend

A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.

1, 1, 3, 1, 5, 3, 1, 7, 9, 3, 1, 9, 19, 9, 5, 1, 11, 33, 27, 13, 5, 1, 13, 51, 65, 33, 21, 5, 1, 15, 73, 131, 89, 57, 21, 5, 1, 17, 99, 233, 221, 137, 81, 21, 5, 1, 19, 129, 379, 485, 333, 233, 81, 25, 7, 1, 21, 163, 577, 953, 797, 573, 297, 93, 29, 7, 1, 23, 201, 835, 1713, 1793

Offset: 1

R. J. Mathar, Oct 29 2006, Oct 31 2006

Number of solutions to sum_(i=1,..,d) x[i]^2 <= n, x[i] in Z.

			T(2,2)=9 counts 1 pair (0,0) with sum 0, 4 pairs (-1,0),(1,0),(0,-1),(0,1) with sum 1 and 4 pairs (-1,-1),(-1,1),(1,1),(1,-1) with sum 2.
Array T(d,n) with rows d=1,2,3... and columns n=0,1,2,3.. reads
  1  3   3    3    5     5     5     5      5      7      7
  1  5   9    9   13    21    21    21     25     29     37
  1  7  19   27   33    57    81    81     93    123    147
  1  9  33   65   89   137   233   297    321    425    569
  1 11  51  131  221   333   573   893   1093   1343   1903
  1 13  73  233  485   797  1341  2301   3321   4197   5757
  1 15  99  379  953  1793  3081  5449   8893  12435  16859
  1 17 129  577 1713  3729  6865 12369  21697  33809  47921
  1 19 163  835 2869  7189 14581 27253  49861  84663 129303
  1 21 201 1161 4541 12965 29285 58085 110105 198765 327829

Index entries for sequences related to sums of squares

Rows d=1..10 give A001650, A057655, A117609, A046895, A175360, A175361, A341396, A341397, A341398, A341399.

Cf. A005408 (column 1), A058331 (column 2), A161712 (column 3), A055426 (column 4), A055427 (column 9)

T := proc(d,n) local i,cnts ; cnts := 0 ; for i from -trunc(sqrt(n)) to trunc(sqrt(n)) do if n-i^2 >= 0 then if d > 1 then cnts := cnts+T(d-1,n-i^2) ; else cnts := cnts+1 ; fi ; fi ; od ; RETURN(cnts) ; end: for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d,",T(d,n)) ; od ; od;

t[d_, n_] := t[d, n] = t[d, n-1] + SquaresR[d, n]; t[d_, 0] = 1; Table[t[d-n, n], {d, 1, 12}, {n, 0, d-1}] // Flatten (* Jean-François Alcover, Jun 13 2013 *)

Recurrence along rows: T(d,n)=T(d,n-1)+A122141(d,n) for n>=1; T(d,n)=sum_{i=0..n} A122141(d,i). Recurrence along columns: cf. A123937.

A161706 a(n) = (-11n^5 + 145n^4 - 635n^3 + 1115n^2 - 494*n + 120)/120.

Original entry on oeis.org

1, 2, 4, 5, 10, 20, 21, -27, -201, -626, -1486, -3035, -5608, -9632, -15637, -24267, -36291, -52614, -74288, -102523, -138698, -184372, -241295, -311419, -396909, -500154, -623778, -770651, -943900, -1146920, -1383385, -1657259, -1972807

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 20:

a(n) = A027750(A006218(19) + k + 1), 0 <= k < A000005(20).

			Differences of divisors of 20 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     4     5    10    20
     1     2     1     5    10
        1    -1     4     5
          -2     5     1
              7    -4
               -11

G. C. Greubel, Table of n, a(n) for n = 0..1000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261.

Cf. A005018, A161700, A161856. - Reinhard Zumkeller, Jun 21 2009

[(-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161706:=n->(-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120: seq(A161706(n), n=0..50); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1 - 4*x + 7*x^2 - 9*x^3 + 15*x^4 - 21*x^5)/(1 - x)^6, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

a(n)=(-11*n^5+145*n^4-635*n^3+1115*n^2-494*n+120)/120 \\ Charles R Greathouse IV, Sep 24 2015

def A161706(n): return (n*(n*(n*(n*(145 - 11*n) - 635) + 1115) - 494) + 120)//15>>3 # Chai Wah Wu, Oct 23 2023

a(n) = C(n,0) + C(n,1) + C(n,2) - 2*C(n,3) + 7*C(n,4) - 11*C(n,5).

G.f.: (1-4*x+7*x^2-9*x^3+15*x^4-21*x^5)/(1-x)^6. - Colin Barker, Apr 25 2012

A161710 a(n) = (-6n^7 + 154n^6 - 1533n^5 + 7525n^4 - 18879n^3 + 22561n^2 - 7302*n + 2520)/2520.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 24, 39, -2, -295, -1308, -3980, -9996, -22150, -44808, -84483, -150534, -256001, -418588, -661806, -1016288, -1521288, -2226376, -3193341, -4498314, -6234123, -8512892, -11468896, -15261684, -20079482, -26142888

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 8} = divisors of 24:

a(n) = A027750(A006218(23) + k + 1), 0 <= k < A000005(24).

			Differences of divisors of 24 to compute the coefficients of their interpolating polynomial, see formula:
1 ... 2 ... 3 ... 4 ... 6 ... 8 .. 12 .. 24
.. 1 ... 1 ... 1 ... 2 ... 2 ... 4 .. 12
..... 0 ... 0 ... 1 ... 0 ... 2 ... 8
........ 0 ... 1 .. -1 ... 2 ... 6
........... 1 .. -2 ... 3 ... 4
............. -3 ... 5 ... 1
................. 8 .. -4
.................. -12.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A080856, A161711, A161712, A161713, A161715, A006261, A018253, A161700, A161856.

[(-6*n^7 + 154*n^6 - 1533*n^5 + 7525*n^4 - 18879*n^ 3 + 22561*n^2 - 7302*n + 2520)/2520: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[(-6n^7+154n^6-1533n^5+7525n^4-18879n^3+22561n^2-7302n+2520)/2520,{n,0,40}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,2,3,4,6,8,12,24},40] (* Harvey P. Dale, Jul 15 2012 *)

a(n)=(-6*n^7+154*n^6-1533*n^5+7525*n^4-18879*n^3+22561*n^2-7302*n+2520)/2520 \\ Charles R Greathouse IV, Sep 24 2015

A161710_list, m = [1], [-12, 80, -223, 333, -281, 127, -23, 1]
for _ in range(1,10**2):
    for i in range(7):
        m[i+1]+= m[i]
    A161710_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = C(n,0) + C(n,1) + C(n,4) - 3*C(n,5) + 8*C(n,6) - 12*C(n,7).
G.f.: (1-6*x+15*x^2-20*x^3+16*x^4-12*x^5+18*x^6-24*x^7)/(1-x)^8. - Bruno Berselli, Jul 17 2011
a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=6, a(5)=8, a(6)=12, a(7)=24, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+ 8*a(n-7)- a(n-8). - Harvey P. Dale, Jul 15 2012

A161713 a(n) = (-n^5 + 15n^4 - 65n^3 + 125n^2 - 34n + 40)/40.

Original entry on oeis.org

1, 2, 4, 7, 14, 28, 49, 71, 79, 46, -70, -329, -812, -1624, -2897, -4793, -7507, -11270, -16352, -23065, -31766, -42860, -56803, -74105, -95333, -121114, -152138, -189161, -233008, -284576, -344837, -414841, -495719, -588686, -695044

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 28:

a(n) = A027750(A006218(27) + k + 1), 0 <= k < A000005(28).

			Differences of divisors of 28 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     4     7    14    28
     1     2     3     7    14
        1     1     4     7
           0     3     3
              3     0
                -3

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A018254, A058331, A080856, A086514, A161700, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161715, A161856.

[(-n^5 + 15*n^4 - 65*n^3 + 125*n^2 - 34*n + 40)/40: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[(-n^5+15n^4-65n^3+125n^2-34n)/40+1,{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,4,7,14,28},40] (* Harvey P. Dale, Jan 14 2014 *)

a(n)=(-n^5+15*n^4-65*n^3+125*n^2-34*n+40)/40 \\ Charles R Greathouse IV, Sep 24 2015

def A161713(n): return n*(n*(n*(n*(15 - n) - 65) + 125) - 34)//40 + 1 # Chai Wah Wu, Dec 16 2021

a(n) = C(n,0) + C(n,1) + C(n,2) + 3*C(n,4) - 3*C(n,5).
G.f.: -(-1+4*x-7*x^2+7*x^3-7*x^4+7*x^5)/(-1+x)^6. - R. J. Mathar, Jun 18 2009
a(0)=1, a(1)=2, a(2)=4, a(3)=7, a(4)=14, a(5)=28, a(n)=6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jan 14 2014

A161715 a(n) = (50n^7 - 1197n^6 + 11333n^5 - 53655n^4 + 132125n^3 - 156828n^2 + 73212*n + 5040)/5040.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 15, 30, 171, 886, 3359, 10143, 26072, 59502, 123931, 240048, 438261, 761754, 1270123, 2043641, 3188202, 4840994, 7176951, 10416034, 14831391, 20758446, 28604967, 38862163, 52116860, 69064806, 90525155, 117456180

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 8} = divisors of 30:

a(n) = A027750(A006218(29) + k + 1), 0 <= k < A000005(30).

			Differences of divisors of 30 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     3     5     6    10    15    30
     1     1     2     1     4     5    15
        0     1    -1     3     1    10
           1    -2     4    -2     9
             -3     6    -6    11
                 9   -12    17
                  -21    29
                      50

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713.

Cf. A018255, A161700, A161856. - Reinhard Zumkeller, Jun 21 2009

[(50*n^7 - 1197*n^6 + 11333*n^5 - 53655*n^4 + 132125*n^3 - 156828*n^2 + 73212*n + 5040)/5040: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

CoefficientList[Series[(1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

x='x+O('x^50); Vec((1 -6*x +15*x^2 -19*x^3 +8*x^4 +18*x^5 -51*x^6 +84*x^7) /(-1+x)^8) \\ G. C. Greubel, Jul 16 2017

A161710_list, m = [1], [50, -321, 864, -1249, 1024, -452, 85, 1]
for _ in range(1,10**2):
    for i in range(7):
        m[i+1]+= m[i]
    A161710_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = C(n,0) + C(n,1) + C(n,3) - 3*C(n,4) + 9*C(n,5) - 21*C(n,6) + 50*C(n,7).
G.f.: (1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8. - R. J. Mathar, Jun 18 2009
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Apr 26 2021

A161701 a(n) = (n^5 - 5n^4 + 5n^3 + 5n^2 + 114n + 120)/120.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 28, 64, 135, 262, 473, 804, 1300, 2016, 3018, 4384, 6205, 8586, 11647, 15524, 20370, 26356, 33672, 42528, 53155, 65806, 80757, 98308, 118784, 142536, 169942, 201408, 237369, 278290, 324667, 377028, 435934, 501980, 575796, 658048

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 12:

a(n) = A027750(A006218(11) + k + 1), 0 <= k < A000005(12).

			Differences of divisors of 12 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     3     4     6    12
     1     1     1     2     6
        0     0     1     4
           0     1     3
              1     2
                 1

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715.

[(n^5 - 5*n^4 + 5*n^3 + 5*n^2 + 114*n + 120)/120: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161701:=n->(n^5 - 5*n^4 + 5*n^3 + 5*n^2 + 114*n + 120)/120: seq(A161701(n), n=0..60); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1-4*x+6*x^2-4*x^3+2*x^4)/(1-x)^6, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

a(n)=(n^5-5*n^4+5*n^3+5*n^2+114*n+120)/120 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = C(n,0) + C(n,1) + C(n,4) + C(n,5).

G.f.: (1-4*x+6*x^2-4*x^3+2*x^4)/(1-x)^6. - Colin Barker, Aug 20 2012

A161704 a(n) = (3n^5 - 35n^4 + 145n^3 - 235n^2 + 152*n + 30)/30.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 59, 190, 513, 1186, 2435, 4566, 7977, 13170, 20763, 31502, 46273, 66114, 92227, 125990, 168969, 222930, 289851, 371934, 471617, 591586, 734787, 904438, 1104041, 1337394, 1608603, 1922094, 2282625, 2695298, 3165571, 3699270

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 18:

a(n) = A027750(A006218(17) + k + 1), 0 <= k < A000005(18).

			Differences of divisors of 18 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     3     6     9    18
     1     1     3     3     9
        0     2     0     6
           2    -2     6
             -4     8
                12

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A018251, A058331, A080856, A086514, A161701, A161702, A161703, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A161856.

[(3*n^5 - 35*n^4 + 145*n^3 - 235*n^2 + 152*n + 30)/30: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161704:=n->(3*n^5 - 35*n^4 + 145*n^3 - 235*n^2 + 152*n + 30)/30: seq(A161704(n), n=0..50); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1 - 4*x + 6*x^2 - 2*x^3 - 7*x^4 + 18*x^5)/(x - 1)^6, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

a(n)=n*(3*n^4-35*n^3+145*n^2-235*n+152)/30+1

a(n) = C(n,0) + C(n,1) + 2*C(n,3) - 4*C(n,4) + 12*C(n,5).

G.f.: ( 1-4*x+6*x^2-2*x^3-7*x^4+18*x^5 ) / (x-1)^6. - R. J. Mathar, Jul 12 2016

A161708 a(n) = -n^3 + 7n^2 - 5n + 1.

Original entry on oeis.org

1, 2, 11, 22, 29, 26, 7, -34, -103, -206, -349, -538, -779, -1078, -1441, -1874, -2383, -2974, -3653, -4426, -5299, -6278, -7369, -8578, -9911, -11374, -12973, -14714, -16603, -18646, -20849, -23218, -25759, -28478, -31381, -34474, -37763, -41254

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 22:

a(n) = A027750(A006218(21) + k + 1), 0 <= k < A000005(22).

			Differences of divisors of 22 to compute the coefficients of their interpolating polynomial, see formula:
  1     2    11    22
     1     9    11
        8     2
          -6

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161710, A080856, A161711, A161712, A161713, A161715, A006261.

[-n^3 + 7*n^2 - 5*n + 1: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[-n^3+7n^2-5n+1,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,2,11,22},40] (* Harvey P. Dale, Nov 12 2013 *)

a(n)=-n^3+7*n^2-5*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = C(n,0) + C(n,1) + 8*C(n,2) - 6*C(n,3).
G.f.: -(-1+2*x-9*x^2+14*x^3)/(-1+x)^4. - R. J. Mathar, Jun 18 2009
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) with a(0)=1, a(1)=2, a(2)=11, a(3)=22. - Harvey P. Dale, Nov 12 2013
E.g.f.: (-x^3 + 4*x^2 + x + 1)*exp(x). - G. C. Greubel, Jul 16 2017

A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

Original entry on oeis.org

1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375

Offset: 1

Clark Kimberling, Feb 03 2011

This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
  1....2.....4.....7...11...16...22...29...
  3....5.....8....12...17...23...30...38...
  6....9....13....18...24...31...39...48...
  10...14...19....25...32...40...49...59...
  15...20...26....33...41...50...60...71...
  21...27...34....42...51...61...72...84...
  28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787.  Deleting the main diagonal and then summing give A185787.  Analogous treatments to the left of the main diagonal give A100182 and A101165.  Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1:  A000124
row 2:  A022856
row 3:  A016028
row 4:  A145018
row 5:  A077169
col 1:  A000217
col 2:  A000096
col 3:  A034856
col 4:  A055998
col 5:  A046691
col 6:  A052905
col 7:  A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817.
Let S(n,k) denote the n-th partial sum of column k.  Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000027, A185788, A100182, A101165, A079824.

[n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k}];
Factor[s[k]]
Table[s[k],{k,1,70}]  (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

a(n)=n*(7*n^2-6*n+5)/6.

G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012

Edited by Clark Kimberling, Feb 25 2023

A061925 a(n) = ceiling(n^2/2) + 1.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406

Offset: 0

Henry Bottomley, May 17 2001

a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004
First term in each group in A074148. - Amarnath Murthy, Aug 28 2002
From Christian Barrientos, Jan 01 2021: (Start)
For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.
For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
      _       _       _       _       _      
   |||_|   |||_|   |||_|   |||_|   |||_|   |||_
   |||_|   |||     || ||       ||     ||       |||
(End)
Except for a(2), a(n) agrees with the lower matching number of the (n+1) X (n+1) bishop graph up to at least n = 13. - Eric W. Weisstein, Dec 23 2024

Harry J. Smith, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Eric Weisstein's World of Mathematics, Bishop Graph.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A100795, A074147-A074149.

seq(floor((n^2+3)/2),n=0..25); # Gary Detlefs, Feb 13 2010

Table[Ceiling[n^2/2]+1,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
LinearRecurrence[{2,0,-2,1},{1,2,3,6},60] (* Harvey P. Dale, Jan 03 2024 *)

a(n) = { ceil(n^2/2) + 1 } \\ Harry J. Smith, Jul 29 2009

a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.
a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010
From R. J. Mathar, Feb 19 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011
a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013
a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - R. J. Mathar, Jul 20 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007

cp's OEIS Frontend

A122510 Array T(d,n) = number of integer lattice points inside the d-dimensional hypersphere of radius sqrt(n), read by ascending antidiagonals.

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A161706 a(n) = (-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120.

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A161710 a(n) = (-6*n^7 + 154*n^6 - 1533*n^5 + 7525*n^4 - 18879*n^3 + 22561*n^2 - 7302*n + 2520)/2520.

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A161713 a(n) = (-n^5 + 15*n^4 - 65*n^3 + 125*n^2 - 34*n + 40)/40.

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A161715 a(n) = (50*n^7 - 1197*n^6 + 11333*n^5 - 53655*n^4 + 132125*n^3 - 156828*n^2 + 73212*n + 5040)/5040.

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A161701 a(n) = (n^5 - 5*n^4 + 5*n^3 + 5*n^2 + 114*n + 120)/120.

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A161706 a(n) = (-11n^5 + 145n^4 - 635n^3 + 1115n^2 - 494*n + 120)/120.

A161710 a(n) = (-6n^7 + 154n^6 - 1533n^5 + 7525n^4 - 18879n^3 + 22561n^2 - 7302*n + 2520)/2520.

A161713 a(n) = (-n^5 + 15n^4 - 65n^3 + 125n^2 - 34n + 40)/40.

A161715 a(n) = (50n^7 - 1197n^6 + 11333n^5 - 53655n^4 + 132125n^3 - 156828n^2 + 73212*n + 5040)/5040.

A161701 a(n) = (n^5 - 5n^4 + 5n^3 + 5n^2 + 114n + 120)/120.

A161704 a(n) = (3n^5 - 35n^4 + 145n^3 - 235n^2 + 152*n + 30)/30.

A161708 a(n) = -n^3 + 7n^2 - 5n + 1.