A098547 -id:A098547 - cp's OEIS frontend

A104257 Square array T(a,n) read by antidiagonals: replace 2^i with a^i in binary representation of n, where a,n >= 2.

2, 3, 3, 4, 4, 4, 5, 5, 9, 5, 6, 6, 16, 10, 6, 7, 7, 25, 17, 12, 7, 8, 8, 36, 26, 20, 13, 8, 9, 9, 49, 37, 30, 21, 27, 9, 10, 10, 64, 50, 42, 31, 64, 28, 10, 11, 11, 81, 65, 56, 43, 125, 65, 30, 11, 12, 12, 100, 82, 72, 57, 216, 126, 68, 31, 12, 13, 13, 121, 101, 90, 73, 343

Offset: 2

Ralf Stephan, Mar 05 2005

Sums of distinct powers of a. Numbers having only {0,1} in a-ary representation.

			Array begins:
  2,  3,  4,  5,  6,  7,   8,   9, ...
  3,  4,  9, 10, 12, 13,  27,  28, ...
  4,  5, 16, 17, 20, 21,  64,  65, ...
  5,  6, 25, 26, 30, 31, 125, 126, ...
  6,  7, 36, 37, 42, 43, 216, 217, ...
  7,  8, 49, 50, 56, 57, 343, 344, ...
  8,  9, 64, 65, 72, 73, 512, 513, ...
  9, 10, 81, 82, 90, 91, 729, 730, ...
  ...

John Tyler Rascoe, Antidiagonals n = 2..140, flattened

Rows include (essentially) A005836, A000695, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A033050, A033051, A033052.
Columns include A000290, A002522, A002378, A000578, A001093, A034262, A071568, A011379, A098547, A027444.
Main diagonal is A104258.

T[, 0] = 0; T[2, n] := n; T[a_, 2] := a;
T[a_, n_] := T[a, n] = If[EvenQ[n], a T[a, n/2], a T[a, (n-1)/2]+1];
Table[T[a-n+2, n], {a, 2, 13}, {n, 2, a}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)

T(a, n) = fromdigits(binary(n), a); \\ Michel Marcus, Aug 19 2022

def T(a, n): return n if n < 2 else (max(a, n) if min(a, n) == 2 else a*T(a, n//2) + n%2)
print([T(a-n+2, n) for a in range(2, 14) for n in range(2, a+1)]) # Michael S. Branicky, Aug 02 2022

T(a, n) = (1/(a-1))*Sum_{j>=1} floor((n+2^(j-1))/2^j) * ((a-2)*a^(j-1) + 1).
T(a, n) = (1/(a-1))*Sum_{j=1..n} ((a-2)*a^A007814(j) + 1).
G.f. of a-th row: (1/(1-x)) * Sum_{k>=0} a^k*x^2^k/(1+x^2^k).
Recurrence: T(a, 2n) = a*T(a, n), T(a, 2n+1) = a*T(a, n) + 1, T(a, 0) = 0.

A099721 a(n) = n^2(2n+1).

Original entry on oeis.org

0, 3, 20, 63, 144, 275, 468, 735, 1088, 1539, 2100, 2783, 3600, 4563, 5684, 6975, 8448, 10115, 11988, 14079, 16400, 18963, 21780, 24863, 28224, 31875, 35828, 40095, 44688, 49619, 54900, 60543, 66560, 72963, 79764, 86975, 94608, 102675, 111188, 120159, 129600

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 07 2004

For a right triangle with sides of lengths 8*n^3 + 12*n^2 + 8*n + 2, 4*n^4 + 8*n^3 + 4*n^2, and 4*n^4 + 8*n^3 + 12*n^2 + 8*n + 2, dividing the area by the perimeter gives a(n). - J. M. Bergot, Jul 30 2013
This sequence is the difference between the centered icosahedral (or cuboctahedral) numbers (A005902(n)) and the centered octagonal pyramidal numbers (A000447(n+1)). - Peter M. Chema, Jan 09 2016
a(n) is the sum of the integers in the closed interval (n-1)*n to n*(n+1). - J. M. Bergot, Apr 19 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Row n=3 of A229079.
Cf. A000578, A001093, A011379, A015237, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568, A098547.
Cf. A000290, A000447, A002378, A005408, A005902, A024196.

[n^2*(2*n+1): n in [0..50]]; // Vincenzo Librandi, May 01 2011

A099721 := proc(n) n^2*(2*n+1) ; end proc:
seq(A099721(n),n=0..10) ;

a[n_]:=2*n^3+n^2; (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,3,20,63},40] (* Harvey P. Dale, Aug 19 2022 *)

a(n) = ceil(sum(i=n^2-(n-1), n^2+(n-1), if(!issquare(4*i+1), (2*i+1+sqrt(4*i+1))/2, 0))); \\ Michel Marcus, Nov 14 2014, after Richard R. Forberg

G.f.: x*(3 + 8*x + x^2)/(x-1)^4.
a(n) = A024196(n) - A024196(n-1). - Philippe Deléham, May 07 2012
a(n) = ceiling(Sum_{i=n^2-(n-1)..n^2+(n-1)} s(i)), for n > 0 and integer i, where s(i) are the real solutions to x = i + sqrt(x), and the summation range excludes the integer solutions which occur where i is an oblong number (A002378). The fractional portion of the summation converges to 2/3 for large n. If s(i) is replaced with i, then the summation equals n^2*(2*n-1) = A015237. - Richard R. Forberg, Oct 15 2014
a(n) = A005902(n) - A000447(n+1). - Peter M. Chema, Jan 09 2016
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 + 4*log(2) - 4.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - Pi - 2*log(2) + 4. (End)
From Elmo R. Oliveira, Aug 08 2025: (Start)
E.g.f.: x*(1 + 2*x)*(3 + x)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A000290(n)*A005408(n). (End)

A120479 Primes of the form k^3 + k^2 + 1.

Original entry on oeis.org

3, 13, 37, 151, 577, 811, 1453, 1873, 12697, 14401, 18253, 27901, 44101, 75853, 87121, 93151, 106033, 151687, 178753, 188443, 242173, 291853, 319057, 333271, 362953, 410701, 643453, 666073, 712891, 787153, 1040503, 1379953, 1742401, 1830733, 1875997, 1968751

Offset: 1

Jonathan Vos Post, Jul 21 2006

Primes in A098547. - Michel Marcus, Jan 21 2015

			1^3 + 1^2 + 1 = 3 (prime), so 3 is in the sequence.
2^3 + 2^2 + 1 = 13 (prime), so 13 is in the sequence.
3^3 + 3^2 + 1 = 37 (prime), so 37 is in the sequence.
4^3 + 4^2 + 1 = 81 = 3^4, so 81 is not in the sequence.
5^3 + 5^2 + 1 = 151 (prime), so 151 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A119863 (corresponding k).

Cf. A000040, A098547.

Select[Table[n^3+n^2+1,{n,200}],PrimeQ] (* Harvey P. Dale, Oct 23 2020 *)

for(n=1,10^3,if(isprime(p=n^3+n^2+1),print1(p,", "))) \\ Derek Orr, Jan 21 2015

a(n) = A098547(A119863(n)). - Elmo R. Oliveira, Apr 20 2025

A100119 a(n) = n-th centered n-gonal number.

Original entry on oeis.org

1, 2, 7, 19, 41, 76, 127, 197, 289, 406, 551, 727, 937, 1184, 1471, 1801, 2177, 2602, 3079, 3611, 4201, 4852, 5567, 6349, 7201, 8126, 9127, 10207, 11369, 12616, 13951, 15377, 16897, 18514, 20231, 22051, 23977, 26012, 28159, 30421, 32801, 35302

Offset: 0

Jonathan Vos Post, Dec 26 2004

a(n) is n times the n-th triangular number plus 1. - Thomas M. Green, Nov 16 2009
From Gary W. Adamson, Jul 31 2010: (Start)
Equals (1, 2, 3, 4, ...) convolved with (1, 0, 4, 7, 10, 13, ...).
Example: a(5) = 76 = (6, 5, 4, 3, 2, 1) dot (1, 0, 4, 7, 10, 13) = (6 + 0 + 16 + 21 + 20 + 13). (End)

			a(2) = 2*3 + 1 = 7, a(3) = 3*6 + 1 = 19, a(4) = 4*10 + 1 = 41. - _Thomas M. Green_, Nov 16 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

See also A101357 (Cumulative sums of the n-th n-gonal numbers).
A diagonal of A101321.
Cf. A060354, A098547, A101357.
Cf. A006003, A005920.

I:=[1, 2, 7, 19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 25 2012

Table[(n^3+n^2)/2+1,{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)
LinearRecurrence[{4,-6,4,-1},{1,2,7,19},40] (* Vincenzo Librandi, Jun 25 2012 *)

a(n) = n^2*(n+1)/2+1; \\ Altug Alkan, Sep 21 2018

a(n) = 1 + n*(n + n^2)/2 = 1 + (1/2)*n^2 + (1/2) * n^3 = 1 + mean(n^2, n^3). - Joshua Zucker, May 03 2006
Equals A002411(n) + 1. - Olivier Gérard, Jun 20 2007
G.f.: (1 - 2*x + 5*x^2 - x^3) / (x-1)^4. - R. J. Mathar, Apr 04 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 25 2012
a(n) = (A098547(n)+1)/2. - Richard Turk, Jul 18 2017
a(n) = A060354(n+2) - A000290(n+1) = A006003(n+1) - A005563(n) and for n>0 A005920(n) - A068601(n+1). - Bruce J. Nicholson, Jun 23 2018

Corrected and extended by Joshua Zucker, May 03 2006

A123111 1+n^2+n^3+n^5+n^7; 10101101 in base n.

Original entry on oeis.org

5, 173, 2467, 17489, 81401, 287965, 840743, 2130497, 4842829, 10101101, 19649675, 36082513, 63122177, 105954269, 171622351, 269488385, 411763733, 614115757, 896355059, 1283208401, 1805182345, 2499522653, 3411274487, 4594448449

Offset: 1

Jonathan Vos Post, Sep 28 2006

4th row, A(4,n), of the infinite array A(k,n) = 1 + SUM[i=1..k]n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = SUM[i=0..k]n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. Row 4 (the current sequence) is prime for n = 1, 2, 3, 4, 5, 7, 10, 18, 19, 23, 25.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000040, A002522, A098547.

seq(1 + n^2 + n^3 + n^5 + n^7, n=1..100); # Robert Israel, Sep 02 2014

Table[Total[n^Prime[Range[4]]]+1,{n,30}] (* Harvey P. Dale, Jan 01 2014 *)

Vec(-x*(x^7-9*x^6-127*x^5-1227*x^4-2317*x^3-1223*x^2-133*x-5)/(x-1)^8 + O(x^100)) \\ Colin Barker, Sep 02 2014

a(n) = 1 + n^2 + n^3 + n^5 + n^7 = 10101101 (base n) = 1 + SUM[i=1..4]n^prime(i).
G.f.: -x*(x^7-9*x^6-127*x^5-1227*x^4-2317*x^3-1223*x^2-133*x-5) / (x-1)^8. - Colin Barker, Sep 02 2014
a(n+7)-7*a(n+6)+21*a(n+5)-35*a(n+4)+35*a(n+3)-21*a(n+2)+7*a(n+1)-a(n)=5040. - Robert Israel, Sep 02 2014

A123650 a(n) = 1 + n^2 + n^3 + n^5.

Original entry on oeis.org

4, 45, 280, 1105, 3276, 8029, 17200, 33345, 59860, 101101, 162504, 250705, 373660, 540765, 762976, 1052929, 1425060, 1895725, 2483320, 3208401, 4093804, 5164765, 6449040, 7977025, 9781876, 11899629, 14369320, 17233105, 20536380, 24327901

Offset: 1

Jonathan Vos Post, Oct 04 2006

3rd row, A(3,n), of the infinite array A(k,n) = 1 + Sum_{i=1..k} n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = Sum_{i=0..k} n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. The 4th row, A(4,n), is A123111 1 + n^2 + n^3 + n^5 + n^7. 10101101 (base n). A(n,n) is A123113 Main diagonal of prime power sum array. The current sequence, A(3,n), can never be prime because of the polynomial factorization a(n) = 1 + n^2 + n^3 + n^5 = +/- (n+1)*(n^2-n+1)*(n^2+1). Its fewest prime factors are 2 for the semiprime a(1) = 4. We similarly have polynomial factorizations for A123651 = A(7,n) = 1+n^2+n^3+n^5+n^7+n^11+n^13+n^17 and A123652 = A(13,n) = 1+n^2+n^3+n^5+...+n^41.

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

Cf. A002522, A098547, A123111, A123113, A123651, A123652.

[1+n^2+n^3+n^5: n in [1..25]]; // G. C. Greubel, Oct 17 2017

Table[1+n^2+n^3+n^5,{n,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{4,45,280,1105,3276,8029},30] (* Harvey P. Dale, Jan 18 2014 *)

for(n=1,25, print1(1+n^2+n^3+n^5, ", ")) \\ G. C. Greubel, Oct 17 2017

a(n) = 1 + n^2 + n^3 + n^5 = 101101 (base n) = +/- (n+1)*(n^2-n+1)*(n^2+1).

G.f.: x*(4 +21*x +70*x^2 +20*x^3 +6*x^4 -x^5)/(1-x)^6. - Colin Barker, May 25 2012

A123651 a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17.

Original entry on oeis.org

8, 141485, 130914100, 17251189841, 764209065776, 16940083223773, 232729381165100, 2252358161564225, 16679754951397336, 100010100010101101, 505481836542757988, 2218718842990269265, 8650720586711446400

Offset: 1

Jonathan Vos Post, Oct 04 2006

7th row, A(7,n), of the infinite array A(k,n) = 1 + Sum_{i=1..k} n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = Sum_{i=0..k} n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. The 3rd row, A(3,n), is A123650. The 4th row, A(4,n), is A123111 1 + n^2 + n^3 + n^5 + n^7. 10101101 (base n). A(n,n) is A123113 Main diagonal of prime power sum array. The current sequence, A(7,n), can never be prime, because of the polynomial factorization a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 = +/- (n^2+1)*(n^15 -n^13 +2n^11 -n^9 +n^7 +n^3 +1). It can be semiprime, as with a(2) and with a(10) = 100010100010101101 = 101 * 990199010001001 and a(14). We similarly have polynomial factorization for A123652 = A(13,n) = 1 +n^2 +n^3 +n^5 +...+ n^41.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002522, A098547, A123111, A123113, A123650, A123652.

[1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17: n in [1..25]]; // G. C. Greubel, Oct 17 2017

Table[Total[n^Prime[Range[7]]]+1,{n,20}] (* Harvey P. Dale, Aug 22 2012 *)

for(n=1,25, print1(1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17, ", ")) \\ G. C. Greubel, Oct 17 2017

a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 = 100010100010101101 (base n) = +/- (n^2+1)*(n^15-n^13+2n^11-n^9+n^7+n^3+1).

A123652 a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41.

Original entry on oeis.org

14, 2339155617965, 36923966682271786990, 4854597644377050732053585, 45547499507677574921923909526, 80266855145143309588022024772829, 44586202603279528645530450127574150

Offset: 1

Jonathan Vos Post, Oct 04 2006

13th row, A(13,n), of the infinite array A(k,n) = 1 + Sum_{i=1..k} n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = Sum_{i=0..k} n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. The 3rd row, A(3,n), is A123650. The 4th row, A(4,n), is A123111 1 +n^2 +n^3 +n^5 +n^7. 10101101 (base n). A(n,n) is A123113 Main diagonal of prime power sum array. The sequence A(13,n) = a(n) can never be prime because of the polynomial factorization. It can be semiprime, as with a(1) = 14 and a(2) = 2339155617965 = 5 * 467831123593 and a(6) and 100010000010100000100010100010100010101101 = 101 * 990198019901980199010000990199010001001. We similarly have polynomial factorization for the 7th row, A123651 = A(7,n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 = +/- (n^2+1)*(n^15-n^13+2n^11-n^9+n^7+n^3+1).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002522, A098547, A123111, A123113, A123650, A123651.

[1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41: n in [1..25]]; // G. C. Greubel, Oct 17 2017

Table[1+Total[n^Prime[Range[PrimePi[41]]]],{n,8}] (* Harvey P. Dale, Dec 20 2010 *)

for(n=1,25, print1(1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41, ", ")) \\ G. C. Greubel, Oct 17 2017

a(n) = 1+n^2+n^3+n^5+n^7+n^11+n^13+n^17+n^19+n^23+n^29+n^31+n^37+n^41 = 100010000010100000100010100010100010101101(base n) = +/-(n^2+1)*(n^39-n^37+2n^35-2n^33+2n^31-n^29+2n^27-2n^25+2n^23-n^21+n^19+n^15-n^13+2n^11-n^9+n^7+n^3+1).

A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 0, 1, 1, 3, 3, 1, 0, 0, 2, 4, 4, 4, 1, 0, 1, 2, 5, 9, 5, 5, 1, 0, 0, 3, 6, 10, 16, 6, 6, 1, 0, 1, 1, 7, 12, 17, 25, 7, 7, 1, 0, 0, 2, 8, 13, 20, 26, 36, 8, 8, 1, 0, 1, 2, 9, 27, 21, 30, 37, 49, 9, 9, 1, 0, 0, 3, 10, 28, 64, 31, 42, 50, 64, 10, 10, 1, 0

Offset: 0

Rémy Sigrist, Jun 04 2021

For any n >= 0, the n-th row, k -> T(n, k), corresponds to a polynomial in k with coefficients in {0, 1}.

For any k > 1, the k-th column, n -> T(n, k), corresponds to sums of distinct powers of k.

			Array T(n, k) begins:
  n\k|  0  1   2   3   4    5    6    7    8    9    10    11    12
  ---+-------------------------------------------------------------
    0|  0  0   0   0   0    0    0    0    0    0     0     0     0
    1|  1  1   1   1   1    1    1    1    1    1     1     1     1
    2|  0  1   2   3   4    5    6    7    8    9    10    11    12
    3|  1  2   3   4   5    6    7    8    9   10    11    12    13
    4|  0  1   4   9  16   25   36   49   64   81   100   121   144
    5|  1  2   5  10  17   26   37   50   65   82   101   122   145
    6|  0  2   6  12  20   30   42   56   72   90   110   132   156
    7|  1  3   7  13  21   31   43   57   73   91   111   133   157
    8|  0  1   8  27  64  125  216  343  512  729  1000  1331  1728
    9|  1  2   9  28  65  126  217  344  513  730  1001  1332  1729
   10|  0  2  10  30  68  130  222  350  520  738  1010  1342  1740
   11|  1  3  11  31  69  131  223  351  521  739  1011  1343  1741
   12|  0  2  12  36  80  150  252  392  576  810  1100  1452  1872

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Index entries for sequences related to binary expansion of n

See A342707 for a similar sequence.

Cf. A000035, A000120, A000583, A000695, A001093, A002061, A002378, A002522, A002523, A005836, A007088, A011379, A027444, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A034262, A053698, A071568, A098547, A104258.

T(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n,2); v+=k^e); v }

T(n, n) = A104258(n).
T(n, 0) = A000035(n).
T(n, 1) = A000120(n).
T(n, 2) = n.
T(n, 3) = A005836(n).
T(n, 4) = A000695(n).
T(n, 5) = A033042(n).
T(n, 6) = A033043(n).
T(n, 7) = A033044(n).
T(n, 8) = A033045(n).
T(n, 9) = A033046(n).
T(n, 10) = A007088(n).
T(n, 11) = A033047(n).
T(n, 12) = A033048(n).
T(n, 13) = A033049(n).
T(0, k) = 0.
T(1, k) = 1.
T(2, k) = k.
T(3, k) = k + 1.
T(4, k) = k^2.
T(5, k) = k^2 + 1 = A002522(k).
T(6, k) = k^2 + k = A002378(k).
T(7, k) = k^2 + k + 1 = A002061(k).
T(8, k) = k^3.
T(9, k) = k^3 + 1 = A001093(k).
T(10, k) = k^3 + k = A034262(k).
T(11, k) = k^3 + k + 1 = A071568(k).
T(12, k) = k^3 + k^2 = A011379(k).
T(13, k) = k^3 + k^2 + 1 = A098547(k).
T(14, k) = k^3 + k^2 + k = A027444(k).
T(15, k) = k^3 + k^2 + k + 1 = A053698(k).
T(16, k) = k^4 = A000583(k).
T(17, k) = k^4 + 1 = A002523(k).
T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).

A124054 Array(d,n) = number of ordered ways to write n as the sum of d squares less than d, read by rows, through last nonzero value per row.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 0, 3, 3, 0, 0, 1, 1, 4, 6, 4, 5, 12, 12, 4, 6, 16, 18, 12, 8, 16, 24, 12, 1, 12, 18, 12, 6, 4, 12, 12, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 1, 1, 5, 10, 10, 10, 21, 30, 20, 15, 35, 50, 40, 30, 45, 70, 60, 30, 55, 100, 80, 56

Offset: 1

Jonathan Vos Post, Nov 03 2006

Rows terminate with last nonzero element. Row length of row n = A098547 n^3+n^2+1. Row 4 = A123999 Number of ordered ways of writing n as a sum of 4 squares of nonnegative numbers less than 4. Row 5 = A123337 Number of ordered ways to write n as the sum of 5 squares less than 5. Column 0 = A000012 The simplest sequence of positive numbers: the all 1's sequence. Column 1 = A000027 The natural numbers. Column 2 = A000217(n-2) = Triangular numbers C(n-1,2) = n(n-1)/2. Column 3 = A000292(n-2) Tetrahedral numbers = C(n,3).

			A(1,n) = 1 because the unique ordered way to write 1 as the sum of 0 squares less than 0 is the null set {}.
a(2,n) = 1, 2, 1 = Card{0=0^2+0^2}; Card{1=0^2+1^2,1=1^2+0^2}; Card{2=1^2+1^2}.
a(3,n) = 1, 3, 3, 1, 3, 6, 3, 0, 3, 3, 0, 0, 1.
a(4,n) = 1, 4, 6, 4, 5, 12, 12, 4,  6, 16, 18, ... = A123999.
a(5,n) = 1, 5, 10, 10, 10, 21, 30, 20, 15, 35, ... = A123337.
a(6,n) = 1, 6, 15, 20, 21, 36, 61, 60, 45, 72, ...
a(7,n) = 1, 7, 21, 35, 42, 63, 112, 141, 126, 154, ...
a(8,n) = 1, 8, 28, 56, 78, 112, 196, 288, 309, 344, ...
a(9,n) = 1, 9, 36, 84, 135, 198, 336, 540, 675, 766, ...
a(10,n) = 1, 10, 45, 120, 220, 342, 570, 960, 1350, 1640, ...

Cf. A000012, A000027, A000217, A000292, A098547, A123337, A123999.

cntper[v_] := Length[v]!/Times @@ ((Last /@ Tally[v])!); sqq[d_, n_] := Total[ cntper /@ IntegerPartitions[n, {d}, Range[0, d - 1]^2]]; Flatten[ Table[ sqq[d, #] & /@ Range[0, d (d - 1)^2], {d, 1, 6}]] (* Giovanni Resta, Jun 16 2016 *)

A(d,n) for fixed d = Row d = Card{(c_1,c_2,...,c_d) such that 0<=c_i

Data corrected by Giovanni Resta, Jun 16 2016

Showing 1-10 of 10 results.

cp's OEIS Frontend

A104257 Square array T(a,n) read by antidiagonals: replace 2^i with a^i in binary representation of n, where a,n >= 2.

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A099721 a(n) = n^2*(2*n+1).

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A120479 Primes of the form k^3 + k^2 + 1.

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A100119 a(n) = n-th centered n-gonal number.

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A123111 1+n^2+n^3+n^5+n^7; 10101101 in base n.

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A123650 a(n) = 1 + n^2 + n^3 + n^5.

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A123651 a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17.

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A099721 a(n) = n^2(2n+1).